hypothesis testing a level biology

Chi-Squared Test

Overview of chi-squared.

The chi-squared test is used in genetics to compare the goodness of fit of observed data with expected data. It tests if the difference between observed and expected values is due to chance.

Inheritance

• Genetic diagrams are used to predict the expected phenotypic ratio of offspring.
• Predictions are rarely 100% accurate because of the random nature of gametes fusing during fertilisation.
• Chi-squared is used to compare observed phenotypic ratios with expected ratios.
• Chi-squared tells us if the difference between the observed and expected ratios are due to chance.

Requirements

• Variation is discrete not continuous. This means the data are in categories (e.g. Aa and aa).
• Data show absolute numbers (whole numbers), normally frequencies.

Null hypothesis

• Before using chi-squared, a null hypothesis is stated.
• 'There is no significant difference between observed and expected data, the difference is due to chance'.
• The chi-squared test is used to reject or accept the null hypothesis.

• O = observed values.
• E = expected values.

The steps involved in applying the chi-squared test are:

• The equation for chi-squared is:

1) Calculate expected values

• To use the chi-squared equation, the expected values need to be calculated.
• Expected values are predicted using genetic diagrams.
• The expected values are the phenotypic ratios given by the genetic diagram.
• Compare the expected values with observed values and use these numbers in the equation.

2) Calculate chi-squared

• Using the chi-squared equation, calculate the chi-squared value.

3) Find the critical value

• Degrees of freedom = the number of categories (e.g. phenotypes) − 1.
• Find the critical value that corresponds to the degrees of freedom in a probability distribution table at 0.05 significance level.

4) Accept the null hypothesis?

• Compare the chi-squared value to the critical value.
• The difference between observed and expected data is due to chance.

Reject the null hypothesis?

• The difference between observed and expected data is NOT due to chance.
• This means we would get this chi-squared value in less than 5% of cases, which is very unlikely.

1 Biological Molecules

1.1 Monomers & Polymers

1.1.1 Monomers & Polymers

1.1.2 Condensation & Hydrolysis Reactions

1.2 Carbohydrates

1.2.1 Structure of Carbohydrates

1.2.2 Types of Polysaccharides

1.2.3 End of Topic Test - Monomers, Polymers and Carbs

1.2.4 Exam-Style Question - Carbohydrates

1.2.5 A-A* (AO3/4) - Carbohydrates

1.3.1 Triglycerides & Phospholipids

1.3.2 Types of Fatty Acids

1.3.3 Testing for Lipids

1.3.4 Exam-Style Question - Fats

1.3.5 A-A* (AO3/4) - Lipids

1.4 Proteins

1.4.1 The Peptide Chain

1.4.2 Investigating Proteins

1.4.3 Primary & Secondary Protein Structure

1.4.4 Tertiary & Quaternary Protein Structure

1.4.5 Enzymes

1.4.6 Factors Affecting Enzyme Activity

1.4.7 Enzyme-Controlled Reactions

1.4.8 End of Topic Test - Lipids & Proteins

1.4.9 A-A* (AO3/4) - Enzymes

1.4.10 A-A* (AO3/4) - Proteins

1.5 Nucleic Acids

1.5.1 DNA & RNA

1.5.2 Nucleotides

1.5.3 Polynucleotides

1.5.4 DNA Replication

1.5.5 Exam-Style Question - Nucleic Acids

1.5.6 A-A* (AO3/4) - Nucleic Acids

1.6.1 Structure of ATP

1.6.2 Hydrolysis of ATP

1.6.3 Resynthesis of ATP

1.6.4 End of Topic Test - Nucleic Acids & ATP

1.7.1 Importance of Water

1.7.2 Structure of Water

1.7.3 Properties of Water

1.7.4 A-A* (AO3/4) - Water

1.8 Inorganic Ions

1.8.1 Inorganic Ions

1.8.2 End of Topic Test - Water & Inorganic Ions

2.1 Cell Structure

2.1.1 Introduction to Cells

2.1.2 Eukaryotic Cells & Organelles

2.1.3 Eukaryotic Cells & Organelles 2

2.1.4 Prokaryotes

2.1.5 A-A* (AO3/4) - Organelles

2.1.6 Methods of Studying Cells

2.1.7 Microscopes

2.1.8 End of Topic Test - Cell Structure

2.1.9 Exam-Style Question - Cells

2.1.10 A-A* (AO3/4) - Cells

2.2 Mitosis & Cancer

2.2.1 Mitosis

2.2.2 Stages of Mitosis

2.2.3 Investigating Mitosis

2.2.4 Cancer

2.2.5 A-A* (AO3/4) - The Cell Cycle

2.3 Transport Across Cell Membrane

2.3.1 Cell Membrane Structure

2.3.2 A-A* (AO3/4) - Membrane Structure

2.3.3 Diffusion

2.3.4 Osmosis

2.3.5 Active Transport

2.3.6 End of Topic Test - Mitosis, Cancer & Transport

2.3.7 Exam-Style Question - Membranes

2.3.8 A-A* (AO3/4) - Membranes & Transport

2.3.9 A-A*- Mitosis, Cancer & Transport

2.4 Cell Recognition & the Immune System

2.4.1 Immune System

2.4.2 Phagocytosis

2.4.3 T Lymphocytes

2.4.4 B Lymphocytes

2.4.5 Antibodies

2.4.6 Primary & Secondary Response

2.4.7 Vaccines

2.4.9 Ethical Issues

2.4.10 End of Topic Test - Immune System

2.4.11 Exam-Style Question - Immune System

2.4.12 A-A* (AO3/4) - Immune System

3 Substance Exchange

3.1 Surface Area to Volume Ratio

3.1.1 Size & Surface Area

3.1.2 A-A* (AO3/4) - Cell Size

3.2 Gas Exchange

3.2.1 Single-Celled Organisms

3.2.2 Multicellular Organisms

3.2.3 Control of Water Loss

3.2.4 Human Gas Exchange

3.2.5 Ventilation

3.2.6 Dissection

3.2.7 Measuring Gas Exchange

3.2.8 Lung Disease

3.2.9 Lung Disease Data

3.2.10 End of Topic Test - Gas Exchange

3.2.11 A-A* (AO3/4) - Gas Exchange

3.3 Digestion & Absorption

3.3.1 Overview of Digestion

3.3.2 Digestion in Mammals

3.3.3 Absorption

3.3.4 End of Topic Test - Substance Exchange & Digestion

3.3.5 A-A* (AO3/4) - Substance Ex & Digestion

3.4 Mass Transport

3.4.1 Haemoglobin

3.4.2 Oxygen Transport

3.4.3 The Circulatory System

3.4.4 The Heart

3.4.5 Blood Vessels

3.4.6 Cardiovascular Disease

3.4.7 Heart Dissection

3.4.8 Xylem

3.4.9 Phloem

3.4.10 Investigating Plant Transport

3.4.11 End of Topic Test - Mass Transport

3.4.12 A-A* (AO3/4) - Mass Transport

4 Genetic Information & Variation

4.1 DNA, Genes & Chromosomes

4.1.2 Genes

4.1.3 Non-Coding Genes

4.1.4 The Genetic Code

4.1.5 A-A* (AO3/4) - DNA

4.2 DNA & Protein Synthesis

4.2.1 Protein Synthesis

4.2.2 Transcription & Translation

4.2.3 End of Topic Test - DNA, Genes & Protein Synthesis

4.2.4 Exam-Style Question - Protein Synthesis

4.2.5 A-A* (AO3/4) - Coronavirus Translation

4.2.6 A-A* (AO3/4) - Transcription

4.2.7 A-A* (AO3/4) - Translation

4.3 Mutations & Meiosis

4.3.1 Mutations

4.3.2 Meiosis

4.3.3 A-A* (AO3/4) - Meiosis

4.3.4 Meiosis vs Mitosis

4.3.5 End of Topic Test - Mutations, Meiosis

4.3.6 A-A* (AO3/4) - DNA,Genes, CellDiv & ProtSynth

4.4 Genetic Diversity & Adaptation

4.4.1 Genetic Diversity

4.4.2 Natural Selection

4.4.3 A-A* (AO3/4) - Natural Selection

4.4.4 Adaptations

4.4.5 Investigating Natural Selection

4.4.6 End of Topic Test - Genetic Diversity & Adaptation

4.4.7 A-A* (AO3/4) - Genetic Diversity & Adaptation

4.5 Species & Taxonomy

4.5.1 Courtship Behaviour

4.5.2 Phylogeny

4.5.3 Classification

4.5.4 DNA Technology

4.5.5 A-A* (AO3/4) - Species & Taxonomy

4.6 Biodiversity Within a Community

4.6.1 Biodiversity

4.6.2 Index of diversity

4.6.3 Agriculture

4.6.4 End of Topic Test - Species,Taxonomy& Biodiversity

4.6.5 A-A* (AO3/4) - Species,Taxon&Biodiversity

4.7 Investigating Diversity

4.7.1 Genetic Diversity

4.7.2 Quantitative Investigation

5 Energy Transfers (A2 only)

5.1 Photosynthesis

5.1.1 Overview of Photosynthesis

5.1.2 Photoionisation of Chlorophyll

5.1.3 Production of ATP & Reduced NADP

5.1.4 Cyclic Photophosphorylation

5.1.5 Light-Independent Reaction

5.1.6 A-A* (AO3/4) - Photosynthesis Reactions

5.1.7 Limiting Factors

5.1.8 Photosynthesis Experiments

5.1.9 End of Topic Test - Photosynthesis

5.1.10 A-A* (AO3/4) - Photosynthesis

5.2 Respiration

5.2.1 Overview of Respiration

5.2.2 Anaerobic Respiration

5.2.3 A-A* (AO3/4) - Anaerobic Respiration

5.2.4 The Link Reaction

5.2.5 The Krebs Cycle

5.2.6 Oxidative Phosphorylation

5.2.7 Respiration Experiments

5.2.8 End of Topic Test - Respiration

5.2.9 A-A* (AO3/4) - Respiration

5.3 Energy & Ecosystems

5.3.1 Biomass

5.3.2 Production & Productivity

5.3.3 Agricultural Practices

5.4 Nutrient Cycles

5.4.1 Nitrogen Cycle

5.4.2 Phosphorous Cycle

5.4.3 Fertilisers & Eutrophication

5.4.4 End of Topic Test - Nutrient Cycles

5.4.5 A-A* (AO3/4) - Energy,Ecosystems&NutrientCycles

6 Responding to Change (A2 only)

6.1 Nervous Communication

6.1.1 Survival

6.1.2 Plant Responses

6.1.3 Animal Responses

6.1.4 Reflexes

6.1.5 End of Topic Test - Reflexes, Responses & Survival

6.1.6 Receptors

6.1.7 The Human Retina

6.1.8 Control of Heart Rate

6.1.9 End of Topic Test - Receptors, Retina & Heart Rate

6.2 Nervous Coordination

6.2.1 Neurones

6.2.2 Action Potentials

6.2.3 Speed of Transmission

6.2.4 End of Topic Test - Neurones & Action Potentials

6.2.5 Synapses

6.2.6 Types of Synapse

6.2.7 Medical Application

6.2.8 End of Topic Test - Synapses

6.2.9 A-A* (AO3/4) - Nervous Comm&Coord

6.3 Muscle Contraction

6.3.1 Skeletal Muscle

6.3.2 Sliding Filament Theory

6.3.3 Contraction

6.3.4 Slow & Fast Twitch Fibres

6.3.5 End of Topic Test - Muscles

6.3.6 A-A* (AO3/4) - Muscle Contraction

6.4 Homeostasis

6.4.1 Overview of Homeostasis

6.4.2 Blood Glucose Concentration

6.4.3 Controlling Blood Glucose Concentration

6.4.4 End of Topic Test - Blood Glucose

6.4.5 Primary & Secondary Messengers

6.4.6 Diabetes Mellitus

6.4.7 Measuring Glucose Concentration

6.4.8 Osmoregulation

6.4.9 Controlling Blood Water Potential

6.4.11 End of Topic Test - Diabetes & Osmoregulation

6.4.12 A-A* (AO3/4) - Homeostasis

7 Genetics & Ecosystems (A2 only)

7.1 Genetics

7.1.1 Key Terms in Genetics

7.1.2 Inheritance

7.1.3 Linkage

7.1.4 Multiple Alleles & Epistasis

7.1.5 Chi-Squared Test

7.1.6 End of Topic Test - Genetics

7.1.7 A-A* (AO3/4) - Genetics

7.2 Populations

7.2.1 Populations

7.2.2 Hardy-Weinberg Principle

7.3 Evolution

7.3.1 Variation

7.3.2 Natural Selection & Evolution

7.3.3 End of Topic Test - Populations & Evolution

7.3.4 Types of Selection

7.3.5 Types of Selection Summary

7.3.6 Overview of Speciation

7.3.7 Causes of Speciation

7.3.8 Diversity

7.3.9 End of Topic Test - Selection & Speciation

7.3.10 A-A* (AO3/4) - Populations & Evolution

7.4 Populations in Ecosystems

7.4.1 Overview of Ecosystems

7.4.2 Niche

7.4.3 Population Size

7.4.4 Investigating Population Size

7.4.5 End of Topic Test - Ecosystems & Population Size

7.4.6 Succession

7.4.7 Conservation

7.4.8 End of Topic Test - Succession & Conservation

7.4.9 A-A* (AO3/4) - Ecosystems

8 The Control of Gene Expression (A2 only)

8.1 Mutation

8.1.1 Mutations

8.1.2 Effects of Mutations

8.1.3 Causes of Mutations

8.2 Gene Expression

8.2.1 Stem Cells

8.2.2 Stem Cells in Disease

8.2.3 End of Topic Test - Mutation & Gene Epression

8.2.4 A-A* (AO3/4) - Mutation & Stem Cells

8.2.5 Regulating Transcription

8.2.6 Epigenetics

8.2.7 Epigenetics & Disease

8.2.8 Regulating Translation

8.2.9 Experimental Data

8.2.10 End of Topic Test - Transcription & Translation

8.2.11 Tumours

8.2.12 Correlations & Causes

8.2.13 Prevention & Treatment

8.2.14 End of Topic Test - Cancer

8.2.15 A-A* (AO3/4) - Gene Expression & Cancer

8.3 Genome Projects

8.3.1 Using Genome Projects

8.4 Gene Technology

8.4.1 Recombinant DNA

8.4.2 Producing Fragments

8.4.3 Amplification

8.4.4 End of Topic Test - Genome Project & Amplification

8.4.5 Using Recombinant DNA

8.4.6 Medical Diagnosis

8.4.7 Genetic Fingerprinting

8.4.8 End of Topic Test - Gene Technologies

8.4.9 A-A* (AO3/4) - Gene Technology

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Multiple Alleles & Epistasis

End of Topic Test - Genetics

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Statistics for A level biology - summary and practice questions

Subject: Biology

Age range: 16+

Resource type: Worksheet/Activity

Last updated

16 May 2018

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This resource summarises the four statistical tests required for A level biology (Standard Deviation, T-test, Spearman Rank, Chi-squared). There is a practice question and mark scheme for each test. The questions are Edexcel SNAB but hopefully they’re relevant to other boards.

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• Knowledge Base

Methodology

• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1. Ask a question

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

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 .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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McCombes, S. (2023, November 20). How to Write a Strong Hypothesis | Steps & Examples. Scribbr. Retrieved September 26, 2024, from https://www.scribbr.com/methodology/hypothesis/

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Hypothesis Examples

A hypothesis is a prediction of the outcome of a test. It forms the basis for designing an experiment in the scientific method . A good hypothesis is testable, meaning it makes a prediction you can check with observation or experimentation. Here are different hypothesis examples.

Null Hypothesis Examples

The null hypothesis (H 0 ) is also known as the zero-difference or no-difference hypothesis. It predicts that changing one variable ( independent variable ) will have no effect on the variable being measured ( dependent variable ). Here are null hypothesis examples:

• Plant growth is unaffected by temperature.
• If you increase temperature, then solubility of salt will increase.
• Incidence of skin cancer is unrelated to ultraviolet light exposure.
• All brands of light bulb last equally long.
• Cats have no preference for the color of cat food.
• All daisies have the same number of petals.

Sometimes the null hypothesis shows there is a suspected correlation between two variables. For example, if you think plant growth is affected by temperature, you state the null hypothesis: “Plant growth is not affected by temperature.” Why do you do this, rather than say “If you change temperature, plant growth will be affected”? The answer is because it’s easier applying a statistical test that shows, with a high level of confidence, a null hypothesis is correct or incorrect.

Research Hypothesis Examples

A research hypothesis (H 1 ) is a type of hypothesis used to design an experiment. This type of hypothesis is often written as an if-then statement because it’s easy identifying the independent and dependent variables and seeing how one affects the other. If-then statements explore cause and effect. In other cases, the hypothesis shows a correlation between two variables. Here are some research hypothesis examples:

• If you leave the lights on, then it takes longer for people to fall asleep.
• If you refrigerate apples, they last longer before going bad.
• If you keep the curtains closed, then you need less electricity to heat or cool the house (the electric bill is lower).
• If you leave a bucket of water uncovered, then it evaporates more quickly.
• Goldfish lose their color if they are not exposed to light.
• Workers who take vacations are more productive than those who never take time off.

Is It Okay to Disprove a Hypothesis?

Yes! You may even choose to write your hypothesis in such a way that it can be disproved because it’s easier to prove a statement is wrong than to prove it is right. In other cases, if your prediction is incorrect, that doesn’t mean the science is bad. Revising a hypothesis is common. It demonstrates you learned something you did not know before you conducted the experiment.

Test yourself with a Scientific Method Quiz .

• Mellenbergh, G.J. (2008). Chapter 8: Research designs: Testing of research hypotheses. In H.J. Adèr & G.J. Mellenbergh (eds.), Advising on Research Methods: A Consultant’s Companion . Huizen, The Netherlands: Johannes van Kessel Publishing.
• Popper, Karl R. (1959). The Logic of Scientific Discovery . Hutchinson & Co. ISBN 3-1614-8410-X.
• Schick, Theodore; Vaughn, Lewis (2002). How to think about weird things: critical thinking for a New Age . Boston: McGraw-Hill Higher Education. ISBN 0-7674-2048-9.
• Tobi, Hilde; Kampen, Jarl K. (2018). “Research design: the methodology for interdisciplinary research framework”. Quality & Quantity . 52 (3): 1209–1225. doi: 10.1007/s11135-017-0513-8

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• Knowledge Base
• Methodology
• How to Write a Strong Hypothesis | Guide & Examples

How to Write a Strong Hypothesis | Guide & Examples

Published on 6 May 2022 by Shona McCombes .

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more variables . An independent variable is something the researcher changes or controls. A dependent variable is something the researcher observes and measures.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

Prevent plagiarism, run a free check.

Step 1: ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2: Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalise more complex constructs.

Step 3: Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

Step 4: Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

Step 5: Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in if … then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

Step 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 .

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis is not just a guess. It should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

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.

Cite this Scribbr article

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.

McCombes, S. (2022, May 06). How to Write a Strong Hypothesis | Guide & Examples. Scribbr. Retrieved 23 September 2024, from https://www.scribbr.co.uk/research-methods/hypothesis-writing/

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Shona McCombes

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Chapter 3: Hypothesis Testing

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?

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.

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

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 H 0 : 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 H 1 : 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 (H 0 ). This means that you have enough statistical evidence to support the alternative claim (H 1 ).

Option 2) Fail to reject the null hypothesis (H 0 ). 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:

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.

H o : 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.

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.

• H o : 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.

H o : 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.

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.

• H o : μ = 15 lakes
• H 1 : 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.

H o : 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.

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.

• H o : 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.

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.

The probability that at a fixed level α significance test will reject H 0 , 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 .

Hypothesis Test about the Population Mean ( μ ) when the Population Standard Deviation ( σ ) is Known

We are going to examine two equivalent ways to perform a hypothesis test: the classical approach and the p-value approach. The classical approach is based on standard deviations. This method compares the test statistic (Z-score) to a critical value (Z-score) from the standard normal table. If the test statistic falls in the rejection zone, you reject the null hypothesis. The p-value approach is based on area under the normal curve. This method compares the area associated with the test statistic to alpha ( α ), the level of significance (which is also area under the normal curve). If the p-value is less than alpha, you would reject the null hypothesis.

As a past student poetically said: If the p-value is a wee value, Reject Ho

Both methods must have:

• Data from a random sample.
• Verification of the assumption of normality.
• A null and alternative hypothesis.
• A criterion that determines if we reject or fail to reject the null hypothesis.
• A conclusion that answers the question.

There are four steps required for a hypothesis test:

• State the null and alternative hypotheses.
• State the level of significance and the critical value.
• Compute the test statistic.
• State a conclusion.

The Classical Method for Testing a Claim about the Population Mean ( μ ) when the Population Standard Deviation ( σ ) is Known

A forester studying diameter growth of red pine believes that the mean diameter growth will be different from the known mean growth of 1.35 inches/year if a fertilization treatment is applied to the stand. He conducts his experiment, collects data from a sample of 32 plots, and gets a sample mean diameter growth of 1.6 in./year. The population standard deviation for this stand is known to be 0.46 in./year. Does he have enough evidence to support his claim?

Step 1) State the null and alternative hypotheses.

• H o : μ = 1.35 in./year
• H 1 : μ ≠ 1.35 in./year

Step 2) State the level of significance and the critical value.

• We will choose a level of significance of 5% ( α = 0.05).
• For a two-sided question, we need a two-sided critical value – Z α /2 and + Z α /2 .
• The level of significance is divided by 2 (since we are only testing “not equal”). We must have two rejection zones that can deal with either a greater than or less than outcome (to the right (+) or to the left (-)).
• We need to find the Z-score associated with the area of 0.025. The red areas are equal to α /2 = 0.05/2 = 0.025 or 2.5% of the area under the normal curve.
• Go into the body of values and find the negative Z-score associated with the area 0.025.

• The negative critical value is -1.96. Since the curve is symmetric, we know that the positive critical value is 1.96.
• ±1.96 are the critical values. These values set up the rejection zone. If the test statistic falls within these red rejection zones, we reject the null hypothesis.

Step 3) Compute the test statistic.

• The test statistic is the number of standard deviations the sample mean is from the known mean. It is also a Z-score, just like the critical value.

• For this problem, the test statistic is

Step 4) State a conclusion.

• Compare the test statistic to the critical value. If the test statistic falls into the rejection zones, reject the null hypothesis. In other words, if the test statistic is greater than +1.96 or less than -1.96, reject the null hypothesis.

In this problem, the test statistic falls in the red rejection zone. The test statistic of 3.07 is greater than the critical value of 1.96.We will reject the null hypothesis. We have enough evidence to support the claim that the mean diameter growth is different from (not equal to) 1.35 in./year.

A researcher believes that there has been an increase in the average farm size in his state since the last study five years ago. The previous study reported a mean size of 450 acres with a population standard deviation ( σ ) of 167 acres. He samples 45 farms and gets a sample mean of 485.8 acres. Is there enough information to support his claim?

• H o : μ = 450 acres
• H 1 : μ >450 acres
• For a one-sided question, we need a one-sided positive critical value Z α .
• The level of significance is all in the right side (the rejection zone is just on the right side).
• We need to find the Z-score associated with the 5% area in the right tail.

• Go into the body of values in the standard normal table and find the Z-score that separates the lower 95% from the upper 5%.
• The critical value is 1.645. This value sets up the rejection zone.

• Compare the test statistic to the critical value.

• The test statistic does not fall in the rejection zone. It is less than the critical value.

We fail to reject the null hypothesis. We do not have enough evidence to support the claim that the mean farm size has increased from 450 acres.

A researcher believes that there has been a reduction in the mean number of hours that college students spend preparing for final exams. A national study stated that students at a 4-year college spend an average of 23 hours preparing for 5 final exams each semester with a population standard deviation of 7.3 hours. The researcher sampled 227 students and found a sample mean study time of 19.6 hours. Does this indicate that the average study time for final exams has decreased? Use a 1% level of significance to test this claim.

• H o : μ = 23 hours
• H 1 : μ < 23 hours
• This is a left-sided test so alpha (0.01) is all in the left tail.

• Go into the body of values in the standard normal table and find the Z-score that defines the lower 1% of the area.
• The critical value is -2.33. This value sets up the rejection zone.

• The test statistic falls in the rejection zone. The test statistic of -7.02 is less than the critical value of -2.33.

We reject the null hypothesis. We have sufficient evidence to support the claim that the mean final exam study time has decreased below 23 hours.

Testing a Hypothesis using P-values

The p-value is the probability of observing our sample mean given that the null hypothesis is true. It is the area under the curve to the left or right of the test statistic. If the probability of observing such a sample mean is very small (less than the level of significance), we would reject the null hypothesis. Computations for the p-value depend on whether it is a one- or two-sided test.

Steps for a hypothesis test using p-values:

• State the level of significance.
• Compute the test statistic and find the area associated with it (this is the p-value).
• Compare the p-value to alpha ( α ) and state a conclusion.

Instead of comparing Z-score test statistic to Z-score critical value, as in the classical method, we compare area of the test statistic to area of the level of significance.

The Decision Rule: If the p-value is less than alpha, we reject the null hypothesis

Computing P-values

If it is a two-sided test (the alternative claim is ≠), the p-value is equal to two times the probability of the absolute value of the test statistic. If the test is a left-sided test (the alternative claim is “<”), then the p-value is equal to the area to the left of the test statistic. If the test is a right-sided test (the alternative claim is “>”), then the p-value is equal to the area to the right of the test statistic.

Let’s look at Example 6 again.

A forester studying diameter growth of red pine believes that the mean diameter growth will be different from the known mean growth of 1.35 in./year if a fertilization treatment is applied to the stand. He conducts his experiment, collects data from a sample of 32 plots, and gets a sample mean diameter growth of 1.6 in./year. The population standard deviation for this stand is known to be 0.46 in./year. Does he have enough evidence to support his claim?

Step 2) State the level of significance.

• For this problem, the test statistic is:

The p-value is two times the area of the absolute value of the test statistic (because the alternative claim is “not equal”).

• Look up the area for the Z-score 3.07 in the standard normal table. The area (probability) is equal to 1 – 0.9989 = 0.0011.
• Multiply this by 2 to get the p-value = 2 * 0.0011 = 0.0022.

Step 4) Compare the p-value to alpha and state a conclusion.

• Use the Decision Rule (if the p-value is less than α , reject H 0 ).
• In this problem, the p-value (0.0022) is less than alpha (0.05).
• We reject the H 0 . We have enough evidence to support the claim that the mean diameter growth is different from 1.35 inches/year.

Let’s look at Example 7 again.

The p-value is the area to the right of the Z-score 1.44 (the hatched area).

• This is equal to 1 – 0.9251 = 0.0749.
• The p-value is 0.0749.

• Use the Decision Rule.
• In this problem, the p-value (0.0749) is greater than alpha (0.05), so we Fail to Reject the H 0 .
• The area of the test statistic is greater than the area of alpha ( α ).

We fail to reject the null hypothesis. We do not have enough evidence to support the claim that the mean farm size has increased.

Let’s look at Example 8 again.

• H 0 : μ = 23 hours

The p-value is the area to the left of the test statistic (the little black area to the left of -7.02). The Z-score of -7.02 is not on the standard normal table. The smallest probability on the table is 0.0002. We know that the area for the Z-score -7.02 is smaller than this area (probability). Therefore, the p-value is <0.0002.

• In this problem, the p-value (p<0.0002) is less than alpha (0.01), so we Reject the H 0 .
• The area of the test statistic is much less than the area of alpha ( α ).

We reject the null hypothesis. We have enough evidence to support the claim that the mean final exam study time has decreased below 23 hours.

Both the classical method and p-value method for testing a hypothesis will arrive at the same conclusion. In the classical method, the critical Z-score is the number on the z-axis that defines the level of significance ( α ). The test statistic converts the sample mean to units of standard deviation (a Z-score). If the test statistic falls in the rejection zone defined by the critical value, we will reject the null hypothesis. In this approach, two Z-scores, which are numbers on the z-axis, are compared. In the p-value approach, the p-value is the area associated with the test statistic. In this method, we compare α (which is also area under the curve) to the p-value. If the p-value is less than α , we reject the null hypothesis. The p-value is the probability of observing such a sample mean when the null hypothesis is true. If the probability is too small (less than the level of significance), then we believe we have enough statistical evidence to reject the null hypothesis and support the alternative claim.

Software Solutions

(referring to Ex. 8)

One-Sample Z

Excel does not offer 1-sample hypothesis testing.

Hypothesis Test about the Population Mean ( μ ) when the Population Standard Deviation ( σ ) is Unknown

Frequently, the population standard deviation (σ) is not known. We can estimate the population standard deviation (σ) with the sample standard deviation (s). However, the test statistic will no longer follow the standard normal distribution. We must rely on the student’s t-distribution with n-1 degrees of freedom. Because we use the sample standard deviation (s), the test statistic will change from a Z-score to a t-score.

Steps for a hypothesis test are the same that we covered in Section 2.

Just as with the hypothesis test from the previous section, the data for this test must be from a random sample and requires either that the population from which the sample was drawn be normal or that the sample size is sufficiently large (n≥30). A t-test is robust, so small departures from normality will not adversely affect the results of the test. That being said, if the sample size is smaller than 30, it is always good to verify the assumption of normality through a normal probability plot.

We will still have the same three pairs of null and alternative hypotheses and we can still use either the classical approach or the p-value approach.

Selecting the correct critical value from the student’s t-distribution table depends on three factors: the type of test (one-sided or two-sided alternative hypothesis), the sample size, and the level of significance.

For a two-sided test (“not equal” alternative hypothesis), the critical value (t α /2 ), is determined by alpha ( α ), the level of significance, divided by two, to deal with the possibility that the result could be less than OR greater than the known value.

• If your level of significance was 0.05, you would use the 0.025 column to find the correct critical value (0.05/2 = 0.025).
• If your level of significance was 0.01, you would use the 0.005 column to find the correct critical value (0.01/2 = 0.005).

For a one-sided test (“a less than” or “greater than” alternative hypothesis), the critical value (t α ) , is determined by alpha ( α ), the level of significance, being all in the one side.

• If your level of significance was 0.05, you would use the 0.05 column to find the correct critical value for either a left or right-side question. If you are asking a “less than” (left-sided question, your critical value will be negative. If you are asking a “greater than” (right-sided question), your critical value will be positive.

Find the critical value you would use to test the claim that μ ≠ 112 with a sample size of 18 and a 5% level of significance.

In this case, the critical value (t α /2 ) would be 2.110. This is a two-sided question (≠) so you would divide alpha by 2 (0.05/2 = 0.025) and go down the 0.025 column to 17 degrees of freedom.

What would the critical value be if you wanted to test that μ < 112 for the same data?

In this case, the critical value would be 1.740. This is a one-sided question (<) so alpha would be divided by 1 (0.05/1 = 0.05). You would go down the 0.05 column with 17 degrees of freedom to get the correct critical value.

In 2005, the mean pH level of rain in a county in northern New York was 5.41. A biologist believes that the rain acidity has changed. He takes a random sample of 11 rain dates in 2010 and obtains the following data. Use a 1% level of significance to test his claim.

4.70, 5.63, 5.02, 5.78, 4.99, 5.91, 5.76, 5.54, 5.25, 5.18, 5.01

The sample size is small and we don’t know anything about the distribution of the population, so we examine a normal probability plot. The distribution looks normal so we will continue with our test.

The sample mean is 5.343 with a sample standard deviation of 0.397.

• H o : μ = 5.41
• H 1 : μ ≠ 5.41
• This is a two-sided question so alpha is divided by two.

• t α /2 is found by going down the 0.005 column with 14 degrees of freedom.
• t α /2 = ±3.169.
• The test statistic is a t-score.

• The test statistic does not fall in the rejection zone.

We will fail to reject the null hypothesis. We do not have enough evidence to support the claim that the mean rain pH has changed.

A One-sided Test

Cadmium, a heavy metal, is toxic to animals. Mushrooms, however, are able to absorb and accumulate cadmium at high concentrations. The government has set safety limits for cadmium in dry vegetables at 0.5 ppm. Biologists believe that the mean level of cadmium in mushrooms growing near strip mines is greater than the recommended limit of 0.5 ppm, negatively impacting the animals that live in this ecosystem. A random sample of 51 mushrooms gave a sample mean of 0.59 ppm with a sample standard deviation of 0.29 ppm. Use a 5% level of significance to test the claim that the mean cadmium level is greater than the acceptable limit of 0.5 ppm.

The sample size is greater than 30 so we are assured of a normal distribution of the means.

• H o : μ = 0.5 ppm
• H 1 : μ > 0.5 ppm
• This is a right-sided question so alpha is all in the right tail.

• t α is found by going down the 0.05 column with 50 degrees of freedom.
• t α = 1.676

Step 4) State a Conclusion.

The test statistic falls in the rejection zone. We will reject the null hypothesis. We have enough evidence to support the claim that the mean cadmium level is greater than the acceptable safe limit.

BUT, what happens if the significance level changes to 1%?

The critical value is now found by going down the 0.01 column with 50 degrees of freedom. The critical value is 2.403. The test statistic is now LESS THAN the critical value. The test statistic does not fall in the rejection zone. The conclusion will change. We do NOT have enough evidence to support the claim that the mean cadmium level is greater than the acceptable safe limit of 0.5 ppm.

The level of significance is the probability that you, as the researcher, set to decide if there is enough statistical evidence to support the alternative claim. It should be set before the experiment begins.

P-value Approach

We can also use the p-value approach for a hypothesis test about the mean when the population standard deviation ( σ ) is unknown. However, when using a student’s t-table, we can only estimate the range of the p-value, not a specific value as when using the standard normal table. The student’s t-table has area (probability) across the top row in the table, with t-scores in the body of the table.

• To find the p-value (the area associated with the test statistic), you would go to the row with the number of degrees of freedom.
• Go across that row until you find the two values that your test statistic is between, then go up those columns to find the estimated range for the p-value.

Estimating P-value from a Student’s T-table

If your test statistic is 3.789 with 3 degrees of freedom, you would go across the 3 df row. The value 3.789 falls between the values 3.482 and 4.541 in that row. Therefore, the p-value is between 0.02 and 0.01. The p-value will be greater than 0.01 but less than 0.02 (0.01<p<0.02).

If your level of significance is 5%, you would reject the null hypothesis as the p-value (0.01-0.02) is less than alpha ( α ) of 0.05.

If your level of significance is 1%, you would fail to reject the null hypothesis as the p-value (0.01-0.02) is greater than alpha ( α ) of 0.01.

Software packages typically output p-values. It is easy to use the Decision Rule to answer your research question by the p-value method.

(referring to Ex. 12)

One-Sample T

Test of mu = 0.5 vs. > 0.5

Additional example: www.youtube.com/watch?v=WwdSjO4VUsg .

Hypothesis Test for a Population Proportion ( p )

Frequently, the parameter we are testing is the population proportion.

• We are studying the proportion of trees with cavities for wildlife habitat.
• We need to know if the proportion of people who support green building materials has changed.
• Has the proportion of wolves that died last year in Yellowstone increased from the year before?

Recall that the best point estimate of p , the population proportion, is given by

when np (1 – p )≥10. We can use both the classical approach and the p-value approach for testing.

The steps for a hypothesis test are the same that we covered in Section 2.

The test statistic follows the standard normal distribution. Notice that the standard error (the denominator) uses p instead of p̂ , which was used when constructing a confidence interval about the population proportion. In a hypothesis test, the null hypothesis is assumed to be true, so the known proportion is used.

• The critical value comes from the standard normal table, just as in Section 2. We will still use the same three pairs of null and alternative hypotheses as we used in the previous sections, but the parameter is now p instead of μ :

• For a two-sided test, alpha will be divided by 2 giving a ± Z α /2 critical value.
• For a left-sided test, alpha will be all in the left tail giving a – Z α critical value.
• For a right-sided test, alpha will be all in the right tail giving a Z α critical value.

A botanist has produced a new variety of hybrid soy plant that is better able to withstand drought than other varieties. The botanist knows the seed germination for the parent plants is 75%, but does not know the seed germination for the new hybrid. He tests the claim that it is different from the parent plants. To test this claim, 450 seeds from the hybrid plant are tested and 321 have germinated. Use a 5% level of significance to test this claim that the germination rate is different from 75%.

• H o : p = 0.75
• H 1 : p ≠ 0.75

This is a two-sided question so alpha is divided by 2.

• Alpha is 0.05 so the critical values are ± Z α /2 = ± Z .025 .
• Look on the negative side of the standard normal table, in the body of values for 0.025.
• The critical values are ± 1.96.

The test statistic does not fall in the rejection zone. We fail to reject the null hypothesis. We do not have enough evidence to support the claim that the germination rate of the hybrid plant is different from the parent plants.

Let’s answer this question using the p-value approach. Remember, for a two-sided alternative hypothesis (“not equal”), the p-value is two times the area of the test statistic. The test statistic is -1.81 and we want to find the area to the left of -1.81 from the standard normal table.

• On the negative page, find the Z-score -1.81. Find the area associated with this Z-score.
• The area = 0.0351.
• This is a two-sided test so multiply the area times 2 to get the p-value = 0.0351 x 2 = 0.0702.

Now compare the p-value to alpha. The Decision Rule states that if the p-value is less than alpha, reject the H 0 . In this case, the p-value (0.0702) is greater than alpha (0.05) so we will fail to reject H 0 . We do not have enough evidence to support the claim that the germination rate of the hybrid plant is different from the parent plants.

You are a biologist studying the wildlife habitat in the Monongahela National Forest. Cavities in older trees provide excellent habitat for a variety of birds and small mammals. A study five years ago stated that 32% of the trees in this forest had suitable cavities for this type of wildlife. You believe that the proportion of cavity trees has increased. You sample 196 trees and find that 79 trees have cavities. Does this evidence support your claim that there has been an increase in the proportion of cavity trees?

Use a 10% level of significance to test this claim.

• H o : p = 0.32
• H 1 : p > 0.32

This is a one-sided question so alpha is divided by 1.

• Alpha is 0.10 so the critical value is Z α = Z .10
• Look on the positive side of the standard normal table, in the body of values for 0.90.
• The critical value is 1.28.

• The test statistic is the number of standard deviations the sample proportion is from the known proportion. It is also a Z-score, just like the critical value.

The test statistic is larger than the critical value (it falls in the rejection zone). We will reject the null hypothesis. We have enough evidence to support the claim that there has been an increase in the proportion of cavity trees.

Now use the p-value approach to answer the question. This is a right-sided question (“greater than”), so the p-value is equal to the area to the right of the test statistic. Go to the positive side of the standard normal table and find the area associated with the Z-score of 2.49. The area is 0.9936. Remember that this table is cumulative from the left. To find the area to the right of 2.49, we subtract from one.

p-value = (1 – 0.9936) = 0.0064

The p-value is less than the level of significance (0.10), so we reject the null hypothesis. We have enough evidence to support the claim that the proportion of cavity trees has increased.

(referring to Ex. 15)

Test and CI for One Proportion

Test of p = 0.32 vs. p > 0.32

Hypothesis Test about a Variance

When people think of statistical inference, they usually think of inferences involving population means or proportions. However, the particular population parameter needed to answer an experimenter’s practical questions varies from one situation to another, and sometimes a population’s variability is more important than its mean. Thus, product quality is often defined in terms of low variability.

Sample variance S 2 can be used for inferences concerning a population variance σ 2 . For a random sample of n measurements drawn from a normal population with mean μ and variance σ 2 , the value S 2 provides a point estimate for σ 2 . In addition, the quantity ( n – 1) S 2 / σ 2 follows a Chi-square ( χ 2 ) distribution, with df = n – 1.

The properties of Chi-square ( χ 2 ) distribution are:

• Unlike Z and t distributions, the values in a chi-square distribution are all positive.
• The chi-square distribution is asymmetric, unlike the Z and t distributions.
• There are many chi-square distributions. We obtain a particular one by specifying the degrees of freedom (df = n – 1) associated with the sample variances S 2 .

One-sample χ 2 test for testing the hypotheses:

Alternative hypothesis:

where the χ 2 critical value in the rejection region is based on degrees of freedom df = n – 1 and a specified significance level of α .

As with previous sections, if the test statistic falls in the rejection zone set by the critical value, you will reject the null hypothesis.

A forester wants to control a dense understory of striped maple that is interfering with desirable hardwood regeneration using a mist blower to apply an herbicide treatment. She wants to make sure that treatment has a consistent application rate, in other words, low variability not exceeding 0.25 gal./acre (0.06 gal. 2 ). She collects sample data (n = 11) on this type of mist blower and gets a sample variance of 0.064 gal. 2 Using a 5% level of significance, test the claim that the variance is significantly greater than 0.06 gal. 2

H 0 : σ 2 = 0.06

H 1 : σ 2 >0.06

The critical value is 18.307. Any test statistic greater than this value will cause you to reject the null hypothesis.

The test statistic is

We fail to reject the null hypothesis. The forester does NOT have enough evidence to support the claim that the variance is greater than 0.06 gal. 2 You can also estimate the p-value using the same method as for the student t-table. Go across the row for degrees of freedom until you find the two values that your test statistic falls between. In this case going across the row 10, the two table values are 4.865 and 15.987. Now go up those two columns to the top row to estimate the p-value (0.1-0.9). The p-value is greater than 0.1 and less than 0.9. Both are greater than the level of significance (0.05) causing us to fail to reject the null hypothesis.

(referring to Ex. 16)

Test and CI for One Variance

The chi-square method is only for the normal distribution.

Excel does not offer 1-sample χ 2 testing.

Putting it all Together Using the Classical Method

To test a claim about μ when σ is known.

• Write the null and alternative hypotheses.
• State the level of significance and get the critical value from the standard normal table.

• Compare the test statistic to the critical value (Z-score) and write the conclusion.

To Test a Claim about μ When σ is Unknown

• State the level of significance and get the critical value from the student’s t-table with n-1 degrees of freedom.

• Compare the test statistic to the critical value (t-score) and write the conclusion.

To Test a Claim about p

• State the level of significance and get the critical value from the standard normal distribution.

To Test a Claim about Variance

• State the level of significance and get the critical value from the chi-square table using n-1 degrees of freedom.

• Compare the test statistic to the critical value and write the conclusion.

Natural Resources Biometrics Copyright © 2014 by Diane Kiernan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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