greater than (>) less than (<)
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
H a : 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
H a : 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
H a : 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
H a : 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
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: Evaluate the null hypothesis , typically denoted with H 0 . The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ≤ or ≥) Always write the alternative hypothesis , typically denoted with H a or H 1 , using less than, greater than, or not equals symbols, i.e., (≠, >, or <). If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.
H 0 and H a are contradictory.
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Published on 5 October 2022 by Shaun Turney . Revised on 6 December 2022.
The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :
The effect is usually the effect of the independent variable on the dependent variable .
Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, differences between null and alternative hypotheses, how to write null and alternative hypotheses, frequently asked questions about null and alternative hypotheses.
The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”, the null hypothesis (H 0 ) answers “No, there’s no effect in the population.” On the other hand, the alternative hypothesis (H A ) answers “Yes, there is an effect in the population.”
The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample.
You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.
The null hypothesis is the claim that there’s no effect in the population.
If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.
Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept. Be careful not to say you “prove” or “accept” the null hypothesis.
Null hypotheses often include phrases such as “no effect”, “no difference”, or “no relationship”. When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).
The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.
( ) | ||
Does tooth flossing affect the number of cavities? | Tooth flossing has on the number of cavities. | test: The mean number of cavities per person does not differ between the flossing group (µ ) and the non-flossing group (µ ) in the population; µ = µ . |
Does the amount of text highlighted in the textbook affect exam scores? | The amount of text highlighted in the textbook has on exam scores. | : There is no relationship between the amount of text highlighted and exam scores in the population; β = 0. |
Does daily meditation decrease the incidence of depression? | Daily meditation the incidence of depression.* | test: The proportion of people with depression in the daily-meditation group ( ) is greater than or equal to the no-meditation group ( ) in the population; ≥ . |
*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .
The alternative hypothesis (H A ) is the other answer to your research question . It claims that there’s an effect in the population.
Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.
The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.
Alternative hypotheses often include phrases such as “an effect”, “a difference”, or “a relationship”. When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes > or <). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.
The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.
Does tooth flossing affect the number of cavities? | Tooth flossing has an on the number of cavities. | test: The mean number of cavities per person differs between the flossing group (µ ) and the non-flossing group (µ ) in the population; µ ≠ µ . |
Does the amount of text highlighted in a textbook affect exam scores? | The amount of text highlighted in the textbook has an on exam scores. | : There is a relationship between the amount of text highlighted and exam scores in the population; β ≠ 0. |
Does daily meditation decrease the incidence of depression? | Daily meditation the incidence of depression. | test: The proportion of people with depression in the daily-meditation group ( ) is less than the no-meditation group ( ) in the population; < . |
Null and alternative hypotheses are similar in some ways:
However, there are important differences between the two types of hypotheses, summarized in the following table.
A claim that there is in the population. | A claim that there is in the population. | |
| ||
Equality symbol (=, ≥, or ≤) | Inequality symbol (≠, <, or >) | |
Rejected | Supported | |
Failed to reject | Not supported |
To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.
The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:
Does independent variable affect dependent variable ?
Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.
( ) | ||
test
with two groups | The mean dependent variable does not differ between group 1 (µ ) and group 2 (µ ) in the population; µ = µ . | The mean dependent variable differs between group 1 (µ ) and group 2 (µ ) in the population; µ ≠ µ . |
with three groups | The mean dependent variable does not differ between group 1 (µ ), group 2 (µ ), and group 3 (µ ) in the population; µ = µ = µ . | The mean dependent variable of group 1 (µ ), group 2 (µ ), and group 3 (µ ) are not all equal in the population. |
There is no correlation between independent variable and dependent variable in the population; ρ = 0. | There is a correlation between independent variable and dependent variable in the population; ρ ≠ 0. | |
There is no relationship between independent variable and dependent variable in the population; β = 0. | There is a relationship between independent variable and dependent variable in the population; β ≠ 0. | |
Two-proportions test | The dependent variable expressed as a proportion does not differ between group 1 ( ) and group 2 ( ) in the population; = . | The dependent variable expressed as a proportion differs between group 1 ( ) and group 2 ( ) in the population; ≠ . |
Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.
The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).
The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).
A research hypothesis is your proposed answer to your research question. 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. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.
If you want to cite this source, you can copy and paste the citation or click the ‘Cite this Scribbr article’ button to automatically add the citation to our free Reference Generator.
Turney, S. (2022, December 06). Null and Alternative Hypotheses | Definitions & Examples. Scribbr. Retrieved 9 September 2024, from https://www.scribbr.co.uk/stats/null-and-alternative-hypothesis/
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Updated: July 5, 2023 by Ken Feldman
Hypothesis testing is a branch of statistics in which, using data from a sample, an inference is made about a population parameter or a population probability distribution .
First, a hypothesis statement and assumption is made about the population parameter or probability distribution. This initial statement is called the Null Hypothesis and is denoted by H o.
An alternative or alternate hypothesis (denoted Ha ) is then stated, which will be the opposite of the Null Hypothesis.
The hypothesis testing process and analysis involves using sample data to determine whether or not you can be statistically confident that you can reject or fail to reject the H o. If the H o is rejected, the statistical conclusion is that the alternative or alternate hypothesis Ha is true.
Hypothesis testing applies to all forms of statistical inquiry. For example, it can be used to determine whether there are differences between population parameters or an understanding about slopes of regression lines or equality of probability distributions.
In all cases, the first thing you do is state the Null Hypothesis. The word “null” in the context of hypothesis testing means “nothing” or “zero.”
If we wanted to test whether there was a difference in two population means based on the calculations from two samples, we would state the Null Hypothesis in the form of:
Ho: mu1 = mu2 or mu1 – mu2 = 0
In other words, there is no difference, or the difference is zero. Note that the notation is in the form of a population parameter, not a sample statistic.
The analysis of the Null Hypothesis is designed to test the Null, which will determine whether the Null should be rejected so that the Alternate Hypothesis is defaulted to and assumed to be true, or not to reject the Null so it is assumed to be the true condition.
A classic example is when you get the results back from your doctor after taking a blood test. The Null is written to state that there is no infection. Remember, the Null is always in the form of “nothingness.” The alternate hypothesis is that you have an infection. Once the test is done, the lab will determine whether the Null can be rejected or not. If the test shows an infection, the Null will be rejected, and the Alternate will be assumed to be true. If the test shows no infection, we cannot reject the Null.
While the Null can only be written in one form — “equal to” or “no difference” — the Alternate can be written for three conditions. For example, a marketing director wants to improve sales. She designs and launches a new social media campaign, collecting sample data for sales activity prior to the new campaign. After six months, sample sales data was collected to determine whether the campaign was successful. Hypothesis testing was used to statistically confirm whether the campaign was successful or not. The Null Hypothesis was written as: Ho: muBefore = muAfter, where the claim was that the population average sales before the campaign is the same as the population average sales after the campaign. In other words, the campaign had no effect on sales.
The Alternate Hypothesis can now be written in one of three forms:
The stating and testing of the Null and the default to the Alternate hypothesis is the foundation of hypothesis testing. By doing so, you set the parameters for your statistical inference.
Just looking at the mathematical difference between the means of two samples and making a decision is woefully inadequate. By statistically testing the Null hypothesis, you will have more confidence in any inferences you want to make about populations based on your samples. If you reject the Null, you’ll know which Alternate is most appropriate.
Many statistical tests require assumptions of specific distributions. Many of these tests assume that the population follows the normal distribution . If it doesn’t, the test may be invalid. Rejecting the Null will establish whether the estimated distribution fulfills any test assumption or not.
Hypothesis testing calculations will provide some relative strength to your decisions as to whether you reject or fail to reject the Null hypothesis and, therefore, the Alternate.
It is the interpretation of the statistics relative to the Null and Alternate hypotheses that is important.
The Alternate can take the form of “not equal,” “less than,” or “greater than.”
You don’t want to place too big of a hurdle (or burde)n on your decision-making relative to action on the Null hypothesis by selecting an alpha value that is too high or too low. The Alternate can really reflect the true condition of the population, so failing to reject the Null too often can mask the truth.
Since your decision relative to rejecting or not rejecting the Null impacts the Alternate Hypothesis, it’s important to understand how that decision works.
The director of manufacturing at a medium-size window manufacturer recently had an older machine retrofitted to increase run speed of the equipment. The supplier, after doing the retrofit, claimed that the machine was now running significantly faster. He showed a comparison of the sample average run speed before the retrofit and the sample average run speed after the retrofit. Looking at the two averages, it appeared that the supplier was correct and that the retrofit did indeed increase the speed.
However, having had some training in Lean Six Sigma, the director asked his local Black Belt for some help in doing a hypothesis test on the data to see if there was truly a statistically significant improvement of at least 100 rpm. The Null hypothesis was written as:
mu1-mu2= -100 rpms
That is the Before speed minus the After speed equaled -100 rpms. This was slightly different than the format he was used to seeing, where the Null would be reflective of “no difference” rather than a value of interest.
The Alternate hypothesis was written to reflect an increase of at least 100 rpms. That form was written as:
mu1 – mu2 greater -100
That would mean the After speed was more than 100 rpms faster than the Before speed. Fortunately, the results indicated that the Null hypothesis should be rejected and that in actuality, the difference was greater than the 100 rpms he wanted.
Using hypothesis testing to help make better data-driven decisions requires that you properly address the Null and Alternate Hypotheses.
The Alternate should be in the form of “not equal to,” “greater than,” “less than,” or “equal to some value of interest.”
The writing of the Alternate Hypothesis can vary, so be sure you understand exactly what condition you are testing against.
Being too cautious will lead you to not reject the Null enough so you will never learn anything about your population data.
What form should the Alternate Hypothesis be written in?
The Alternate Hypothesis should be in the form of “not equal to,” “greater than,” “less than,” or “equal to some value of interest.”
When do you default to the Alternate Hypothesis?
If your analysis of the sample data suggests that you should reject the Null Hypothesis, you will default to the statement of the Alternate Hypothesis.
Should I be disappointed if I default to the Alternate Hypothesis?
Usually no. Typically, you want the actions you took to have had some impact or effect. Rejecting the Null and defaulting to the Alternate signals that something did, in fact, happen — and that might be a good thing.
The Alternate Hypothesis is the default should you reject the Null. It is an indication that something has happened that is significant. Often, that is what you want to see if you’re comparing a before and after situation.
While the form of the Null Hypothesis is usually written in a single format, the format of the Alternate can be written a number of different ways. This provides more sensitivity to your interpretation of the population data and will therefore provide a richer insight for decision-making.
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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.
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., =).
Research Question | Is the population mean different from \( \mu_{0} \)? | Is the population mean greater than \(\mu_{0}\)? | Is the population mean less than \(\mu_{0}\)? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\mu=\mu_{0} \) | \(\mu=\mu_{0} \) | \(\mu=\mu_{0} \) |
Alternative Hypothesis, \(H_{a}\) | \(\mu\neq \mu_{0} \) | \(\mu> \mu_{0} \) | \(\mu<\mu_{0} \) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Is there a difference in the population? | Is there a mean increase in the population? | Is there a mean decrease in the population? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\mu_d=0 \) | \(\mu_d =0 \) | \(\mu_d=0 \) |
Alternative Hypothesis, \(H_{a}\) | \(\mu_d \neq 0 \) | \(\mu_d> 0 \) | \(\mu_d<0 \) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Is the population proportion different from \(p_0\)? | Is the population proportion greater than \(p_0\)? | Is the population proportion less than \(p_0\)? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(p=p_0\) | \(p= p_0\) | \(p= p_0\) |
Alternative Hypothesis, \(H_{a}\) | \(p\neq p_0\) | \(p> p_0\) | \(p< p_0\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Are the population means different? | Is the population mean in group 1 greater than the population mean in group 2? | Is the population mean in group 1 less than the population mean in groups 2? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\mu_1=\mu_2\) | \(\mu_1 = \mu_2 \) | \(\mu_1 = \mu_2 \) |
Alternative Hypothesis, \(H_{a}\) | \(\mu_1 \ne \mu_2 \) | \(\mu_1 \gt \mu_2 \) | \(\mu_1 \lt \mu_2\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Are the population proportions different? | Is the population proportion in group 1 greater than the population proportion in groups 2? | Is the population proportion in group 1 less than the population proportion in group 2? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(p_1 = p_2 \) | \(p_1 = p_2 \) | \(p_1 = p_2 \) |
Alternative Hypothesis, \(H_{a}\) | \(p_1 \ne p_2\) | \(p_1 \gt p_2 \) | \(p_1 \lt p_2\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Is the slope in the population different from 0? | Is the slope in the population positive? | Is the slope in the population negative? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\beta =0\) | \(\beta= 0\) | \(\beta = 0\) |
Alternative Hypothesis, \(H_{a}\) | \(\beta\neq 0\) | \(\beta> 0\) | \(\beta< 0\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
Research Question | Is the correlation in the population different from 0? | Is the correlation in the population positive? | Is the correlation in the population negative? |
---|---|---|---|
Null Hypothesis, \(H_{0}\) | \(\rho=0\) | \(\rho= 0\) | \(\rho = 0\) |
Alternative Hypothesis, \(H_{a}\) | \(\rho \neq 0\) | \(\rho > 0\) | \(\rho< 0\) |
Type of Hypothesis Test | Two-tailed, non-directional | Right-tailed, directional | Left-tailed, directional |
All Subjects
In ap statistics.
The alternative hypothesis, denoted as Ha, is a statement that contradicts or challenges the null hypothesis. It suggests that there is a significant relationship or difference between variables being studied.
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Null Hypothesis : The null hypothesis, denoted as H0, is the opposite of the alternative hypothesis. It assumes that any observed differences or relationships are due to random chance.
Type I Error : A type I error occurs when we reject the null hypothesis when it is actually true. In other words, we mistakenly conclude there is an effect or relationship when there isn't one.
Type II Error : A type II error occurs when we fail to reject the null hypothesis when it is actually false. In other words, we miss detecting an effect or relationship that truly exists.
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Hypothesis Testing with One Sample
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 :
equal (=) | not equal (≠) greater than (>) less than (<) |
greater than or equal to (≥) | less than (<) |
less than or equal to (≤) | more than (>) |
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.
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.
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.
<!– ??? –>
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.
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:
H 0 and H a are contradictory.
has: | equal (=) | greater than or equal to (≥) | less than or equal to (≤) |
has: | not equal (≠) greater than (>) less than (<) | less than (<) | greater than (>) |
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?
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 ).
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:
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:
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:
Data from the National Institute of Mental Health. Available online at http://www.nimh.nih.gov/publicat/depression.cfm.
Null and Alternative Hypotheses Copyright © 2013 by OpenStaxCollege is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.
Hypothesis testing, the null and alternative hypothesis.
In order to undertake hypothesis testing you need to express your research hypothesis as a null and alternative hypothesis. The null hypothesis and alternative hypothesis are statements regarding the differences or effects that occur in the population. You will use your sample to test which statement (i.e., the null hypothesis or alternative hypothesis) is most likely (although technically, you test the evidence against the null hypothesis). So, with respect to our teaching example, the null and alternative hypothesis will reflect statements about all statistics students on graduate management courses.
The null hypothesis is essentially the "devil's advocate" position. That is, it assumes that whatever you are trying to prove did not happen ( hint: it usually states that something equals zero). For example, the two different teaching methods did not result in different exam performances (i.e., zero difference). Another example might be that there is no relationship between anxiety and athletic performance (i.e., the slope is zero). The alternative hypothesis states the opposite and is usually the hypothesis you are trying to prove (e.g., the two different teaching methods did result in different exam performances). Initially, you can state these hypotheses in more general terms (e.g., using terms like "effect", "relationship", etc.), as shown below for the teaching methods example:
Null Hypotheses (H ): | Undertaking seminar classes has no effect on students' performance. |
Alternative Hypothesis (H ): | Undertaking seminar class has a positive effect on students' performance. |
Depending on how you want to "summarize" the exam performances will determine how you might want to write a more specific null and alternative hypothesis. For example, you could compare the mean exam performance of each group (i.e., the "seminar" group and the "lectures-only" group). This is what we will demonstrate here, but other options include comparing the distributions , medians , amongst other things. As such, we can state:
Null Hypotheses (H ): | The mean exam mark for the "seminar" and "lecture-only" teaching methods is the same in the population. |
Alternative Hypothesis (H ): | The mean exam mark for the "seminar" and "lecture-only" teaching methods is not the same in the population. |
Now that you have identified the null and alternative hypotheses, you need to find evidence and develop a strategy for declaring your "support" for either the null or alternative hypothesis. We can do this using some statistical theory and some arbitrary cut-off points. Both these issues are dealt with next.
The level of statistical significance is often expressed as the so-called p -value . Depending on the statistical test you have chosen, you will calculate a probability (i.e., the p -value) of observing your sample results (or more extreme) given that the null hypothesis is true . Another way of phrasing this is to consider the probability that a difference in a mean score (or other statistic) could have arisen based on the assumption that there really is no difference. Let us consider this statement with respect to our example where we are interested in the difference in mean exam performance between two different teaching methods. If there really is no difference between the two teaching methods in the population (i.e., given that the null hypothesis is true), how likely would it be to see a difference in the mean exam performance between the two teaching methods as large as (or larger than) that which has been observed in your sample?
So, you might get a p -value such as 0.03 (i.e., p = .03). This means that there is a 3% chance of finding a difference as large as (or larger than) the one in your study given that the null hypothesis is true. However, you want to know whether this is "statistically significant". Typically, if there was a 5% or less chance (5 times in 100 or less) that the difference in the mean exam performance between the two teaching methods (or whatever statistic you are using) is as different as observed given the null hypothesis is true, you would reject the null hypothesis and accept the alternative hypothesis. Alternately, if the chance was greater than 5% (5 times in 100 or more), you would fail to reject the null hypothesis and would not accept the alternative hypothesis. As such, in this example where p = .03, we would reject the null hypothesis and accept the alternative hypothesis. We reject it because at a significance level of 0.03 (i.e., less than a 5% chance), the result we obtained could happen too frequently for us to be confident that it was the two teaching methods that had an effect on exam performance.
Whilst there is relatively little justification why a significance level of 0.05 is used rather than 0.01 or 0.10, for example, it is widely used in academic research. However, if you want to be particularly confident in your results, you can set a more stringent level of 0.01 (a 1% chance or less; 1 in 100 chance or less).
When considering whether we reject the null hypothesis and accept the alternative hypothesis, we need to consider the direction of the alternative hypothesis statement. For example, the alternative hypothesis that was stated earlier is:
Alternative Hypothesis (H ): | Undertaking seminar classes has a positive effect on students' performance. |
The alternative hypothesis tells us two things. First, what predictions did we make about the effect of the independent variable(s) on the dependent variable(s)? Second, what was the predicted direction of this effect? Let's use our example to highlight these two points.
Sarah predicted that her teaching method (independent variable: teaching method), whereby she not only required her students to attend lectures, but also seminars, would have a positive effect (that is, increased) students' performance (dependent variable: exam marks). If an alternative hypothesis has a direction (and this is how you want to test it), the hypothesis is one-tailed. That is, it predicts direction of the effect. If the alternative hypothesis has stated that the effect was expected to be negative, this is also a one-tailed hypothesis.
Alternatively, a two-tailed prediction means that we do not make a choice over the direction that the effect of the experiment takes. Rather, it simply implies that the effect could be negative or positive. If Sarah had made a two-tailed prediction, the alternative hypothesis might have been:
Alternative Hypothesis (H ): | Undertaking seminar classes has an effect on students' performance. |
In other words, we simply take out the word "positive", which implies the direction of our effect. In our example, making a two-tailed prediction may seem strange. After all, it would be logical to expect that "extra" tuition (going to seminar classes as well as lectures) would either have a positive effect on students' performance or no effect at all, but certainly not a negative effect. However, this is just our opinion (and hope) and certainly does not mean that we will get the effect we expect. Generally speaking, making a one-tail prediction (i.e., and testing for it this way) is frowned upon as it usually reflects the hope of a researcher rather than any certainty that it will happen. Notable exceptions to this rule are when there is only one possible way in which a change could occur. This can happen, for example, when biological activity/presence in measured. That is, a protein might be "dormant" and the stimulus you are using can only possibly "wake it up" (i.e., it cannot possibly reduce the activity of a "dormant" protein). In addition, for some statistical tests, one-tailed tests are not possible.
Let's return finally to the question of whether we reject or fail to reject the null hypothesis.
If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above the cut-off value, we fail to reject the null hypothesis and cannot accept the alternative hypothesis. You should note that you cannot accept the null hypothesis, but only find evidence against it.
Hypothesis testing involves the careful construction of two statements: the null hypothesis and the alternative hypothesis. These hypotheses can look very similar but are actually different.
How do we know which hypothesis is the null and which one is the alternative? We will see that there are a few ways to tell the difference.
The null hypothesis reflects that there will be no observed effect in our experiment. In a mathematical formulation of the null hypothesis, there will typically be an equal sign. This hypothesis is denoted by H 0 .
The null hypothesis is what we attempt to find evidence against in our hypothesis test. We hope to obtain a small enough p-value that it is lower than our level of significance alpha and we are justified in rejecting the null hypothesis. If our p-value is greater than alpha, then we fail to reject the null hypothesis.
If the null hypothesis is not rejected, then we must be careful to say what this means. The thinking on this is similar to a legal verdict. Just because a person has been declared "not guilty", it does not mean that he is innocent. In the same way, just because we failed to reject a null hypothesis it does not mean that the statement is true.
For example, we may want to investigate the claim that despite what convention has told us, the mean adult body temperature is not the accepted value of 98.6 degrees Fahrenheit . The null hypothesis for an experiment to investigate this is “The mean adult body temperature for healthy individuals is 98.6 degrees Fahrenheit.” If we fail to reject the null hypothesis, then our working hypothesis remains that the average adult who is healthy has a temperature of 98.6 degrees. We do not prove that this is true.
If we are studying a new treatment, the null hypothesis is that our treatment will not change our subjects in any meaningful way. In other words, the treatment will not produce any effect in our subjects.
The alternative or experimental hypothesis reflects that there will be an observed effect for our experiment. In a mathematical formulation of the alternative hypothesis, there will typically be an inequality, or not equal to symbol. This hypothesis is denoted by either H a or by H 1 .
The alternative hypothesis is what we are attempting to demonstrate in an indirect way by the use of our hypothesis test. If the null hypothesis is rejected, then we accept the alternative hypothesis. If the null hypothesis is not rejected, then we do not accept the alternative hypothesis. Going back to the above example of mean human body temperature, the alternative hypothesis is “The average adult human body temperature is not 98.6 degrees Fahrenheit.”
If we are studying a new treatment, then the alternative hypothesis is that our treatment does, in fact, change our subjects in a meaningful and measurable way.
The following set of negations may help when you are forming your null and alternative hypotheses. Most technical papers rely on just the first formulation, even though you may see some of the others in a statistics textbook.
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 of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.
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 . This is usually what the researcher is trying to prove.
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 :
equal (=) | not equal (≠) greater than (>) less than (<) |
greater than or equal to (≥) | less than (<) |
less than or equal to (≤) | more than (>) |
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 H a : 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.
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 H a : 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.
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 H a : 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.
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 H a : 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.
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.
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VIDEO
COMMENTS
The null hypothesis (H0) answers "No, there's no effect in the population.". The alternative hypothesis (Ha) answers "Yes, there is an effect in the population.". The null and alternative are always claims about the population. That's because the goal of hypothesis testing is to make inferences about a population based on a sample.
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: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.
The alternative hypothesis and null hypothesis are types of conjectures used in statistical tests, which are formal methods of reaching conclusions or making judgments on the basis of data. In statistical hypothesis testing, the null hypothesis and alternative hypothesis are two mutually exclusive statements. "The statement being tested in a test of statistical significance is called the null ...
H0: 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. Ha: The alternative hypothesis: It is a claim about the population that is contradictory to H0 and what we conclude when we reject H0. Since the ...
Learn how to write null and alternative hypotheses for different statistical tests. The null hypothesis is the claim that there's no effect in the population, while the alternative hypothesis is the claim that there's an effect.
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 of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.
Learn what the alternate hypothesis (Ha) is in hypothesis testing, how to formulate it, and why it is important. See guidelines, examples, and benefits of using Ha in statistical inference.
Null hypothesis: µ ≥ 70 inches. Alternative hypothesis: µ < 70 inches. A two-tailed hypothesis involves making an "equal to" or "not equal to" statement. For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches. The null and alternative hypotheses in this case would be: Null hypothesis: µ = 70 inches.
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 ...
G. A statistical test in which the alternative hypothesis specifies that the population parameter lies entirely above or below the value specified in H0 is a one-sided (or one-tailed) test, e.g. H0: µ = 100 HA: µ > 100 H. An alternative hypothesis that specified that the parameter can lie on either side of
The alternative hypothesis, denoted as Ha, is a statement that contradicts or challenges the null hypothesis. It suggests that there is a significant relationship or difference between variables being studied. All Subjects. Light. AP Statistics. Unit 1 - Exploring One-Variable Data. Unit 2 - Exploring Two-Variable Data ...
Always write the alternative hypothesis, typically denoted with Ha or H1, using less than, greater than, or not equals symbols, i.e., (≠, >, or <). If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. Never state that a claim is proven true or false.
In statistics, the hypothesis to be tested is called the null hypothesis and given the symbol H 0. The alternative hypothesis is given the symbol H a. The null hypothesis defines a specific value of the population parameter that is of interest. Therefore, the null hypothesis always includes the possibility of equality. Consider.
The alternative hypothesis (Ha H a) is a claim about the population that is contradictory to H0 H 0 and what we conclude when we reject H0 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.
Intro to hypothesis testing. Write the null hypothesis H0, and the alternative hypothesis H1 (Ha). #vudomath0:00 Meaning of null and alternative hypotheses0:...
The alternative hypothesis is denoted HA, or Ha, or H1. It will also contain a strict inequality. To write it, begin with the name, and a colon. Then, on the left, input the symbol representing ...
Let's return finally to the question of whether we reject or fail to reject the null hypothesis. If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above ...
The null hypothesis would be H0: µ = 70 inches and the alternative hypothesis would be Ha: µ ≠ 70 inches. Note: The "equal" sign is always included in the null hypothesis, whether it is =, ≥, or ≤. Related: What is a Directional Hypothesis? Types of Hypothesis Tests. There are many different types of hypothesis tests you can perform ...
This statistics video tutorial provides a basic introduction into hypothesis testing. It provides examples and practice problems that explains how to state ...
Most technical papers rely on just the first formulation, even though you may see some of the others in a statistics textbook. Null hypothesis: " x is equal to y.". Alternative hypothesis " x is not equal to y.". Null hypothesis: " x is at least y.". Alternative hypothesis " x is less than y.". Null hypothesis: " x is at most ...
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 of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.
Sellke et al (2001) provided a calibration of p-values and, assuming that the prior probabilities of the null hypothesis and the alternative hypothesis are equal (that is, that each have a prior probability of 0.5), by using a formula provided by them (equation 3), we can correct our NHST p-value into a probability that can be interpreted as ...
An alternative hypothesis (Ha) states that there is a statistically significant relationship between the two variables. The null hypothesis says there is no statistical relationship between the two variables. There are many types of hypotheses in statistics. A statistical hypothesis is a working statement that is logically consistent with data.