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5: Hypothesis Testing

Students formulate null and alternative hypotheses, interpret p-values and significance levels, and perform t-tests and ANOVA.

Matching Chi-Square Terms to Definitions

Click to show Matching Chi-Square Terms to Definitions example problem

Match each of the following chi-square (χ²) terms with their corresponding definitions.
Note: Each choice will be used exactly once.

Your Choice Prompt
Drop Your Choice Here 1. chi-square (χ²) test statistic
Drop Your Choice Here 2. critical value
Drop Your Choice Here 3. p-value
Drop Your Choice Here 4. null hypothesis, H0
Drop Your Choice Here 5. level of significance, α

Drag one of the choices below:

  • A. we attempt to find evidence against this hypothesis in our chi-square (χ²) test
  • B. the bigger this number, the smaller the p-value
  • C. if this value is small, then there is stronger evidence to support the alternative hypothesis, Ha
  • D. a constant probability that provides a cutoff for falsification of the null hypothesis, H0
  • E. the boundary of how extreme a test statistic we need to support the null hypothesis, H0
 

Matching Hypothesis Tests to Their Descriptions

Click to show Matching Hypothesis Tests to Their Descriptions example problem

Match each of the following hypothesis tests with their corresponding descriptions.
Note: Each choice will be used exactly once.

Your Choice Prompt
Drop Your Choice Here 1. Two-Sample F-test
Drop Your Choice Here 2. Two-Sample T-test
Drop Your Choice Here 3. Benchmark T-test
Drop Your Choice Here 4. Benchmark Z-test
Drop Your Choice Here 5. Analysis of Variance (ANOVA)

Drag one of the choices below:

  • A. Used when comparing a sample's mean to a large dataset with a known population standard deviation.
  • B. The average tree height in a natural forest is 12 meters, but the standard deviation is unknown. You collect n = 12 samples from a reforested area to test if the average tree height differs from the natural forest.
  • C. You are comparing the variance of protein concentrations between two experimental batches, each with n = 25 samples, to see if their variability differs.
  • D. Used to determine whether the average heights of two different plant species are significantly different.
  • E. You want to compare the mean blood glucose levels of patients in four drug groups, each group has n = 30 participants. You want to test if at least one group's mean differs significantly.
 

Matching Statistical Test Terms to Definitions

Click to show Matching Statistical Test Terms to Definitions example problem

Match each of the following statistical test terms with their corresponding definitions.
Note: Each choice will be used exactly once.

Your Choice Prompt
Drop Your Choice Here 1. alternative hypothesis, Ha
Drop Your Choice Here 2. p-value
Drop Your Choice Here 3. level of significance, α
Drop Your Choice Here 4. degrees of freedom
Drop Your Choice Here 5. test statistic

Drag one of the choices below:

  • A. a function of the data computed to test the null hypothesis, H0
  • B. the probability, under the null hypothesis, H0, of obtaining a result at least as extreme as the one observed
  • C. biologists often use a probability of 0.05 (5%) for this value
  • D. this hypothesis can sometimes be TRUE even when the test fails to detect it (later called a Type II error)
  • E. a parameter based on sample size and estimated parameters or constraints
 

True/False Statements About Chi-Square Tests

Click to show True/False Statements About Chi-Square Tests example problem

Which one of the following statements is TRUE regarding chi-square (χ²) tests?

 

True/False Statements About Statistical Tests

Click to show True/False Statements About Statistical Tests example problem

Which one of the following statements is TRUE concerning statistical tests?

 

Chi-Square Terms from Definitions

Click to show Chi-Square Terms from Definitions example problem

Which one of the following chi-square (χ²) terms correspond to the definition 'a fixed probability cutoff whether the null hypothesis, H0 can be or cannot be supported'.

 

Appropriate Hypothesis Tests for Mean Comparisons

Click to show Appropriate Hypothesis Tests for Mean Comparisons example problem

Which one of the following hypothesis tests correspond to the description 'You want to compare the average soil pH between two agricultural fields, one using organic fertilizer and one using conventional methods, with n = 30 soil samples from each field. You want to test if the means are significantly different.'.

 

Statistical Test Terms from Definitions

Click to show Statistical Test Terms from Definitions example problem

Which one of the following statistical test terms correspond to the definition 'this hypothesis can sometimes be TRUE even when the test fails to detect it (later called a Type II error)'.

 

Two-Sample t-Test P-Values

Click to show Two-Sample t-Test P-Values example problem

Two-Sample T-Test Scenario
Previously, you compared newborn weights at Joe's Hospital of Fried Foods to the national average (7.5 lbs) using a one-sample t-test. Now, Joe faces a challenge from Alex, CEO of Green Veggies Health Center, who believes her babies are just as heavy as Joe's fried-food babies.
Joe's Hospital of Fried Foods (n=29) and Green Veggies Health Center (n=28) each recorded the weights of newborns this month. Joe believes his hospital's babies weigh more on average than those from Green Veggies.
Use a one tailed two sample t test for H1: μJoe > μVeggies.
Assume unequal variances (Welch's t test).

Joe's Hospital (lbs):
9.6
7.7
5.4
6.9
6.3
7.5
8.2
7.8
6.5
6.3
6.0
5.8
7.5
8.6
7.3
9.1
7.5
9.2
9.5
8.8
5.3
7.2
8.5
6.0
8.8
9.2
4.2
7.1
5.5 		Green Veggies (lbs):
5.2
7.9
8.8
7.6
9.7
5.4
7.1
6.5
7.2
7.4
6.1
7.2
8.1
5.2
6.3
7.3
4.0
6.5
8.1
5.7
8.6
8.4
8.3
9.6
4.7
5.7
8.2
8.1

Compute the p value in Google Sheets using the tutorial: link here.
Enter your result as a decimal between 0 and 1 (for example, 0.084, not 8.4%).

 

Biodiversity Differences Using ANOVA

Click to show Biodiversity Differences Using ANOVA example problem

ANOVA of Shannon Diversity: 1994, 2004, 2009, 2014, 2024
Compare microbial diversity across five years using one way ANOVA.
Sample Data Table:
Copy table rows and use regular paste (Ctrl-V or ⌘-V) into Google Sheets.

Sample
Location		 1994
Shannon
Index		 2004
Shannon
Index		 2009
Shannon
Index		 2014
Shannon
Index		 2024
Shannon
Index
Andrews Park 4.14 4.23 2.93
Beisner Road Entrance 4.84 4.37 4.15 3.89
Boat Launch Area 4.28 4.90 3.70 4.08
Busse Lake 4.12 4.09
Debra Park 3.77 3.59 3.78 3.51
Elk Pasture 4.57 3.47 3.96 4.49 3.89
Forest Central Grove 4.91 3.90 4.22 3.82 4.88
Forest North Grove 3.65 4.04 4.63 3.81 3.71
Forest South Grove 3.54 3.99
Forest West Grove 4.49
Lake Boating Center 3.70 3.87 3.90
Large Event Area 4.53 3.93
Main Dam 3.93 4.69 4.83 3.46
Main Pool 4.13 4.89 3.76
Marshall Park 4.07 5.92 4.56 3.72 3.97
Model Airplane Field 4.31 4.40
Nature Preserve 4.48 3.51 3.88 3.84 3.76
Ned Brown Meadow 3.69 4.21 3.58 3.59 3.05
North Pool 4.38 4.26 3.03
Osborn Park 4.71
Salt Creek Trail 3.41 4.28 3.68
South Pool 5.83 3.98 4.12 3.98
Wildlife Refuge 3.15 4.09 3.85 3.49 3.55
Woodland Meadow 4.42 4.93 4.34 4.17 3.30

Alternate Copyable Format:
Copy this text and use Data → Split text to columns → Comma in Google Sheets.
Sample Location,1994 Shannon Index,2004 Shannon Index,2009 Shannon Index,2014 Shannon Index,2024 Shannon Index
Andrews Park,,,4.14,4.23,2.93
Beisner Road Entrance,,4.84,4.37,4.15,3.89
Boat Launch Area,,4.28,4.90,3.70,4.08
Busse Lake,,,,4.12,4.09
Debra Park,,3.77,3.59,3.78,3.51
Elk Pasture,4.57,3.47,3.96,4.49,3.89
Forest Central Grove,4.91,3.90,4.22,3.82,4.88
Forest North Grove,3.65,4.04,4.63,3.81,3.71
Forest South Grove,,,,3.54,3.99
Forest West Grove,,,,,4.49
Lake Boating Center,,,3.70,3.87,3.90
Large Event Area,,,,4.53,3.93
Main Dam,,3.93,4.69,4.83,3.46
Main Pool,,,4.13,4.89,3.76
Marshall Park,4.07,5.92,4.56,3.72,3.97
Model Airplane Field,,,,4.31,4.40
Nature Preserve,4.48,3.51,3.88,3.84,3.76
Ned Brown Meadow,3.69,4.21,3.58,3.59,3.05
North Pool,,,4.38,4.26,3.03
Osborn Park,,,,,4.71
Salt Creek Trail,,,3.41,4.28,3.68
South Pool,,5.83,3.98,4.12,3.98
Wildlife Refuge,3.15,4.09,3.85,3.49,3.55
Woodland Meadow,4.42,4.93,4.34,4.17,3.30

Enter the ANOVA p value as a decimal between 0 and 1.
Workflow: link here.

 

Microbial Diversity Significance Using a Z-Test

Click to show Microbial Diversity Significance Using a Z-Test example problem

Busse Woods Microbial Diversity vs Benchmark

Z-Test Scenario
Use a Z test. Population sd is known.
Fixed values: mu = 4.00, sigma = 0.60.
Alternative H1: diversity is different from the benchmark.
Use a two tailed test.
Sample Data Table:
Copy the table rows and use regular paste (Ctrl-V or ⌘-V) into Google Sheets.

Sample
Location		 2024
Shannon
Index
Boat Launch Area 3.56
Busse Lake 3.80
Debra Park 2.81
Elk Pasture 4.00
Forest Central Grove 2.58
Forest North Grove 4.59
Forest South Grove 5.00
Forest West Grove 3.96
Lake Boating Center 3.79
Large Event Area 2.85
Main Dam 5.28
Main Pool 3.30
Model Airplane Field 3.54
Nature Preserve 3.44
North Pool 4.73
Osborn Park 2.55
South Pool 4.34
Wildlife Refuge 3.62

Alternate Copyable Format:
Copy this text and use Data → Split text to columns → Comma in Google Sheets.
Sample Location 2024 Shannon Index
Boat Launch Area,3.56
Busse Lake,3.80
Debra Park,2.81
Elk Pasture,4.00
Forest Central Grove,2.58
Forest North Grove,4.59
Forest South Grove,5.00
Forest West Grove,3.96
Lake Boating Center,3.79
Large Event Area,2.85
Main Dam,5.28
Main Pool,3.30
Model Airplane Field,3.54
Nature Preserve,3.44
North Pool,4.73
Osborn Park,2.55
South Pool,4.34
Wildlife Refuge,3.62

Follow the workflow in the tutorial: link here.
Enter the p value as a decimal between 0 and 1.

 

Statistical Significance Using a Two-Sample F-Test

Click to show Statistical Significance Using a Two-Sample F-Test example problem

F-Test of Variances: 2014 vs 2024
Use a one tailed F test for H1: variance_2024 < variance_2014.
Sample Data Table:
Copy the table rows and use regular paste (Ctrl-V or ⌘-V) into Google Sheets.

Sample
Location		 2014
Shannon
Index		 2024
Shannon
Index
Andrews Park 2.86 3.58
Boat Launch Area 3.59 4.01
Busse Lake 5.09 3.99
Debra Park 4.80 3.83
Forest Central Grove 4.08 4.39
Forest North Grove 3.44 3.87
Forest South Grove 3.66 2.80
Forest West Grove 3.23 3.77
Lake Boating Center 4.54 4.71
Large Event Area 3.39 4.29
Main Pool 5.37 2.70
Marshall Park 3.19 4.04
Model Airplane Field 3.61 4.31
Nature Preserve 4.37 3.78
North Pool 2.61 4.51
Osborn Park 4.57 2.84
Salt Creek Trail 3.83 3.66
South Pool 3.82 4.73
Wildlife Refuge 3.65 3.91
Woodland Meadow 4.17 3.49

Alternate Copyable Format:
Copy this text and use Data → Split text to columns → Comma in Google Sheets.
Sample Location,2014 Shannon Index,2024 Shannon Index
Andrews Park,2.86,3.58
Boat Launch Area,3.59,4.01
Busse Lake,5.09,3.99
Debra Park,4.80,3.83
Forest Central Grove,4.08,4.39
Forest North Grove,3.44,3.87
Forest South Grove,3.66,2.80
Forest West Grove,3.23,3.77
Lake Boating Center,4.54,4.71
Large Event Area,3.39,4.29
Main Pool,5.37,2.70
Marshall Park,3.19,4.04
Model Airplane Field,3.61,4.31
Nature Preserve,4.37,3.78
North Pool,2.61,4.51
Osborn Park,4.57,2.84
Salt Creek Trail,3.83,3.66
South Pool,3.82,4.73
Wildlife Refuge,3.65,3.91
Woodland Meadow,4.17,3.49

Enter the p value as a decimal between 0 and 1.
Workflow: link here.

 

Statistical Significance Using a Two-Sample t-Test

Click to show Statistical Significance Using a Two-Sample t-Test example problem

Two-Sample Test of Shannon Diversity: 2014 vs 2024
Use a one tailed Welch two sample t test for H1: mean_2024 < mean_2014.
Assume unequal variances.
Sample Data Table:
Copy the table rows and use regular paste (Ctrl-V or ⌘-V) into Google Sheets.

Sample
Location		 2014
Shannon
Index		 2024
Shannon
Index
Andrews Park 3.79 4.51
Beisner Road Entrance 3.89 3.35
Boat Launch Area 4.07 4.23
Busse Lake 4.47 3.79
Debra Park 5.68 3.59
Elk Pasture 3.77 4.15
Forest Central Grove 4.54 4.47
Forest North Grove 3.77 4.46
Forest South Grove 3.87 3.45
Forest West Grove 4.12 2.54
Lake Boating Center 3.91 4.05
Large Event Area 3.01 3.53
Main Dam 3.60 4.54
Main Pool 4.31 4.82
Marshall Park 3.93 3.44
Model Airplane Field 4.67 3.09
Nature Preserve 3.73 3.58
Ned Brown Meadow 3.86 3.18
North Pool 4.83 3.95
Osborn Park 4.85 4.14
Salt Creek Trail 4.36 3.53
South Pool 3.38 4.06
Wildlife Refuge 4.42 4.31
Woodland Meadow 3.69 3.35

Alternate Copyable Format:
Copy this text and use Data → Split text to columns → Comma in Google Sheets.
Sample Location,2014 Shannon Index,2024 Shannon Index
Andrews Park,3.79,4.51
Beisner Road Entrance,3.89,3.35
Boat Launch Area,4.07,4.23
Busse Lake,4.47,3.79
Debra Park,5.68,3.59
Elk Pasture,3.77,4.15
Forest Central Grove,4.54,4.47
Forest North Grove,3.77,4.46
Forest South Grove,3.87,3.45
Forest West Grove,4.12,2.54
Lake Boating Center,3.91,4.05
Large Event Area,3.01,3.53
Main Dam,3.60,4.54
Main Pool,4.31,4.82
Marshall Park,3.93,3.44
Model Airplane Field,4.67,3.09
Nature Preserve,3.73,3.58
Ned Brown Meadow,3.86,3.18
North Pool,4.83,3.95
Osborn Park,4.85,4.14
Salt Creek Trail,4.36,3.53
South Pool,3.38,4.06
Wildlife Refuge,4.42,4.31
Woodland Meadow,3.69,3.35

Compute the p value in Google Sheets using the tutorial: link here.
Enter the p value as a decimal between 0 and 1.

 

Chi-Square Values for Phenotypic Ratios

Click to show Chi-Square Values for Phenotypic Ratios example problem
Data Table
Phenotype Expected Observed Calculation Statistic
 Yellow Round (Y–R–) 90 84 __ __
 Yellow Wrinkled (Y–rr) 30 29 __ __
 Green Round (yyR–) 30 34 __ __
 Green Wrinkled (yyrr) 10 13 __ __
(sum) χ2 =  __

Complete the table and calculate the chi-squared (χ2) value.
Even though not part of the question, ask yourself whether you would reject or fail to reject the null hypothesis
Note: answers need to be within 3% of the correct number to be correct.

 

Hypothesis Decisions from Chi-Square Tests

Click to show Hypothesis Decisions from Chi-Square Tests example problem
Table of Chi-Squared (χ²) Critical Values
Degrees of Freedom Probability
0.95 0.90 0.75 0.50 0.25 0.10 0.05 0.01
1 0.00 0.02 0.10 0.45 1.32 2.71 3.84 6.63
2 0.10 0.21 0.58 1.39 2.77 4.61 5.99 9.21
3 0.35 0.58 1.21 2.37 4.11 6.25 7.81 11.34
4 0.71 1.06 1.92 3.36 5.39 7.78 9.49 13.28

Table 1
Phenotype Expected Observed Calculation Statistic
 Yellow Round (Y–R–) 90 73 (73-90)²⁄ 73 3.959
 Yellow Wrinkled (Y–rr) 30 20 (20-30)²⁄ 20 5.000
 Green Round (yyR–) 30 49 (49-30)²⁄ 49 7.367
 Green Wrinkled (yyrr) 10 18 (18-10)²⁄ 18 3.556
(sum) χ² =  19.882

Table 2
Phenotype Expected Observed Calculation Statistic
 Yellow Round (Y–R–) 90 73 (73-90)²⁄ 73² 0.054
 Yellow Wrinkled (Y–rr) 30 20 (20-30)²⁄ 20² 0.250
 Green Round (yyR–) 30 49 (49-30)²⁄ 49² 0.150
 Green Wrinkled (yyrr) 10 18 (18-10)²⁄ 18² 0.198
(sum) χ² =  0.652

Table 3
Phenotype Expected Observed Calculation Statistic
 Yellow Round (Y–R–) 90 73 (73-90)²⁄ 90 3.211
 Yellow Wrinkled (Y–rr) 30 20 (20-30)²⁄ 30 3.333
 Green Round (yyR–) 30 49 (49-30)²⁄ 30 12.033
 Green Wrinkled (yyrr) 10 18 (18-10)²⁄ 10 6.400
(sum) χ² =  24.978

Your lab partner is trying again (eye roll) and did another a chi-squared (χ²) test on the F2 generation in a dihybid cross based on your lab data (above). They wanted to know if the results confirm the expected phenotype ratios.
You helped them set up the null hypothesis, so you know that part is correct, but they got confused and were unsure about how to calculate the chi-squared (χ²) value. So much so that they did it three (3) different ways.
Before you ask your instructor for a new lab partner, tell them which table is correct AND whether they can reject or fail to reject the null hypothesis using the information provided.

 

Errors in Chi-Square Calculations and Hypothesis Decisions

Click to show Errors in Chi-Square Calculations and Hypothesis Decisions example problem
Table of Chi-Squared (χ²) Critical Values
Degrees of Freedom Probability
0.95 0.90 0.75 0.50 0.25 0.10 0.05 0.01
1 0.00 0.02 0.10 0.45 1.32 2.71 3.84 6.63
2 0.10 0.21 0.58 1.39 2.77 4.61 5.99 9.21
3 0.35 0.58 1.21 2.37 4.11 6.25 7.81 11.34
4 0.71 1.06 1.92 3.36 5.39 7.78 9.49 13.28

Phenotype Expected Observed Calculation Statistic
 Yellow Round (Y–R–) 90 87 (87-90)²⁄ 90 0.100
 Yellow Wrinkled (Y–rr) 30 26 (26-30)²⁄ 30 0.533
 Green Round (yyR–) 30 39 (39-30)²⁄ 30 2.700
 Green Wrinkled (yyrr) 10 8 (8-10)²⁄ 10 0.400
(sum) χ² =  3.733

The final result gives the chi-squared (χ²) test value of 3.73 with 3 degrees of freedom. Consulting the Table of χ² Critical Values and a level of significance α=0.50, we obtain a critical value of 2.37.
Since the chi-squared value of 3.73 is greater than the critical value of 2.37, the null hypothesis has BEEN REJECTED.

Your lab partner completed a chi-squared (χ²) test on your lab data (above) for the F2 generation in a standard dihybrid cross. The goal was to verify if the observed results matched the expected phenotype ratios.
However, it appears they made an error. What did they do wrong?

 

Chi-Square Tests for Hardy-Weinberg Equilibrium

Click to show Chi-Square Tests for Hardy-Weinberg Equilibrium example problem
Table of Chi-Squared (χ2) Critical Values
Degrees of Freedom Probability
0.95 0.90 0.75 0.50 0.25 0.10 0.05 0.01
1 0.00 0.02 0.10 0.45 1.32 2.71 3.84 6.63
2 0.10 0.21 0.58 1.39 2.77 4.61 5.99 9.21
3 0.35 0.58 1.21 2.37 4.11 6.25 7.81 11.34
4 0.71 1.06 1.92 3.36 5.39 7.78 9.49 13.28

Table 1
Phenotype Observed Expected Calculation Statistic
 Red Flowers 93 81.3 (93-81.3)2⁄ 81.3 1.684
 Pink Flowers 217 240.7 (217-240.7)2⁄ 240.7 2.334
 White Flowers 190 178.0 (190-178.0)2⁄ 178.0 0.809
(sum) χ2 =  4.826

You finally have a new competent lab partner that you trust.
This lab partner calculated the allele frequencies of p=0.40 and q=0.60. Then they did a chi-squared (χ2) test for your Hardy-Weinberg data.
They need you to decide whether you reject or accept the null hypothesis using the information provided.

 

Null and Alternative Hypotheses for Genetic Crosses

Click to show Null and Alternative Hypotheses for Genetic Crosses example problem

You perform a dihybrid testcross (AaBb × aabb) and count the offspring phenotypes.
Total offspring scored: 200

Observed data
Category Ratio Expected Observed
 A–B– 1 50 50
 A–bb 1 50 57
 aaB– 1 50 47
 aabb 1 50 46

For a chi-squared (χ2) goodness-of-fit test, which option correctly states the null hypothesis (H0) and the alternative hypothesis (HA)?

 

Misstated Null Hypotheses for Genetic Ratios

Click to show Misstated Null Hypotheses for Genetic Ratios example problem

Your lab partner is trying again (eye roll).
In a plant species with incomplete dominance, you cross two pink individuals (Rr × Rr) and score flower color.
Total offspring scored: 224

Observed data
Category Ratio Expected Observed
 Red flowers (RR) 1 56 54
 Pink flowers (Rr) 2 112 114
 White flowers (rr) 1 56 56

They are setting up a chi-squared (χ2) goodness-of-fit test, but they wrote the hypotheses below:
H0: The offspring proportions are consistent with the expected 1:1:1 ratio (any differences from the expected ratio are due to chance).
HA: The offspring proportions are not consistent with the expected 1:1:1 ratio (the differences are too large to explain by chance alone).
What is the main problem with their hypotheses?

 

Flaws in Statistical Hypothesis Testing

Click to show Flaws in Statistical Hypothesis Testing example problem

Your lab partner is trying again (eye roll).
Scenario: ANOVA (five years of Shannon diversity)
Ecologists measure Shannon Diversity Index at the same set of sites across five years (1994, 2004, 2009, 2014, 2024).
Research question: Are any of the group means different?

They wrote the hypotheses below:
H0: All group means are equal.
In words: All group means are equal.
HA: At least one group mean is different.
In words: At least one group mean is different.
What is the main problem with their hypotheses?

 

Null and Alternative Hypotheses in Statistical Tests

Click to show Null and Alternative Hypotheses in Statistical Tests example problem

Hypotheses practice: One-sample mean test (baby weights)
Joe's Hospital of Fried Foods claims their newborns are heavier than the national average.
The national average birth weight is 7.5 lbs.
Your task is to correctly identify the null hypothesis (H0) and the alternative hypothesis (HA).

Research question: Is the mean higher than 7.5 lbs?
Which option correctly states H0 and HA?

 

Population Z-Test Using Google Sheets Data

Click to show Population Z-Test Using Google Sheets Data example problem

Joe's Hospital of Fried Foods vs National Average
Use a one tailed Z test for H1: mu_hospital > mu.
Fixed population values: μ = 7.5 lbs, σ = 1.5 lbs.
Sample weights (lbs), enter into a single column in Google Sheets:
8.5
7.5
7.7
8.2
8.4
9.2
9.3
9.4
6.2
7.9
6.9
8.4
9.4
7.3
5.2
7.9
9.2
8.3
8.1
9.6
7.1
7.0
6.7
8.2
7.8
7.9
7.5
9.0
8.0
6.3
Compute the one-tailed p-value using the tutorial workflow from last week: link here.
Enter your result as a decimal between 0 and 1 (for example, 0.084, not 8.4%).