Question

The following problem is based on information from an article by N. Keyfitz in The American Journal of Sociology (Vol. 53, pp. 470-480). Let x = age in years of a rural Quebec woman at the time of her first marriage. In the year 1941, the population variance of x was approximately σ² = 5.1. Suppose a recent study of age at first marriage for a random sample of 31 women in rural Quebec gave a sample variance s² = 3.1. Use a 5% level of significance to test the claim that the current variance is less than 5.1.

(A) What is the level of significance?

(B) State the null and alternate hypotheses.

H₀: σ² = 5.1; H₁: σ² < 5.1

H₀: σ² = 5.1; H₁: σ² > 5.1    

H₀: σ² > 5.1; H₁: σ² = 5.1

H₀: σ² = 5.1; H₁: σ² ≠ 5.1

H₀: σ² < 5.1; H₁: σ² = 5.1

(C) Find the value of the chi-square statistic for the sample. (Round your answer to three decimal places.)

(D) What are the degrees of freedom?

(E) What assumptions are you making about the original distribution?

We assume a uniform population distribution.

We assume a binomial population distribution.    

We assume a normal population distribution.

We assume a exponential population distribution.

(F) Find or estimate the P-value of the sample test statistic. (Round your answer to three decimal places.)

(G) Based on your answers in parts (a) to (f), will you reject or fail to reject the null hypothesis of independence?

Since the P-value > α, we fail to reject the null hypothesis.

Since the P-value > α, we reject the null hypothesis.    

Since the P-value ≤ α, we reject the null hypothesis.

Since the P-value ≤ α, we fail to reject the null hypothesis.

(H) Interpret your conclusion in the context of the application.

At the 5% level of significance, there is sufficient evidence to conclude that the variance of age at first marriage is less than 5.1.

At the 5% level of significance, there is insufficient evidence to conclude that the variance of age at first marriage is less than 5.1.    

Answer 1