Chi-Square Analysis Worksheet

1. Why Do Scientists Use Hypothesis Testing? (6 points)

Scientists use hypothesis testing to:

Make objective decisions about research results
Determine if differences in data are due to the independent variable or due to chance
Reject or fail to reject statistical hypotheses based on evidence

Which of the following statements best describes hypothesis testing? (2 points)

A method to prove theories with 100% certainty
A statistical method to determine if results are due to chance or a specific cause
A way to ensure all experiments have the same outcome

What is the difference between null and alternative hypotheses? (2 points)

Select ALL correct statements about hypothesis testing: (2 points)

(Partial credit will be given for partially correct answers)

Hypothesis testing can determine if results are statistically significant
A null hypothesis always states that there is no difference or relationship
Researchers typically try to prove the null hypothesis
A p-value of 0.05 means there's a 5% chance the null hypothesis is true

2. Purpose of Chi-Square Goodness-of-Fit Test (6 points)

The chi-square goodness-of-fit test:

Determines if there is a significant relationship between observed and expected data
Compares observed outcomes to expected outcomes to see if variations could be due to chance
Is often used to test genetic crosses and other categorical data

When would you use a chi-square goodness-of-fit test? (2 points)

To compare means of two different groups
To determine if observed categorical data matches expected distribution
To measure correlation between two continuous variables

Explain when a researcher would reject the null hypothesis in a chi-square test. (2 points)

Select ALL correct statements about chi-square tests: (2 points)

(Partial credit will be given for partially correct answers)

Chi-square tests are appropriate for categorical data
The chi-square statistic increases as the difference between observed and expected values increases
A chi-square test can determine causation between variables
Degrees of freedom in a goodness-of-fit test equals the number of categories minus one
Expected frequencies should always be greater than 5 for reliable results

3. Steps to Perform a Chi-Square Goodness-of-Fit Test (12 points)

Step 1: Establish the research question

Step 2: Determine the null hypothesis (H₀)

Step 3: Determine the alternative hypothesis (Hₐ)

Step 4: Count observed values

Step 5: Determine expected values

Step 6: Calculate the chi-square value

Step 7: Calculate the degrees of freedom

Step 8: Select the p-value (typically 0.05)

Step 9: Identify the critical value

Step 10: Compare the chi-square value to the critical value to draw conclusions

Chi-Square Formula

\[χ^2 = \sum \frac{(O - E)^2}{E}\]

Where:

O = observed results
E = expected results
Σ = sum of all categories

Degrees of freedom = number of categories - 1

Chi-Square Critical Values Table

p-value	df=1	df=2	df=3	df=4	df=5	df=6	df=7	df=8
0.05	3.84	5.99	7.81	9.49	11.07	12.59	14.07	15.51
0.01	6.63	9.21	11.34	13.28	15.09	16.81	18.48	20.09

Practice Problem: Genetic Cross (12 points)

In a genetic experiment, you're studying the inheritance of seed shape in pea plants. According to Mendelian genetics, you expect a 3:1 ratio of round seeds to wrinkled seeds. You observe 85 round seeds and 35 wrinkled seeds in your experiment.

Step 1: State Hypotheses (2 points)

Null Hypothesis (H₀):

Alternative Hypothesis (Hₐ):

Step 2: Calculate Expected Values (2 points)

Total seeds: 120

Expected ratio: 3:1 (round:wrinkled)

Expected round seeds:

Expected wrinkled seeds:

Step 3: Calculate Chi-Square Value (4 points)

Phenotype	Observed (O)	Expected (E)
Round	85	?
Wrinkled	35	?
Total χ² =

Step 4: Determine Degrees of Freedom and Critical Value (2 points)

Degrees of freedom =

Using p = 0.05, the critical value is:

Step 5: Draw Conclusion (2 points)

Compare your calculated χ² value to the critical value:

Calculated χ²: ?

Critical value: ?

Conclusion:

Chi-Square Analysis Interactive Worksheet

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