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3.1: The Fundamentals of Hypothesis Testing

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Diane Kiernan
SUNY College of Environmental Science and Forestry via OpenSUNY

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The previous two chapters introduced methods for organizing and summarizing sample data, and using sample statistics to estimate population parameters. This chapter introduces the next major topic of inferential statistics: hypothesis testing.

A hypothesis is a statement or claim about a property of a population.

The Fundamentals of Hypothesis Testing

When conducting scientific research, typically there is some known information, perhaps from some past work or from a long accepted idea. We want to test whether this claim is believable. This is the basic idea behind a hypothesis test:

State what we think is true.
Quantify how confident we are about our claim.
Use sample statistics to make inferences about population parameters.

For example, past research tells us that the average life span for a hummingbird is about four years. You have been studying the hummingbirds in the southeastern United States and find a sample mean lifespan of 4.8 years. Should you reject the known or accepted information in favor of your results? How confident are you in your estimate? At what point would you say that there is enough evidence to reject the known information and support your alternative claim? How far from the known mean of four years can the sample mean be before we reject the idea that the average lifespan of a hummingbird is four years?

Definition: hypothesis testing

Hypothesis testing is a procedure, based on sample evidence and probability, used to test claims regarding a characteristic of a population.

A hypothesis is a claim or statement about a characteristic of a population of interest to us. A hypothesis test is a way for us to use our sample statistics to test a specific claim.

Example $\PageIndex{1}$:

The population mean weight is known to be 157 lb. We want to test the claim that the mean weight has increased.

Example $\PageIndex{2}$:

Two years ago, the proportion of infected plants was 37%. We believe that a treatment has helped, and we want to test the claim that there has been a reduction in the proportion of infected plants.

Components of a Formal Hypothesis Test

The null hypothesis is a statement about the value of a population parameter, such as the population mean (µ) or the population proportion ( p ). It contains the condition of equality and is denoted as H 0 (H-naught).

H 0 : µ = 157 or H0 : p = 0.37

The alternative hypothesis is the claim to be tested, the opposite of the null hypothesis. It contains the value of the parameter that we consider plausible and is denoted as H 1 .

H 1 : µ > 157 or H1 : p ≠ 0.37

The test statistic is a value computed from the sample data that is used in making a decision about the rejection of the null hypothesis. The test statistic converts the sample mean ( x̄ ) or sample proportion ( p̂ ) to a Z- or t-score under the assumption that the null hypothesis is true. It is used to decide whether the difference between the sample statistic and the hypothesized claim is significant.

The p-value is the area under the curve to the left or right of the test statistic. It is compared to the level of significance (α).

The critical value is the value that defines the rejection zone (the test statistic values that would lead to rejection of the null hypothesis). It is defined by the level of significance.

The level of significance (α) is the probability that the test statistic will fall into the critical region when the null hypothesis is true. This level is set by the researcher.

The conclusion is the final decision of the hypothesis test. The conclusion must always be clearly stated, communicating the decision based on the components of the test. It is important to realize that we never prove or accept the null hypothesis. We are merely saying that the sample evidence is not strong enough to warrant the rejection of the null hypothesis. The conclusion is made up of two parts:

1) Reject or fail to reject the null hypothesis, and 2) there is or is not enough evidence to support the alternative claim.

Option 1) Reject the null hypothesis (H0). This means that you have enough statistical evidence to support the alternative claim (H 1 ).

Option 2) Fail to reject the null hypothesis (H0). This means that you do NOT have enough evidence to support the alternative claim (H 1 ).

Another way to think about hypothesis testing is to compare it to the US justice system. A defendant is innocent until proven guilty (Null hypothesis—innocent). The prosecuting attorney tries to prove that the defendant is guilty (Alternative hypothesis—guilty). There are two possible conclusions that the jury can reach. First, the defendant is guilty (Reject the null hypothesis). Second, the defendant is not guilty (Fail to reject the null hypothesis). This is NOT the same thing as saying the defendant is innocent! In the first case, the prosecutor had enough evidence to reject the null hypothesis (innocent) and support the alternative claim (guilty). In the second case, the prosecutor did NOT have enough evidence to reject the null hypothesis (innocent) and support the alternative claim of guilty.

The Null and Alternative Hypotheses

There are three different pairs of null and alternative hypotheses:

Table $PageIndex{1}$: The rejection zone for a two-sided hypothesis test.

where c is some known value.

A Two-sided Test

This tests whether the population parameter is equal to, versus not equal to, some specific value.

Ho: μ = 12 vs. H 1 : μ ≠ 12

The critical region is divided equally into the two tails and the critical values are ± values that define the rejection zones.

Example $\PageIndex{3}$:

A forester studying diameter growth of red pine believes that the mean diameter growth will be different if a fertilization treatment is applied to the stand.

Ho: μ = 1.2 in./ year
H 1 : μ ≠ 1.2 in./ year

This is a two-sided question, as the forester doesn’t state whether population mean diameter growth will increase or decrease.

A Right-sided Test

This tests whether the population parameter is equal to, versus greater than, some specific value.

Ho: μ = 12 vs. H 1 : μ > 12

The critical region is in the right tail and the critical value is a positive value that defines the rejection zone.

Example $\PageIndex{4}$:

A biologist believes that there has been an increase in the mean number of lakes infected with milfoil, an invasive species, since the last study five years ago.

Ho: μ = 15 lakes
H1: μ >15 lakes

This is a right-sided question, as the biologist believes that there has been an increase in population mean number of infected lakes.

A Left-sided Test

This tests whether the population parameter is equal to, versus less than, some specific value.

Ho: μ = 12 vs. H 1 : μ < 12

The critical region is in the left tail and the critical value is a negative value that defines the rejection zone.

Example $\PageIndex{5}$:

A scientist’s research indicates that there has been a change in the proportion of people who support certain environmental policies. He wants to test the claim that there has been a reduction in the proportion of people who support these policies.

Ho: p = 0.57
H 1 : p < 0.57

This is a left-sided question, as the scientist believes that there has been a reduction in the true population proportion.

Statistically Significant

When the observed results (the sample statistics) are unlikely (a low probability) under the assumption that the null hypothesis is true, we say that the result is statistically significant, and we reject the null hypothesis. This result depends on the level of significance, the sample statistic, sample size, and whether it is a one- or two-sided alternative hypothesis.

Types of Errors

When testing, we arrive at a conclusion of rejecting the null hypothesis or failing to reject the null hypothesis. Such conclusions are sometimes correct and sometimes incorrect (even when we have followed all the correct procedures). We use incomplete sample data to reach a conclusion and there is always the possibility of reaching the wrong conclusion. There are four possible conclusions to reach from hypothesis testing. Of the four possible outcomes, two are correct and two are NOT correct.

Table $\PageIndex{2}$. Possible outcomes from a hypothesis test.

A Type I error is when we reject the null hypothesis when it is true. The symbol α (alpha) is used to represent Type I errors. This is the same alpha we use as the level of significance. By setting alpha as low as reasonably possible, we try to control the Type I error through the level of significance.

A Type II error is when we fail to reject the null hypothesis when it is false. The symbol β(beta) is used to represent Type II errors.

In general, Type I errors are considered more serious. One step in the hypothesis test procedure involves selecting the significance level ( α ), which is the probability of rejecting the null hypothesis when it is correct. So the researcher can select the level of significance that minimizes Type I errors. However, there is a mathematical relationship between α, β, and n (sample size).

As α increases, β decreases
As α decreases, β increases
As sample size increases (n), both α and β decrease

The natural inclination is to select the smallest possible value for α, thinking to minimize the possibility of causing a Type I error. Unfortunately, this forces an increase in Type II errors. By making the rejection zone too small, you may fail to reject the null hypothesis, when, in fact, it is false. Typically, we select the best sample size and level of significance, automatically setting β.

Power of the Test

A Type II error (β) is the probability of failing to reject a false null hypothesis. It follows that 1-β is the probability of rejecting a false null hypothesis. This probability is identified as the power of the test, and is often used to gauge the test’s effectiveness in recognizing that a null hypothesis is false.

Definition: power of the test

The probability that at a fixed level α significance test will reject H0, when a particular alternative value of the parameter is true is called the power of the test.

Power is also directly linked to sample size. For example, suppose the null hypothesis is that the mean fish weight is 8.7 lb. Given sample data, a level of significance of 5%, and an alternative weight of 9.2 lb., we can compute the power of the test to reject μ = 8.7 lb. If we have a small sample size, the power will be low. However, increasing the sample size will increase the power of the test. Increasing the level of significance will also increase power. A 5% test of significance will have a greater chance of rejecting the null hypothesis than a 1% test because the strength of evidence required for the rejection is less. Decreasing the standard deviation has the same effect as increasing the sample size: there is more information about μ.

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5.2 - writing hypotheses.

The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ($H_0$) and an alternative hypothesis ($H_a$).

When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.

At this point we can write hypotheses for a single mean ($\mu$), paired means($\mu_d$), a single proportion ($p$), the difference between two independent means ($\mu_1-\mu_2$), the difference between two proportions ($p_1-p_2$), a simple linear regression slope ($\beta$), and a correlation ($\rho$).
The research question will give us the information necessary to determine if the test is two-tailed (e.g., "different from," "not equal to"), right-tailed (e.g., "greater than," "more than"), or left-tailed (e.g., "less than," "fewer than").
The research question will also give us the hypothesized parameter value. This is the number that goes in the hypothesis statements (i.e., $\mu_0$ and $p_0$). For the difference between two groups, regression, and correlation, this value is typically 0.

Hypotheses are always written in terms of population parameters (e.g., $p$ and $\mu$). The tables below display all of the possible hypotheses for the parameters that we have learned thus far. Note that the null hypothesis always includes the equality (i.e., =).

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Hypothesis Testing with One Sample

Null and Alternative Hypotheses

OpenStaxCollege

[latexpage]

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 : The null hypothesis: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.

H a : The alternative hypothesis: It is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are “reject H 0 ” if the sample information favors the alternative hypothesis or “do not reject H 0 ” or “decline to reject H 0 ” if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

H 0 : The drug reduces cholesterol by 25%. p = 0.25

H a : The drug does not reduce cholesterol by 25%. p ≠ 0.25

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:

H 0 : μ = 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

H 0 : μ = 66
H a : μ ≠ 66

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:

H 0 : μ ≥ 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

H 0 : μ ≥ 45
H a : μ < 45

In an issue of U. S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.

H 0 : p ≤ 0.066

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

H 0 : p = 0.40
H a : p > 0.40

<!– ??? –>

Bring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

Chapter Review

In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we:

Formula Review

H 0 and H a are contradictory.

If α ≤ p -value, then do not reject H 0 .

If α > p -value, then reject H 0 .

α is preconceived. Its value is set before the hypothesis test starts. The p -value is calculated from the data.

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. What is the random variable? Describe in words.

The random variable is the mean Internet speed in Megabits per second.

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. State the null and alternative hypotheses.

The American family has an average of two children. What is the random variable? Describe in words.

The random variable is the mean number of children an American family has.

The mean entry level salary of an employee at a company is 💲58,000. You believe it is higher for IT professionals in the company. State the null and alternative hypotheses.

A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if the proportion is actually less. What is the random variable? Describe in words.

The random variable is the proportion of people picked at random in Times Square visiting the city.

A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if the claim is correct. State the null and alternative hypotheses.

In a population of fish, approximately 42% are female. A test is conducted to see if, in fact, the proportion is less. State the null and alternative hypotheses.

Suppose that a recent article stated that the mean time spent in jail by a first–time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was 3 years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. If you were conducting a hypothesis test to determine if the mean length of jail time has increased, what would the null and alternative hypotheses be? The distribution of the population is normal.

A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. If you were conducting a hypothesis test to determine if the population mean time on death row could likely be 15 years, what would the null and alternative hypotheses be?

H 0 : __________
H a : __________
H 0 : μ = 15
H a : μ ≠ 15

The National Institute of Mental Health published an article stating that in any one-year period, approximately 9.5 percent of American adults suffer from depression or a depressive illness. Suppose that in a survey of 100 people in a certain town, seven of them suffered from depression or a depressive illness. If you were conducting a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population, what would the null and alternative hypotheses be?

Some of the following statements refer to the null hypothesis, some to the alternate hypothesis.

State the null hypothesis, H 0 , and the alternative hypothesis. H a , in terms of the appropriate parameter ( μ or p ).

The mean number of years Americans work before retiring is 34.
At most 60% of Americans vote in presidential elections.
The mean starting salary for San Jose State University graduates is at least 💲100,000 per year.
Twenty-nine percent of high school seniors get drunk each month.
Fewer than 5% of adults ride the bus to work in Los Angeles.
The mean number of cars a person owns in her lifetime is not more than ten.
About half of Americans prefer to live away from cities, given the choice.
Europeans have a mean paid vacation each year of six weeks.
The chance of developing breast cancer is under 11% for women.
Private universities’ mean tuition cost is more than 💲20,000 per year.
H 0 : μ = 34; H a : μ ≠ 34
H 0 : p ≤ 0.60; H a : p > 0.60
H 0 : μ ≥ 100,000; H a : μ < 100,000
H 0 : p = 0.29; H a : p ≠ 0.29
H 0 : p = 0.05; H a : p < 0.05
H 0 : μ ≤ 10; H a : μ > 10
H 0 : p = 0.50; H a : p ≠ 0.50
H 0 : μ = 6; H a : μ ≠ 6
H 0 : p ≥ 0.11; H a : p < 0.11
H 0 : μ ≤ 20,000; H a : μ > 20,000

Over the past few decades, public health officials have examined the link between weight concerns and teen girls’ smoking. Researchers surveyed a group of 273 randomly selected teen girls living in Massachusetts (between 12 and 15 years old). After four years the girls were surveyed again. Sixty-three said they smoked to stay thin. Is there good evidence that more than thirty percent of the teen girls smoke to stay thin? The alternative hypothesis is:

p < 0.30
p > 0.30

A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening night midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 attended the midnight showing. An appropriate alternative hypothesis is:

p > 0.20
p < 0.20

Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test. The null and alternative hypotheses are:

H o : $\overline{x}$ = 4.5, H a : $\overline{x}$ > 4.5
H o : μ ≥ 4.5, H a : μ < 4.5
H o : μ = 4.75, H a : μ > 4.75
H o : μ = 4.5, H a : μ > 4.5

Data from the National Institute of Mental Health. Available online at http://www.nimh.nih.gov/publicat/depression.cfm.

Home » What is a Hypothesis – Types, Examples and Writing Guide

What is a Hypothesis – Types, Examples and Writing Guide

Table of Contents

Definition:

Hypothesis is an educated guess or proposed explanation for a phenomenon, based on some initial observations or data. It is a tentative statement that can be tested and potentially proven or disproven through further investigation and experimentation.

Hypothesis is often used in scientific research to guide the design of experiments and the collection and analysis of data. It is an essential element of the scientific method, as it allows researchers to make predictions about the outcome of their experiments and to test those predictions to determine their accuracy.

Types of Hypothesis

Types of Hypothesis are as follows:

Research Hypothesis

A research hypothesis is a statement that predicts a relationship between variables. It is usually formulated as a specific statement that can be tested through research, and it is often used in scientific research to guide the design of experiments.

Null Hypothesis

The null hypothesis is a statement that assumes there is no significant difference or relationship between variables. It is often used as a starting point for testing the research hypothesis, and if the results of the study reject the null hypothesis, it suggests that there is a significant difference or relationship between variables.

Alternative Hypothesis

An alternative hypothesis is a statement that assumes there is a significant difference or relationship between variables. It is often used as an alternative to the null hypothesis and is tested against the null hypothesis to determine which statement is more accurate.

Directional Hypothesis

A directional hypothesis is a statement that predicts the direction of the relationship between variables. For example, a researcher might predict that increasing the amount of exercise will result in a decrease in body weight.

Non-directional Hypothesis

A non-directional hypothesis is a statement that predicts the relationship between variables but does not specify the direction. For example, a researcher might predict that there is a relationship between the amount of exercise and body weight, but they do not specify whether increasing or decreasing exercise will affect body weight.

Statistical Hypothesis

A statistical hypothesis is a statement that assumes a particular statistical model or distribution for the data. It is often used in statistical analysis to test the significance of a particular result.

Composite Hypothesis

A composite hypothesis is a statement that assumes more than one condition or outcome. It can be divided into several sub-hypotheses, each of which represents a different possible outcome.

Empirical Hypothesis

An empirical hypothesis is a statement that is based on observed phenomena or data. It is often used in scientific research to develop theories or models that explain the observed phenomena.

Simple Hypothesis

A simple hypothesis is a statement that assumes only one outcome or condition. It is often used in scientific research to test a single variable or factor.

Complex Hypothesis

A complex hypothesis is a statement that assumes multiple outcomes or conditions. It is often used in scientific research to test the effects of multiple variables or factors on a particular outcome.

Applications of Hypothesis

Hypotheses are used in various fields to guide research and make predictions about the outcomes of experiments or observations. Here are some examples of how hypotheses are applied in different fields:

Science : In scientific research, hypotheses are used to test the validity of theories and models that explain natural phenomena. For example, a hypothesis might be formulated to test the effects of a particular variable on a natural system, such as the effects of climate change on an ecosystem.
Medicine : In medical research, hypotheses are used to test the effectiveness of treatments and therapies for specific conditions. For example, a hypothesis might be formulated to test the effects of a new drug on a particular disease.
Psychology : In psychology, hypotheses are used to test theories and models of human behavior and cognition. For example, a hypothesis might be formulated to test the effects of a particular stimulus on the brain or behavior.
Sociology : In sociology, hypotheses are used to test theories and models of social phenomena, such as the effects of social structures or institutions on human behavior. For example, a hypothesis might be formulated to test the effects of income inequality on crime rates.
Business : In business research, hypotheses are used to test the validity of theories and models that explain business phenomena, such as consumer behavior or market trends. For example, a hypothesis might be formulated to test the effects of a new marketing campaign on consumer buying behavior.
Engineering : In engineering, hypotheses are used to test the effectiveness of new technologies or designs. For example, a hypothesis might be formulated to test the efficiency of a new solar panel design.

How to write a Hypothesis

Here are the steps to follow when writing a hypothesis:

Identify the Research Question

The first step is to identify the research question that you want to answer through your study. This question should be clear, specific, and focused. It should be something that can be investigated empirically and that has some relevance or significance in the field.

Conduct a Literature Review

Before writing your hypothesis, it’s essential to conduct a thorough literature review to understand what is already known about the topic. This will help you to identify the research gap and formulate a hypothesis that builds on existing knowledge.

Determine the Variables

The next step is to identify the variables involved in the research question. A variable is any characteristic or factor that can vary or change. There are two types of variables: independent and dependent. The independent variable is the one that is manipulated or changed by the researcher, while the dependent variable is the one that is measured or observed as a result of the independent variable.

Formulate the Hypothesis

Based on the research question and the variables involved, you can now formulate your hypothesis. A hypothesis should be a clear and concise statement that predicts the relationship between the variables. It should be testable through empirical research and based on existing theory or evidence.

Write the Null Hypothesis

The null hypothesis is the opposite of the alternative hypothesis, which is the hypothesis that you are testing. The null hypothesis states that there is no significant difference or relationship between the variables. It is important to write the null hypothesis because it allows you to compare your results with what would be expected by chance.

Refine the Hypothesis

After formulating the hypothesis, it’s important to refine it and make it more precise. This may involve clarifying the variables, specifying the direction of the relationship, or making the hypothesis more testable.

Examples of Hypothesis

Here are a few examples of hypotheses in different fields:

Psychology : “Increased exposure to violent video games leads to increased aggressive behavior in adolescents.”
Biology : “Higher levels of carbon dioxide in the atmosphere will lead to increased plant growth.”
Sociology : “Individuals who grow up in households with higher socioeconomic status will have higher levels of education and income as adults.”
Education : “Implementing a new teaching method will result in higher student achievement scores.”
Marketing : “Customers who receive a personalized email will be more likely to make a purchase than those who receive a generic email.”
Physics : “An increase in temperature will cause an increase in the volume of a gas, assuming all other variables remain constant.”
Medicine : “Consuming a diet high in saturated fats will increase the risk of developing heart disease.”

Purpose of Hypothesis

The purpose of a hypothesis is to provide a testable explanation for an observed phenomenon or a prediction of a future outcome based on existing knowledge or theories. A hypothesis is an essential part of the scientific method and helps to guide the research process by providing a clear focus for investigation. It enables scientists to design experiments or studies to gather evidence and data that can support or refute the proposed explanation or prediction.

The formulation of a hypothesis is based on existing knowledge, observations, and theories, and it should be specific, testable, and falsifiable. A specific hypothesis helps to define the research question, which is important in the research process as it guides the selection of an appropriate research design and methodology. Testability of the hypothesis means that it can be proven or disproven through empirical data collection and analysis. Falsifiability means that the hypothesis should be formulated in such a way that it can be proven wrong if it is incorrect.

In addition to guiding the research process, the testing of hypotheses can lead to new discoveries and advancements in scientific knowledge. When a hypothesis is supported by the data, it can be used to develop new theories or models to explain the observed phenomenon. When a hypothesis is not supported by the data, it can help to refine existing theories or prompt the development of new hypotheses to explain the phenomenon.

When to use Hypothesis

Here are some common situations in which hypotheses are used:

In scientific research , hypotheses are used to guide the design of experiments and to help researchers make predictions about the outcomes of those experiments.
In social science research , hypotheses are used to test theories about human behavior, social relationships, and other phenomena.
I n business , hypotheses can be used to guide decisions about marketing, product development, and other areas. For example, a hypothesis might be that a new product will sell well in a particular market, and this hypothesis can be tested through market research.

Characteristics of Hypothesis

Here are some common characteristics of a hypothesis:

Testable : A hypothesis must be able to be tested through observation or experimentation. This means that it must be possible to collect data that will either support or refute the hypothesis.
Falsifiable : A hypothesis must be able to be proven false if it is not supported by the data. If a hypothesis cannot be falsified, then it is not a scientific hypothesis.
Clear and concise : A hypothesis should be stated in a clear and concise manner so that it can be easily understood and tested.
Based on existing knowledge : A hypothesis should be based on existing knowledge and research in the field. It should not be based on personal beliefs or opinions.
Specific : A hypothesis should be specific in terms of the variables being tested and the predicted outcome. This will help to ensure that the research is focused and well-designed.
Tentative: A hypothesis is a tentative statement or assumption that requires further testing and evidence to be confirmed or refuted. It is not a final conclusion or assertion.
Relevant : A hypothesis should be relevant to the research question or problem being studied. It should address a gap in knowledge or provide a new perspective on the issue.

Advantages of Hypothesis

Hypotheses have several advantages in scientific research and experimentation:

Guides research: A hypothesis provides a clear and specific direction for research. It helps to focus the research question, select appropriate methods and variables, and interpret the results.
Predictive powe r: A hypothesis makes predictions about the outcome of research, which can be tested through experimentation. This allows researchers to evaluate the validity of the hypothesis and make new discoveries.
Facilitates communication: A hypothesis provides a common language and framework for scientists to communicate with one another about their research. This helps to facilitate the exchange of ideas and promotes collaboration.
Efficient use of resources: A hypothesis helps researchers to use their time, resources, and funding efficiently by directing them towards specific research questions and methods that are most likely to yield results.
Provides a basis for further research: A hypothesis that is supported by data provides a basis for further research and exploration. It can lead to new hypotheses, theories, and discoveries.
Increases objectivity: A hypothesis can help to increase objectivity in research by providing a clear and specific framework for testing and interpreting results. This can reduce bias and increase the reliability of research findings.

Limitations of Hypothesis

Some Limitations of the Hypothesis are as follows:

Limited to observable phenomena: Hypotheses are limited to observable phenomena and cannot account for unobservable or intangible factors. This means that some research questions may not be amenable to hypothesis testing.
May be inaccurate or incomplete: Hypotheses are based on existing knowledge and research, which may be incomplete or inaccurate. This can lead to flawed hypotheses and erroneous conclusions.
May be biased: Hypotheses may be biased by the researcher’s own beliefs, values, or assumptions. This can lead to selective interpretation of data and a lack of objectivity in research.
Cannot prove causation: A hypothesis can only show a correlation between variables, but it cannot prove causation. This requires further experimentation and analysis.
Limited to specific contexts: Hypotheses are limited to specific contexts and may not be generalizable to other situations or populations. This means that results may not be applicable in other contexts or may require further testing.
May be affected by chance : Hypotheses may be affected by chance or random variation, which can obscure or distort the true relationship between variables.

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Statistics Made Easy

5 Tips for Interpreting P-Values Correctly in Hypothesis Testing

Hypothesis testing is a critical part of statistical analysis and is often the endpoint where conclusions are drawn about larger populations based on a sample or experimental dataset. Central to this process is the p-value. Broadly, the p-value quantifies the strength of evidence against the null hypothesis. Given the importance of the p-value, it is essential to ensure its interpretation is correct. Here are five essential tips for ensuring the p-value from a hypothesis test is understood correctly.

1. Know What the P-value Represents

First, it is essential to understand what a p-value is. In hypothesis testing, the p-value is defined as the probability of observing your data, or data more extreme, if the null hypothesis is true. As a reminder, the null hypothesis states no difference between your data and the expected population.

For example, in a hypothesis test to see if changing a company’s logo drives more traffic to the website, a null hypothesis would state that the new traffic numbers are equal to the old traffic numbers. In this context, the p-value would be the probability that the data you observed, or data more extreme, would occur if this null hypothesis were true.

Therefore, a smaller p-value indicates that what you observed is unlikely to have occurred if the null were true, offering evidence to reject the null hypothesis. Typically, a cut-off value of 0.05 is used where any p-value below this is considered significant evidence against the null.

2. Understand the Directionality of Your Hypothesis

Based on the research question under exploration, there are two types of hypotheses: one-sided and two-sided. A one-sided test specifies a particular direction of effect, such as traffic to a website increasing after a design change. On the other hand, a two-sided test allows the change to be in either direction and is effective when the researcher wants to see any effect of the change.

Either way, determining the statistical significance of a p-value is the same: if the p-value is below a threshold value, it is statistically significant. However, when calculating the p-value, it is important to ensure the correct sided calculations have been completed.

Additionally, the interpretation of the meaning of a p-value will differ based on the directionality of the hypothesis. If a one-sided test is significant, the researchers can use the p-value to support a statistically significant increase or decrease based on the direction of the test. If a two-sided test is significant, the p-value can only be used to say that the two groups are different, but not that one is necessarily greater.

3. Avoid Threshold Thinking

A common pitfall in interpreting p-values is falling into the threshold thinking trap. The most commonly used cut-off value for whether a calculated p-value is statistically significant is 0.05. Typically, a p-value of less than 0.05 is considered statistically significant evidence against the null hypothesis.

However, this is just an arbitrary value. Rigid adherence to this or any other predefined cut-off value can obscure business-relevant effect sizes. For example, a hypothesis test looking at changes in traffic after a website design may find that an increase of 10,000 views is not statistically significant with a p-value of 0.055 since that value is above 0.05. However, the actual increase of 10,000 may be important to the growth of the business.

Therefore, a p-value can be practically significant while not being statistically significant. Both types of significance and the broader context of the hypothesis test should be considered when making a final interpretation.

4. Consider the Power of Your Study

Similarly, some study conditions can result in a non-significant p-value even if practical significance exists. Statistical power is the ability of a study to detect an effect when it truly exists. In other words, it is the probability that the null hypothesis will be rejected when it is false.

Power is impacted by a lot of factors. These include sample size, the effect size you are looking for, and variability within the data. In the example of website traffic after a design change, if the number of visits overall is too small, there may not be enough views to have enough power to detect a difference.

Simple ways to increase the power of a hypothesis test and increase the chances of detecting an effect are increasing the sample size, looking for a smaller effect size, changing the experiment design to control for variables that can increase variability, or adjusting the type of statistical test being run.

5. Be Aware of Multiple Comparisons

Whenever multiple p-values are calculated in a single study due to multiple comparisons, there is an increased risk of false positives. This is because each individual comparison introduces random fluctuations, and each additional comparison compounds these fluctuations.

For example, in a hypothesis test looking at traffic before and after a website redesign, the team may be interested in making more than one comparison. This can include total visits, page views, and average time spent on the website. Since multiple comparisons are being made, there must be a correction made when interpreting the p-value.

The Bonferroni correction is one of the most commonly used methods to account for this increased probability of false positives. In this method, the significance cut-off value, typically 0.05, is divided by the number of comparisons made. The result is used as the new significance cut-off value. Applying this correction mitigates the risk of false positives and improves the reliability of findings from a hypothesis test.

In conclusion, interpreting p-values requires a nuanced understanding of many statistical concepts and careful consideration of the hypothesis test’s context. By following these five tips, the interpretation of the p-value from a hypothesis test can be more accurate and reliable, leading to better data-driven decision-making.

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Chi-Square (Χ²) Test & How To Calculate Formula Equation

Benjamin Frimodig

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B.A., History and Science, Harvard University

Ben Frimodig is a 2021 graduate of Harvard College, where he studied the History of Science.

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Chi-square (χ2) is used to test hypotheses about the distribution of observations into categories with no inherent ranking.

What Is a Chi-Square Statistic?

The Chi-square test (pronounced Kai) looks at the pattern of observations and will tell us if certain combinations of the categories occur more frequently than we would expect by chance, given the total number of times each category occurred.

It looks for an association between the variables. We cannot use a correlation coefficient to look for the patterns in this data because the categories often do not form a continuum.

There are three main types of Chi-square tests, tests of goodness of fit, the test of independence, and the test for homogeneity. All three tests rely on the same formula to compute a test statistic.

These tests function by deciphering relationships between observed sets of data and theoretical or “expected” sets of data that align with the null hypothesis.

What is a Contingency Table?

Contingency tables (also known as two-way tables) are grids in which Chi-square data is organized and displayed. They provide a basic picture of the interrelation between two variables and can help find interactions between them.

In contingency tables, one variable and each of its categories are listed vertically, and the other variable and each of its categories are listed horizontally.

Additionally, including column and row totals, also known as “marginal frequencies,” will help facilitate the Chi-square testing process.

In order for the Chi-square test to be considered trustworthy, each cell of your expected contingency table must have a value of at least five.

Each Chi-square test will have one contingency table representing observed counts (see Fig. 1) and one contingency table representing expected counts (see Fig. 2).

Figure 1. Observed table (which contains the observed counts).

To obtain the expected frequencies for any cell in any cross-tabulation in which the two variables are assumed independent, multiply the row and column totals for that cell and divide the product by the total number of cases in the table.

Figure 2. Expected table (what we expect the two-way table to look like if the two categorical variables are independent).

To decide if our calculated value for χ2 is significant, we also need to work out the degrees of freedom for our contingency table using the following formula: df= (rows – 1) x (columns – 1).

Formula Calculation

Calculate the chi-square statistic (χ2) by completing the following steps:

Calculate the expected frequencies and the observed frequencies.
For each observed number in the table, subtract the corresponding expected number (O — E).
Square the difference (O —E)².
Divide the squares obtained for each cell in the table by the expected number for that cell (O – E)² / E.
Sum all the values for (O – E)² / E. This is the chi-square statistic.
Calculate the degrees of freedom for the contingency table using the following formula; df= (rows – 1) x (columns – 1).

Once we have calculated the degrees of freedom (df) and the chi-squared value (χ2), we can use the χ2 table (often at the back of a statistics book) to check if our value for χ2 is higher than the critical value given in the table. If it is, then our result is significant at the level given.

Interpretation

The chi-square statistic tells you how much difference exists between the observed count in each table cell to the counts you would expect if there were no relationship at all in the population.

Small Chi-Square Statistic: If the chi-square statistic is small and the p-value is large (usually greater than 0.05), this often indicates that the observed frequencies in the sample are close to what would be expected under the null hypothesis.

The null hypothesis usually states no association between the variables being studied or that the observed distribution fits the expected distribution.

In theory, if the observed and expected values were equal (no difference), then the chi-square statistic would be zero — but this is unlikely to happen in real life.

Large Chi-Square Statistic : If the chi-square statistic is large and the p-value is small (usually less than 0.05), then the conclusion is often that the data does not fit the model well, i.e., the observed and expected values are significantly different. This often leads to the rejection of the null hypothesis.

How to Report

To report a chi-square output in an APA-style results section, always rely on the following template:

χ2 ( degrees of freedom , N = sample size ) = chi-square statistic value , p = p value .

In the case of the above example, the results would be written as follows:

A chi-square test of independence showed that there was a significant association between gender and post-graduation education plans, χ2 (4, N = 101) = 54.50, p < .001.

APA Style Rules

Do not use a zero before a decimal when the statistic cannot be greater than 1 (proportion, correlation, level of statistical significance).
Report exact p values to two or three decimals (e.g., p = .006, p = .03).
However, report p values less than .001 as “ p < .001.”
Put a space before and after a mathematical operator (e.g., minus, plus, greater than, less than, equals sign).
Do not repeat statistics in both the text and a table or figure.

p -value Interpretation

You test whether a given χ2 is statistically significant by testing it against a table of chi-square distributions , according to the number of degrees of freedom for your sample, which is the number of categories minus 1. The chi-square assumes that you have at least 5 observations per category.

If you are using SPSS then you will have an expected p -value.

For a chi-square test, a p-value that is less than or equal to the .05 significance level indicates that the observed values are different to the expected values.

Thus, low p-values (p< .05) indicate a likely difference between the theoretical population and the collected sample. You can conclude that a relationship exists between the categorical variables.

Remember that p -values do not indicate the odds that the null hypothesis is true but rather provide the probability that one would obtain the sample distribution observed (or a more extreme distribution) if the null hypothesis was true.

A level of confidence necessary to accept the null hypothesis can never be reached. Therefore, conclusions must choose to either fail to reject the null or accept the alternative hypothesis, depending on the calculated p-value.

The four steps below show you how to analyze your data using a chi-square goodness-of-fit test in SPSS (when you have hypothesized that you have equal expected proportions).

Step 1 : Analyze > Nonparametric Tests > Legacy Dialogs > Chi-square… on the top menu as shown below:

Step 2 : Move the variable indicating categories into the “Test Variable List:” box.

Step 3 : If you want to test the hypothesis that all categories are equally likely, click “OK.”

Step 4 : Specify the expected count for each category by first clicking the “Values” button under “Expected Values.”

Step 5 : Then, in the box to the right of “Values,” enter the expected count for category one and click the “Add” button. Now enter the expected count for category two and click “Add.” Continue in this way until all expected counts have been entered.

Step 6 : Then click “OK.”

The four steps below show you how to analyze your data using a chi-square test of independence in SPSS Statistics.

Step 1 : Open the Crosstabs dialog (Analyze > Descriptive Statistics > Crosstabs).

Step 2 : Select the variables you want to compare using the chi-square test. Click one variable in the left window and then click the arrow at the top to move the variable. Select the row variable and the column variable.

Step 3 : Click Statistics (a new pop-up window will appear). Check Chi-square, then click Continue.

Step 4 : (Optional) Check the box for Display clustered bar charts.

Step 5 : Click OK.

Goodness-of-Fit Test

The Chi-square goodness of fit test is used to compare a randomly collected sample containing a single, categorical variable to a larger population.

This test is most commonly used to compare a random sample to the population from which it was potentially collected.

The test begins with the creation of a null and alternative hypothesis. In this case, the hypotheses are as follows:

Null Hypothesis (Ho) : The null hypothesis (Ho) is that the observed frequencies are the same (except for chance variation) as the expected frequencies. The collected data is consistent with the population distribution.

Alternative Hypothesis (Ha) : The collected data is not consistent with the population distribution.

The next step is to create a contingency table that represents how the data would be distributed if the null hypothesis were exactly correct.

The sample’s overall deviation from this theoretical/expected data will allow us to draw a conclusion, with a more severe deviation resulting in smaller p-values.

Test for Independence

The Chi-square test for independence looks for an association between two categorical variables within the same population.

Unlike the goodness of fit test, the test for independence does not compare a single observed variable to a theoretical population but rather two variables within a sample set to one another.

The hypotheses for a Chi-square test of independence are as follows:

Null Hypothesis (Ho) : There is no association between the two categorical variables in the population of interest.

Alternative Hypothesis (Ha) : There is no association between the two categorical variables in the population of interest.

The next step is to create a contingency table of expected values that reflects how a data set that perfectly aligns the null hypothesis would appear.

The simplest way to do this is to calculate the marginal frequencies of each row and column; the expected frequency of each cell is equal to the marginal frequency of the row and column that corresponds to a given cell in the observed contingency table divided by the total sample size.

Test for Homogeneity

The Chi-square test for homogeneity is organized and executed exactly the same as the test for independence.

The main difference to remember between the two is that the test for independence looks for an association between two categorical variables within the same population, while the test for homogeneity determines if the distribution of a variable is the same in each of several populations (thus allocating population itself as the second categorical variable).

Null Hypothesis (Ho) : There is no difference in the distribution of a categorical variable for several populations or treatments.

Alternative Hypothesis (Ha) : There is a difference in the distribution of a categorical variable for several populations or treatments.

The difference between these two tests can be a bit tricky to determine, especially in the practical applications of a Chi-square test. A reliable rule of thumb is to determine how the data was collected.

If the data consists of only one random sample with the observations classified according to two categorical variables, it is a test for independence. If the data consists of more than one independent random sample, it is a test for homogeneity.

What is the chi-square test?

The Chi-square test is a non-parametric statistical test used to determine if there’s a significant association between two or more categorical variables in a sample.

It works by comparing the observed frequencies in each category of a cross-tabulation with the frequencies expected under the null hypothesis, which assumes there is no relationship between the variables.

This test is often used in fields like biology, marketing, sociology, and psychology for hypothesis testing.

What does chi-square tell you?

The Chi-square test informs whether there is a significant association between two categorical variables. Suppose the calculated Chi-square value is above the critical value from the Chi-square distribution.

In that case, it suggests a significant relationship between the variables, rejecting the null hypothesis of no association.

How to calculate chi-square?

To calculate the Chi-square statistic, follow these steps:

1. Create a contingency table of observed frequencies for each category.

2. Calculate expected frequencies for each category under the null hypothesis.

3. Compute the Chi-square statistic using the formula: Χ² = Σ [ (O_i – E_i)² / E_i ], where O_i is the observed frequency and E_i is the expected frequency.

4. Compare the calculated statistic with the critical value from the Chi-square distribution to draw a conclusion.

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1.1: Statements and Conditional Statements

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Ted Sundstrom
Grand Valley State University via ScholarWorks @Grand Valley State University

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Much of our work in mathematics deals with statements. In mathematics, a statement is a declarative sentence that is either true or false but not both. A statement is sometimes called a proposition . The key is that there must be no ambiguity. To be a statement, a sentence must be true or false, and it cannot be both. So a sentence such as "The sky is beautiful" is not a statement since whether the sentence is true or not is a matter of opinion. A question such as "Is it raining?" is not a statement because it is a question and is not declaring or asserting that something is true.

Some sentences that are mathematical in nature often are not statements because we may not know precisely what a variable represents. For example, the equation 2$x$+5 = 10 is not a statement since we do not know what $x$ represents. If we substitute a specific value for $x$ (such as $x$ = 3), then the resulting equation, 2$\cdot$3 +5 = 10 is a statement (which is a false statement). Following are some more examples:

There exists a real number $x$ such that 2$x$+5 = 10. This is a statement because either such a real number exists or such a real number does not exist. In this case, this is a true statement since such a real number does exist, namely $x$ = 2.5.
For each real number $x$, $2x +5 = 2 \left( x + \dfrac{5}{2}\right)$. This is a statement since either the sentence $2x +5 = 2 \left( x + \dfrac{5}{2}\right)$ is true when any real number is substituted for $x$ (in which case, the statement is true) or there is at least one real number that can be substituted for $x$ and produce a false statement (in which case, the statement is false). In this case, the given statement is true.
Solve the equation $x^2 - 7x +10 =0$. This is not a statement since it is a directive. It does not assert that something is true.
$(a+b)^2 = a^2+b^2$ is not a statement since it is not known what $a$ and $b$ represent. However, the sentence, “There exist real numbers $a$ and $b$ such that $(a+b)^2 = a^2+b^2$" is a statement. In fact, this is a true statement since there are such integers. For example, if $a=1$ and $b=0$, then $(a+b)^2 = a^2+b^2$.
Compare the statement in the previous item to the statement, “For all real numbers $a$ and $b$, $(a+b)^2 = a^2+b^2$." This is a false statement since there are values for $a$ and $b$ for which $(a+b)^2 \ne a^2+b^2$. For example, if $a=2$ and $b=3$, then $(a+b)^2 = 5^2 = 25$ and $a^2 + b^2 = 2^2 +3^2 = 13$.

Progress Check 1.1: Statements

Which of the following sentences are statements? Do not worry about determining whether a statement is true or false; just determine whether each sentence is a statement or not.

2$\cdot$7 + 8 = 22.
$(x-1) = \sqrt(x + 11)$.
$2x + 5y = 7$.
There are integers $x$ and $y$ such that $2x + 5y = 7$.
There are integers $x$ and $y$ such that $23x + 27y = 52$.
Given a line $L$ and a point $P$ not on that line, there is a unique line through $P$ that does not intersect $L$.
$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.
$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ for all real numbers $a$ and $b$.
The derivative of $f(x) = \sin x$ is $f' (x) = \cos x$.
Does the equation $3x^2 - 5x - 7 = 0$ have two real number solutions?
If $ABC$ is a right triangle with right angle at vertex $B$, and if $D$ is the midpoint of the hypotenuse, then the line segment connecting vertex $B$ to $D$ is half the length of the hypotenuse.
There do not exist three integers $x$, $y$, and $z$ such that $x^3 + y^2 = z^3$.

Add texts here. Do not delete this text first.

How Do We Decide If a Statement Is True or False?

In mathematics, we often establish that a statement is true by writing a mathematical proof. To establish that a statement is false, we often find a so-called counterexample. (These ideas will be explored later in this chapter.) So mathematicians must be able to discover and construct proofs. In addition, once the discovery has been made, the mathematician must be able to communicate this discovery to others who speak the language of mathematics. We will be dealing with these ideas throughout the text.

For now, we want to focus on what happens before we start a proof. One thing that mathematicians often do is to make a conjecture beforehand as to whether the statement is true or false. This is often done through exploration. The role of exploration in mathematics is often difficult because the goal is not to find a specific answer but simply to investigate. Following are some techniques of exploration that might be helpful.

Techniques of Exploration

Guesswork and conjectures . Formulate and write down questions and conjectures. When we make a guess in mathematics, we usually call it a conjecture.

For example, if someone makes the conjecture that $\sin(2x) = 2 \sin(x)$, for all real numbers $x$, we can test this conjecture by substituting specific values for $x$. One way to do this is to choose values of $x$ for which $\sin(x)$is known. Using $x = \frac{\pi}{4}$, we see that

$\sin(2(\frac{\pi}{4})) = \sin(\frac{\pi}{2}) = 1,$ and

$2\sin(\frac{\pi}{4}) = 2(\frac{\sqrt2}{2}) = \sqrt2$.

Since $1 \ne \sqrt2$, these calculations show that this conjecture is false. However, if we do not find a counterexample for a conjecture, we usually cannot claim the conjecture is true. The best we can say is that our examples indicate the conjecture is true. As an example, consider the conjecture that

If $x$ and $y$ are odd integers, then $x + y$ is an even integer.

We can do lots of calculation, such as $3 + 7 = 10$ and $5 + 11 = 16$, and find that every time we add two odd integers, the sum is an even integer. However, it is not possible to test every pair of odd integers, and so we can only say that the conjecture appears to be true. (We will prove that this statement is true in the next section.)

Use of prior knowledge. This also is very important. We cannot start from square one every time we explore a statement. We must make use of our acquired mathematical knowledge. For the conjecture that $\sin (2x) = 2 \sin(x)$, for all real numbers $x$, we might recall that there are trigonometric identities called “double angle identities.” We may even remember the correct identity for $\sin (2x)$, but if we do not, we can always look it up. We should recall (or find) that for all real numbers $x$, \[\sin(2x) = 2 \sin(x)\cos(x).\]
We could use this identity to argue that the conjecture “for all real numbers $x$, $\sin (2x) = 2 \sin(x)$” is false, but if we do, it is still a good idea to give a specific counterexample as we did before.
Cooperation and brainstorming . Working together is often more fruitful than working alone. When we work with someone else, we can compare notes and articulate our ideas. Thinking out loud is often a useful brainstorming method that helps generate new ideas.

Progress Check 1.2: Explorations

Use the techniques of exploration to investigate each of the following statements. Can you make a conjecture as to whether the statement is true or false? Can you determine whether it is true or false?

$(a + b)^2 = a^2 + b^2$, for all real numbers a and b.
There are integers $x$ and $y$ such that $2x + 5y = 41$.
If $x$ is an even integer, then $x^2$ is an even integer.
If $x$ and $y$ are odd integers, then $x \cdot y$ is an odd integer.

Conditional Statements

One of the most frequently used types of statements in mathematics is the so-called conditional statement. Given statements $P$ and $Q$, a statement of the form “If $P$ then $Q$” is called a conditional statement . It seems reasonable that the truth value (true or false) of the conditional statement “If $P$ then $Q$” depends on the truth values of $P$ and $Q$. The statement “If $P$ then $Q$” means that $Q$ must be true whenever $P$ is true. The statement $P$ is called the hypothesis of the conditional statement, and the statement $Q$ is called the conclusion of the conditional statement. Since conditional statements are probably the most important type of statement in mathematics, we give a more formal definition.

A conditional statement is a statement that can be written in the form “If $P$ then $Q$,” where $P$ and $Q$ are sentences. For this conditional statement, $P$ is called the hypothesis and $Q$ is called the conclusion .

Intuitively, “If $P$ then $Q$” means that $Q$ must be true whenever $P$ is true. Because conditional statements are used so often, a symbolic shorthand notation is used to represent the conditional statement “If $P$ then $Q$.” We will use the notation $P \to Q$ to represent “If $P$ then $Q$.” When $P$ and $Q$ are statements, it seems reasonable that the truth value (true or false) of the conditional statement $P \to Q$ depends on the truth values of $P$ and $Q$. There are four cases to consider:

$P$ is true and $Q$ is true.
$P$ is false and $Q$ is true.
$P$ is true and $Q$ is false.
$P$ is false and $Q$ is false.

The conditional statement $P \to Q$ means that $Q$ is true whenever $P$ is true. It says nothing about the truth value of $Q$ when $P$ is false. Using this as a guide, we define the conditional statement $P \to Q$ to be false only when $P$ is true and $Q$ is false, that is, only when the hypothesis is true and the conclusion is false. In all other cases, $P \to Q$ is true. This is summarized in Table 1.1 , which is called a truth table for the conditional statement $P \to Q$. (In Table 1.1 , T stands for “true” and F stands for “false.”)

Table 1.1: Truth Table for $P \to Q$

The important thing to remember is that the conditional statement $P \to Q$ has its own truth value. It is either true or false (and not both). Its truth value depends on the truth values for $P$ and $Q$, but some find it a bit puzzling that the conditional statement is considered to be true when the hypothesis P is false. We will provide a justification for this through the use of an example.

Example 1.3:

Suppose that I say

“If it is not raining, then Daisy is riding her bike.”

We can represent this conditional statement as $P \to Q$ where $P$ is the statement, “It is not raining” and $Q$ is the statement, “Daisy is riding her bike.”

Although it is not a perfect analogy, think of the statement $P \to Q$ as being false to mean that I lied and think of the statement $P \to Q$ as being true to mean that I did not lie. We will now check the truth value of $P \to Q$ based on the truth values of $P$ and $Q$.

Suppose that both $P$ and $Q$ are true. That is, it is not raining and Daisy is riding her bike. In this case, it seems reasonable to say that I told the truth and that$P \to Q$ is true.
Suppose that $P$ is true and $Q$ is false or that it is not raining and Daisy is not riding her bike. It would appear that by making the statement, “If it is not raining, then Daisy is riding her bike,” that I have not told the truth. So in this case, the statement $P \to Q$ is false.
Now suppose that $P$ is false and $Q$ is true or that it is raining and Daisy is riding her bike. Did I make a false statement by stating that if it is not raining, then Daisy is riding her bike? The key is that I did not make any statement about what would happen if it was raining, and so I did not tell a lie. So we consider the conditional statement, “If it is not raining, then Daisy is riding her bike,” to be true in the case where it is raining and Daisy is riding her bike.
Finally, suppose that both $P$ and $Q$ are false. That is, it is raining and Daisy is not riding her bike. As in the previous situation, since my statement was $P \to Q$, I made no claim about what would happen if it was raining, and so I did not tell a lie. So the statement $P \to Q$ cannot be false in this case and so we consider it to be true.

Progress Check 1.4: xplorations with Conditional Statements

1 . Consider the following sentence:

If $x$ is a positive real number, then $x^2 + 8x$ is a positive real number.

Although the hypothesis and conclusion of this conditional sentence are not statements, the conditional sentence itself can be considered to be a statement as long as we know what possible numbers may be used for the variable $x$. From the context of this sentence, it seems that we can substitute any positive real number for $x$. We can also substitute 0 for $x$ or a negative real number for x provided that we are willing to work with a false hypothesis in the conditional statement. (In Chapter 2 , we will learn how to be more careful and precise with these types of conditional statements.)

(a) Notice that if $x = -3$, then $x^2 + 8x = -15$, which is negative. Does this mean that the given conditional statement is false?

(b) Notice that if $x = 4$, then $x^2 + 8x = 48$, which is positive. Does this mean that the given conditional statement is true?

(c) Do you think this conditional statement is true or false? Record the results for at least five different examples where the hypothesis of this conditional statement is true.

2 . “If $n$ is a positive integer, then $n^2 - n +41$ is a prime number.” (Remember that a prime number is a positive integer greater than 1 whose only positive factors are 1 and itself.) To explore whether or not this statement is true, try using (and recording your results) for $n = 1$, $n = 2$, $n = 3$, $n = 4$, $n = 5$, and $n = 10$. Then record the results for at least four other values of $n$. Does this conditional statement appear to be true?

Further Remarks about Conditional Statements

Suppose that Ed has exactly $52 in his wallet. The following four statements will use the four possible truth combinations for the hypothesis and conclusion of a conditional statement.

If Ed has exactly $52 in his wallet, then he has $20 in his wallet. This is a true statement. Notice that both the hypothesis and the conclusion are true.
If Ed has exactly $52 in his wallet, then he has $100 in his wallet. This statement is false. Notice that the hypothesis is true and the conclusion is false.
If Ed has $100 in his wallet, then he has at least $50 in his wallet. This statement is true regardless of how much money he has in his wallet. In this case, the hypothesis is false and the conclusion is true.

This is admittedly a contrived example but it does illustrate that the conventions for the truth value of a conditional statement make sense. The message is that in order to be complete in mathematics, we need to have conventions about when a conditional statement is true and when it is false.

If $n$ is a positive integer, then $(n^2 - n + 41)$ is a prime number.

Perhaps for all of the values you tried for $n$, $(n^2 - n + 41)$ turned out to be a prime number. However, if we try $n = 41$, we ge $n^2 - n + 41 = 41^2 - 41 + 41$ $n^2 - n + 41 = 41^2$ So in the case where $n = 41$, the hypothesis is true (41 is a positive integer) and the conclusion is false $41^2$ is not prime. Therefore, 41 is a counterexample for this conjecture and the conditional statement “If $n$ is a positive integer, then $(n^2 - n + 41)$ is a prime number” is false. There are other counterexamples (such as $n = 42$, $n = 45$, and $n = 50$), but only one counterexample is needed to prove that the statement is false.

Although one example can be used to prove that a conditional statement is false, in most cases, we cannot use examples to prove that a conditional statement is true. For example, in Progress Check 1.4 , we substituted values for $x$ for the conditional statement “If $x$ is a positive real number, then $x^2 + 8x$ is a positive real number.” For every positive real number used for $x$, we saw that $x^2 + 8x$ was positive. However, this does not prove the conditional statement to be true because it is impossible to substitute every positive real number for $x$. So, although we may believe this statement is true, to be able to conclude it is true, we need to write a mathematical proof. Methods of proof will be discussed in Section 1.2 and Chapter 3 .

Progress Check 1.5: Working with a Conditional Statement

The following statement is a true statement, which is proven in many calculus texts.

If the function $f$ is differentiable at $a$, then the function $f$ is continuous at $a$.

Using only this true statement, is it possible to make a conclusion about the function in each of the following cases?

It is known that the function $f$, where $f(x) = \sin x$, is differentiable at 0.
It is known that the function $f$, where $f(x) = \sqrt[3]x$, is not differentiable at 0.
It is known that the function $f$, where $f(x) = |x|$, is continuous at 0.
It is known that the function $f$, where $f(x) = \dfrac{|x|}{x}$ is not continuous at 0.

Closure Properties of Number Systems

The primary number system used in algebra and calculus is the real number system . We usually use the symbol R to stand for the set of all real numbers. The real numbers consist of the rational numbers and the irrational numbers. The rational numbers are those real numbers that can be written as a quotient of two integers (with a nonzero denominator), and the irrational numbers are those real numbers that cannot be written as a quotient of two integers. That is, a rational number can be written in the form of a fraction, and an irrational number cannot be written in the form of a fraction. Some common irrational numbers are $\sqrt2$, $\pi$ and $e$. We usually use the symbol $\mathbb{Q}$ to represent the set of all rational numbers. (The letter $\mathbb{Q}$ is used because rational numbers are quotients of integers.) There is no standard symbol for the set of all irrational numbers.

Perhaps the most basic number system used in mathematics is the set of natural numbers . The natural numbers consist of the positive whole numbers such as 1, 2, 3, 107, and 203. We will use the symbol $\mathbb{N}$ to stand for the set of natural numbers. Another basic number system that we will be working with is the set of integers . The integers consist of zero, the positive whole numbers, and the negatives of the positive whole numbers. If $n$ is an integer, we can write $n = \dfrac{n}{1}$. So each integer is a rational number and hence also a real number.

We will use the letter $\mathbb{Z}$ to stand for the set of integers. (The letter $\mathbb{Z}$ is from the German word, $Zahlen$, for numbers.) Three of the basic properties of the integers are that the set $\mathbb{Z}$ is closed under addition , the set $\mathbb{Z}$ is closed under multiplication , and the set of integers is closed under subtraction. This means that

If $x$ and $y$ are integers, then $x + y$ is an integer;
If $x$ and $y$ are integers, then $x \cdot y$ is an integer; and
If $x$ and $y$ are integers, then $x - y$ is an integer.

Notice that these so-called closure properties are defined in terms of conditional statements. This means that if we can find one instance where the hypothesis is true and the conclusion is false, then the conditional statement is false.

Example 1.6: Closure

In order for the set of natural numbers to be closed under subtraction, the following conditional statement would have to be true: If $x$ and $y$ are natural numbers, then $x - y$ is a natural number. However, since 5 and 8 are natural numbers, $5 - 8 = -3$, which is not a natural number, this conditional statement is false. Therefore, the set of natural numbers is not closed under subtraction.
We can use the rules for multiplying fractions and the closure rules for the integers to show that the rational numbers are closed under multiplication. If $\dfrac{a}{b}$ and $\dfrac{c}{d}$ are rational numbers (so $a$, $b$, $c$, and $d$ are integers and $b$ and $d$ are not zero), then $\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd}.$ Since the integers are closed under multiplication, we know that $ac$ and $bd$ are integers and since $b \ne 0$ and $d \ne 0$, $bd \ne 0$. Hence, $\dfrac{ac}{bd}$ is a rational number and this shows that the rational numbers are closed under multiplication.

Progress Check 1.7: Closure Properties

Answer each of the following questions.

Is the set of rational numbers closed under addition? Explain.
Is the set of integers closed under division? Explain.
Is the set of rational numbers closed under subtraction? Explain.
Which of the following sentences are statements? (a) $3^2 + 4^2 = 5^2.$ (b) $a^2 + b^2 = c^2.$ (c) There exists integers $a$, $b$, and $c$ such that $a^2 + b^2 = c^2.$ (d) If $x^2 = 4$, then $x = 2.$ (e) For each real number $x$, if $x^2 = 4$, then $x = 2.$ (f) For each real number $t$, $\sin^2t + \cos^2t = 1.$ (g) $\sin x < \sin (\frac{\pi}{4}).$ (h) If $n$ is a prime number, then $n^2$ has three positive factors. (i) 1 + $\tan^2 \theta = \text{sec}^2 \theta.$ (j) Every rectangle is a parallelogram. (k) Every even natural number greater than or equal to 4 is the sum of two prime numbers.
Identify the hypothesis and the conclusion for each of the following conditional statements. (a) If $n$ is a prime number, then $n^2$ has three positive factors. (b) If $a$ is an irrational number and $b$ is an irrational number, then $a \cdot b$ is an irrational number. (c) If $p$ is a prime number, then $p = 2$ or $p$ is an odd number. (d) If $p$ is a prime number and $p \ne 2$ or $p$ is an odd number. (e) $p \ne 2$ or $p$ is a even number, then $p$ is not prime.
Determine whether each of the following conditional statements is true or false. (a) If 10 < 7, then 3 = 4. (b) If 7 < 10, then 3 = 4. (c) If 10 < 7, then 3 + 5 = 8. (d) If 7 < 10, then 3 + 5 = 8.
Determine the conditions under which each of the following conditional sentences will be a true statement. (a) If a + 2 = 5, then 8 < 5. (b) If 5 < 8, then a + 2 = 5.
Let $P$ be the statement “Student X passed every assignment in Calculus I,” and let $Q$ be the statement “Student X received a grade of C or better in Calculus I.” (a) What does it mean for $P$ to be true? What does it mean for $Q$ to be true? (b) Suppose that Student X passed every assignment in Calculus I and received a grade of B-, and that the instructor made the statement $P \to Q$. Would you say that the instructor lied or told the truth? (c) Suppose that Student X passed every assignment in Calculus I and received a grade of C-, and that the instructor made the statement $P \to Q$. Would you say that the instructor lied or told the truth? (d) Now suppose that Student X did not pass two assignments in Calculus I and received a grade of D, and that the instructor made the statement $P \to Q$. Would you say that the instructor lied or told the truth? (e) How are Parts ( 5b ), ( 5c ), and ( 5d ) related to the truth table for $P \to Q$?

Theorem If f is a quadratic function of the form $f(x) = ax^2 + bx + c$ and a < 0, then the function f has a maximum value when $x = \dfrac{-b}{2a}$. Using only this theorem, what can be concluded about the functions given by the following formulas? (a) $g (x) = -8x^2 + 5x - 2$ (b) $h (x) = -\dfrac{1}{3}x^2 + 3x$ (c) $k (x) = 8x^2 - 5x - 7$ (d) $j (x) = -\dfrac{71}{99}x^2 +210$ (e) $f (x) = -4x^2 - 3x + 7$ (f) $F (x) = -x^4 + x^3 + 9$

Theorem If $f$ is a quadratic function of the form $f(x) = ax^2 + bx + c$ and ac < 0, then the function $f$ has two x-intercepts.

Using only this theorem, what can be concluded about the functions given by the following formulas? (a) $g (x) = -8x^2 + 5x - 2$ (b) $h (x) = -\dfrac{1}{3}x^2 + 3x$ (c) $k (x) = 8x^2 - 5x - 7$ (d) $j (x) = -\dfrac{71}{99}x^2 +210$ (e) $f (x) = -4x^2 - 3x + 7$ (f) $F (x) = -x^4 + x^3 + 9$

Theorem A. If $f$ is a cubic function of the form $f (x) = x^3 - x + b$ and b > 1, then the function $f$ has exactly one $x$-intercept. Following is another theorem about $x$-intercepts of functions: Theorem B . If $f$ and $g$ are functions with $g (x) = k \cdot f (x)$, where $k$ is a nonzero real number, then $f$ and $g$ have exactly the same $x$-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas? (a) $f (x) = x^3 -x + 7$ (b) $g (x) = x^3 + x +7$ (c) $h (x) = -x^3 + x - 5$ (d) $k (x) = 2x^3 + 2x + 3$ (e) $r (x) = x^4 - x + 11$ (f) $F (x) = 2x^3 - 2x + 7$

(a) Is the set of natural numbers closed under division? (b) Is the set of rational numbers closed under division? (c) Is the set of nonzero rational numbers closed under division? (d) Is the set of positive rational numbers closed under division? (e) Is the set of positive real numbers closed under subtraction? (f) Is the set of negative rational numbers closed under division? (g) Is the set of negative integers closed under addition? Explorations and Activities
Exploring Propositions . In Progress Check 1.2 , we used exploration to show that certain statements were false and to make conjectures that certain statements were true. We can also use exploration to formulate a conjecture that we believe to be true. For example, if we calculate successive powers of $2, (2^1, 2^2, 2^3, 2^4, 2^5, ...)$ and examine the units digits of these numbers, we could make the following conjectures (among others): $\bullet$ If $n$ is a natural number, then the units digit of $2^n$ must be 2, 4, 6, or 8. $\bullet$ The units digits of the successive powers of 2 repeat according to the pattern “2, 4, 8, 6.” (a) Is it possible to formulate a conjecture about the units digits of successive powers of $4 (4^1, 4^2, 4^3, 4^4, 4^5,...)$? If so, formulate at least one conjecture. (b) Is it possible to formulate a conjecture about the units digit of numbers of the form $7^n - 2^n$, where $n$ is a natural number? If so, formulate a conjecture in the form of a conditional statement in the form “If $n$ is a natural number, then ... .” (c) Let $f (x) = e^(2x)$. Determine the first eight derivatives of this function. What do you observe? Formulate a conjecture that appears to be true. The conjecture should be written as a conditional statement in the form, “If n is a natural number, then ... .”

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3.1: The Fundamentals of Hypothesis Testing
Components of a Formal Hypothesis Test. The null hypothesis is a statement about the value of a population parameter, such as the population mean (µ) or the population proportion (p).It contains the condition of equality and is denoted as H 0 (H-naught).. H 0: µ = 157 or H0 : p = 0.37. The alternative hypothesis is the claim to be tested, the opposite of the null hypothesis.
How to Write a Strong Hypothesis
6. Write a null hypothesis. If your research involves statistical hypothesis testing, you will also have to write a null hypothesis. The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0, while the alternative hypothesis is H 1 or H a.
5.2
5.2 - Writing Hypotheses. The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ( H 0) and an alternative hypothesis ( H a ). Null Hypothesis. The statement that there is not a difference in the population (s), denoted as H 0.
Null and Alternative Hypotheses
Hypothesis a statement about the value of a population parameter, in case of two hypotheses, the statement assumed to be true is called the null hypothesis (notation H 0) and the contradictory statement is called the alternative hypothesis (notation H a). Previous/next navigation.
Hypothesis Testing
Step 5: Present your findings. The results of hypothesis testing will be presented in the results and discussion sections of your research paper, dissertation or thesis.. In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p-value).
Research Hypothesis In Psychology: Types, & Examples
A hypothesis is a testable statement or prediction about the relationship between two or more variables. It is a key component of the scientific method. Some key points about hypotheses: ... It merely indicates that a change or effect will occur without predicting which group will have higher or lower values. For example, "There is a ...
What is a Hypothesis
Definition: Hypothesis is an educated guess or proposed explanation for a phenomenon, based on some initial observations or data. It is a tentative statement that can be tested and potentially proven or disproven through further investigation and experimentation. Hypothesis is often used in scientific research to guide the design of experiments ...
How Hypothesis Tests Work: Significance Levels (Alpha) and P values
Using P values and Significance Levels Together. If your P value is less than or equal to your alpha level, reject the null hypothesis. The P value results are consistent with our graphical representation. The P value of 0.03112 is significant at the alpha level of 0.05 but not 0.01.
What is a Research Hypothesis: How to Write it, Types, and Examples
State the null hypothesis: The null hypothesis is a statement that there is no relationship between the variables you are studying. ... This happens when the test statistic (e.g., p-value) is less than the defined significance level (e.g., 0.05). Rejecting the null hypothesis does not necessarily mean that the alternative hypothesis is true; it ...
Statistical Hypothesis
Statistical hypothesis: A statement about the nature of a population.It is often stated in terms of a population parameter. Null hypothesis: A statistical hypothesis that is to be tested.. Alternative hypothesis: The alternative to the null hypothesis.. Test statistic: A function of the sample data.Depending on its value, the null hypothesis will be either rejected or not rejected.
Null & Alternative Hypotheses
The research hypothesis usually includes an explanation ("x affects y because …"). A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses.
10.2: Null and Alternative Hypotheses
The null hypothesis ($H_{0}$) is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a ... Hypothesis a statement about the value of a population parameter, in case of two hypotheses, the statement assumed to be true is called the null hypothesis ...
Stats chapter 9 Flashcards
The null hypothesis is a statement that the value of a population parameter is equal to some claimed value., A hypothesis test is a procedure for testing a claim about a property of a population. and more. Study with Quizlet and memorize flashcards containing terms like The _____ states that if, under a given assumption, the probability of a ...
Chapter 8 Flashcards
Study with Quizlet and memorize flashcards containing terms like A _____________ is a procedure for testing a claim about a property of a population., The _________ hypothesis is a statement that the value of a population parameter is equal to some claimed value., The ___________ is a value used in making a decision about the null hypothesis and is found by converting the sample statistic to a ...
5 Tips for Interpreting P-Values Correctly in Hypothesis Testing
Here are five essential tips for ensuring the p-value from a hypothesis test is understood correctly. 1. Know What the P-value Represents. First, it is essential to understand what a p-value is. In hypothesis testing, the p-value is defined as the probability of observing your data, or data more extreme, if the null hypothesis is true.
What Is Chi Square Test & How To Calculate Formula Equation
The null hypothesis usually states no association between the variables being studied or that the observed distribution fits the expected distribution. In theory, if the observed and expected values were equal (no difference), then the chi-square statistic would be zero — but this is unlikely to happen in real life.
Chapter 8 Stats Flashcards
alternative hypothesis. Study with Quizlet and memorize flashcards containing terms like The (blank) hypothesis is a statement that the value of a population parameter is equal to some claimed value., A (blank) is a procedure for testing a claim about a property of a population., A (blank) is a claim or statement about a population that is true ...
1.1: Statements and Conditional Statements
The conventions for the truth value of conditional statements may seem a bit strange,especially the fact that the conditional statement is true when the hypothesis of the conditional statement is false. The following example is meant to show that this makes sense. Suppose that Ed has exactly $52 in his wallet.
Solved Fill in the blank. The _____ hypothesis is a
The (1) _____ hypothesis is a statement that the value of a population parameter is equal to some claimed value. alternative claim null statistic In a Harris poll, adults were asked if they are in favor of abolishing the penny. Among the responses, 1229 answered "no, " 486 answered "yes, " and 383 had no opinion.
8.1 HW Flashcards
Study with Quizlet and memorize flashcards containing terms like The _________ hypothesis is a statement that the value of a population parameter is equal to some claimed value., Typically, the idea of the _______ hypothesis is that of "no effect," "no difference," or "no change.", Express the null hypothesis and the alternative hypothesis in symbolic form. Use the correct symbol (μ , ρ , σ ...
4
POLS 3314-002 Lab 4 Hypothesis Testing Section 1. Statistical significance 1. Each of the following statements is false. Explain why. a. Stata calculated a p-value of 0.001, so I know that my hypothesis has been proven. b. Stata reported a p-value of 0.02
The evidence contained in the P-value is context dependent
We were surprised and worried, however, by Muff et al.'s suggestion to interpret the p-value as a "gradual notion of evidence". Muff et al. recommend, for example, that a P-value > 0.1 should be reported as "little or no evidence" and a P-value of 0.001 as "strong evidence" in favor of the alternative hypothesis H1.
Statistics Ch 8.1 HW Flashcards
Study with Quizlet and memorize flashcards containing terms like The _________ hypothesis is a statement that the value of a population parameter is equal to some claimed value., Typically, the idea of the _______ hypothesis is that of "no effect," "no difference," or "no change.", Express the null hypothesis and the alternative hypothesis in symbolic form. Use the correct symbol (μ , ρ , σ ...
PREDICT HF: Risk stratification in advanced heart failure using novel
1 INTRODUCTION. Heart failure (HF) is a serious public health concern in the United States, affecting over 6.5 million patients, with 650,000 new cases diagnosed annually. 1 Despite progress in treatment, an estimated 1%−10% of patients progress to advanced HF with dismal 5-year survival rates among Stage C and D patients of 75% and 20%, respectively. 2, 3 Thus, accurate prognostication is ...
chp.8 Flashcards
h0: =p=.00034 h1: p(=/) .00034 z= -2.75 p-value=.006 Fail to reject the null hypothesis because the P-value is greater than the significance level, alpha. There is not sufficient evidence to support the claim that cell phone users develop cancer of the brain or nervous system at a rate that is different from the rate of 0.0340%.

Margin Size

3.1: The Fundamentals of Hypothesis Testing

The Fundamentals of Hypothesis Testing

Definition: hypothesis testing

Example \(\PageIndex{1}\):

Example \(\PageIndex{2}\):

Components of a Formal Hypothesis Test

The Null and Alternative Hypotheses

A Two-sided Test

Example \(\PageIndex{3}\):

A Right-sided Test

Example \(\PageIndex{4}\):

A Left-sided Test

Example \(\PageIndex{5}\):

Statistically Significant

Types of Errors

Power of the Test

Definition: power of the test

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Null and Alternative Hypotheses

Chapter Review

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What is a Hypothesis – Types, Examples and Writing Guide

Types of Hypothesis

Research Hypothesis

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Statistical Hypothesis

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Simple Hypothesis

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How to write a Hypothesis

Identify the Research Question

Conduct a Literature Review

Determine the Variables

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5 Tips for Interpreting P-Values Correctly in Hypothesis Testing

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