Question

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 51 women in rural Quebec gave a sample variance s2 = 3.2. Use a 5% level of significance to test the claim that the current variance is less than 5.1. Find a 90% confidence interval for the population variance.

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

b)Find or estimate the P-value of the sample test statistic.

1)P-value > 0.100 2)0.050 < P-value < 0.100 3)0.025 < P-value < 0.050 4)0.010 < P-value < 0.025 5)0.005 < P-value < 0.010 6)P-value < 0.005

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

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.

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

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.

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

Answer 1

The statistical software output for this problem is:

One sample variance summary hypothesis test:
σ² : Variance of population
H₀ : σ² = 5.1
H_A : σ² < 5.1

Hypothesis test results:

Variance	Sample Var.	DF	Chi-square Stat	P-value
σ2	3.2	50	31.372549	0.0181

Hence,

a) Chi - square statistic = 31.37

b) 0.010 < P-value < 0.025; Option 4 is correct.

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

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