Answer key: Chi-Square Test Activity
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Goal: Use the chi-square goodness-of-fit test to decide whether offspring counts fit an expected Mendelian ratio, and explain what the result does and does not show.
Completed results table
| Data | χ² | Degrees of freedom | Critical value | Verdict |
|---|---|---|---|---|
| Mendel: 705 purple : 224 white | 0.391 | 1 | 3.841 | Fits 3 : 1 |
| Poor fit: 250 : 150 | 33.333 | 1 | 3.841 | Reject the ratio |
| Snapdragons: 28 red : 55 pink : 17 white | 3.42 | 2 | 5.991 | Fits 1 : 2 : 1 |
| Mendel dihybrid: 315 : 108 : 101 : 32 | 0.47 | 3 | 7.815 | Fits 9 : 3 : 3 : 1 |
Answers
For 705 purple : 224 white, calculate the expected counts for a 3 : 1 ratio and then χ².
The total is 929, so expected purple = 929 × 3/4 = 696.75 and white = 929 × 1/4 = 232.25. χ² = (705 − 696.75)²/696.75 + (224 − 232.25)²/232.25 = 0.098 + 0.293 = 0.391.
Why is the 250 : 150 result rejected, even though it may not look far from 3 : 1?
The expected counts are 300 and 100, so both classes are 50 away from expected, which is large relative to the expected 100: χ² = 2,500/300 + 2,500/100 = 8.33 + 25 = 33.33, far above 3.841. A χ² that large would happen by chance less than 0.1% of the time.
How do you find the degrees of freedom? Why does the critical value change between the 3 : 1, 1 : 2 : 1 and 9 : 3 : 3 : 1 tests?
df = number of classes − 1: 1 for 3 : 1, 2 for 1 : 2 : 1 and 3 for 9 : 3 : 3 : 1. With more classes χ² adds up more terms, so a larger total can happen by chance and the critical value rises: 3.841, 5.991 and 7.815 at α = 0.05.
Do the snapdragon counts (28 red : 55 pink : 17 white) support incomplete dominance with a 1 : 2 : 1 ratio?
Yes. χ² = 3.42 with df = 2 is below the critical value 5.991 (p ≈ 0.181), so the differences could be due to chance and the data fit 1 : 2 : 1.
A “fits” verdict does not prove the hypothesis is true. Explain what it does tell you.
It means the differences between observed and expected counts are small enough to be explained by chance (p > 0.05), so we fail to reject the hypothesis. Another ratio might fit as well, and small samples make almost any ratio hard to reject.
Challenge: Mendel’s dihybrid counts give χ² = 0.47. Is that a good fit? What would a χ² of exactly 0 mean?
Yes: 0.47 is far below the critical value 7.815 for df = 3, with p ≈ 0.925. A χ² of 0 would mean the observed counts exactly equal the expected counts, which is very unlikely in real data.