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Chapter 3: Problem 16

A geneticist, in assessing data that fell into two phenotypic classes, observed values of \(250: 150 .\) He decided to perform chi- square analysis using two different null hypotheses: (a) the data fit a 3: 1 ratio; and (b) the data fit a 1: 1 ratio. Calculate the \(\chi^{2}\) values for each hypothesis. What can you conclude about each hypothesis?

Short Answer

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Question: Based on the chi-square analysis performed for the two hypotheses, the geneticist can conclude that the data fits: A) 3:1 ratio B) 1:1 ratio C) Neither the 3:1 nor the 1:1 ratio Answer: C) Neither the 3:1 nor the 1:1 ratio

Step by step solution

01

Calculate the expected values for each hypothesis

For hypothesis (a), the expected ratio is 3:1. As the total number of observations is 400 (250 + 150), the expected numbers for each phenotypic class are: Class 1: \(\frac{3}{4} \times 400 = 300\) Class 2: \(\frac{1}{4} \times 400 = 100\) For hypothesis (b), the expected ratio is 1:1. That means the expected numbers for each phenotypic class are: Class 1: \(\frac{1}{2} \times 400 = 200\) Class 2: \(\frac{1}{2} \times 400 = 200\)
02

Calculate the chi-square (χ²) values

For hypothesis (a), calculate the χ² value with the formula: χ² = Σ [(observed - expected)² / expected] χ² = ( (250 - 300)² / 300 ) + ( (150 - 100)² / 100 ) χ² = (50² / 300) + (50² / 100) χ² = (2500 / 300) + (2500 / 100) χ² = 8.33 + 25 χ² = 33.33 For hypothesis (b), calculate the χ² value similarly: χ² = ( (250 - 200)² / 200 ) + ( (150 - 200)² / 200 ) χ² = (50² / 200) + (-50² / 200) χ² = (2500 / 200) + (2500 / 200) χ² = 12.5 + 12.5 χ² = 25
03

Compare the calculated chi-square values to the critical values

With two phenotypic classes (2-1 = 1 degree of freedom), the critical value for χ² at a significant level of 0.05 (with 95% confidence) is 3.841. In our case: - For hypothesis (a), χ² = 33.33 > 3.841, meaning we reject this hypothesis, as it does not fit the 3:1 ratio. - For hypothesis (b), χ² = 25 > 3.841, meaning we also reject this hypothesis, as it does not fit the 1:1 ratio. In conclusion, neither hypothesis is suitable as both have chi-square values larger than the critical value. Therefore, the given data do not fit either a 3:1 or a 1:1 ratio.

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