Question

A certain genetic characteristic of a particular plant can appear in one of three forms (phenotypes). A researcher has developed a theory, according to which the hypothesized proportions are p₁ = 0.25, p₂ = 0.50, and p₃ = 0.25. A random sample of 200 plants yields X² = 5.01.

(a)

Carry out a test of the null hypothesis that the theory is correct, using level of significance α = 0.05. What is the critical value for the test? (Round your answer to two decimal places.)

Critical Value =

State the conclusion in the problem context.

Fail to reject H₀. There is not convincing evidence that the researcher's theory is incorrect. Reject H₀. There is not convincing evidence that the researcher's theory is incorrect.     Reject H₀. There is convincing evidence that the researcher's theory is incorrect. Fail to reject H₀. There is convincing evidence that the researcher's theory is incorrect.

(b)

Suppose that a random sample of 300 plants had resulted in the same value of X². How would your analysis and conclusion differ from those in part (a)?

The analysis and conclusion would not change. The previous analysis did not yield a significant result and the null hypothesis failed to be rejected. The new analysis would yield a significant result and the null hypothesis would be rejected.     The previous analysis yielded a significant result and the null hypothesis was rejected. The new analysis would not yield a significant result and the null hypothesis would fail to be rejected.

Answer 1

a) For k - 1 = 2 degrees of freedom, and for 0.05 level of significance, we have from chi square distribution tables:

$P(\chi^2_2 > 5.99) = 0.05$

Therefore 5.99 is the required critical value here.

As the test statistic value here is 5.01 < 5.99, therefore the test is not significant here and we cannot reject the null hypothesis here. Therefore we have insufficient evidence here that that the researcher's theory is incorrect

b) As the chi square test statistic is still the same, we still can see here that it is lower than the crtical value and lies in the non rejection region here. The analysis and conclusion would not change. The previous analysis did not yield a significant result and the null hypothesis failed to be rejected.