Letx= age in years of a rural Quebec woman at the time of her first marriage. In the year 1941, the population variance ofxwas approximately²= 5.1. Suppose a recent study of age at first marriage for a random sample of41women in rural Quebec gave a sample variances²=2.5. 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) What is the level of significance?

State the null and alternate hypotheses.

H_o:²= 5.1;H₁:²> 5.1

H_o:²= 5.1;H₁:²< 5.1

H_o:²< 5.1;H₁:²= 5.1

H_o:²= 5.1;H₁:²5.1

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

What are the degrees of freedom?

What assumptions are you making about the original distribution?

We assume a binomial population distribution.

We assume a exponential population distribution.

We assume a uniform population distribution.

We assume a normal population distribution.

(c) Find or estimate theP-value of the sample test statistic.

P-value > 0.100

0.050 <P-value < 0.100

0.025 <P-value < 0.050

0.010 <P-value < 0.025

0.005 <P-value < 0.010

P-value < 0.005

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

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

Since theP-value >, we reject the null hypothesis.

Since theP-value, we reject the null hypothesis.

Since theP-value, we fail to reject the null hypothesis.

(e) 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.

(f) Find the requested confidence interval for the population variance. (Round your answers to two decimal places.)lower limitupper limit

Interpret the results in the context of the application.

We are 90% confident that²lies below this interval.

We are 90% confident that²lies above this interval.

We are 90% confident that²lies outside this interval.

We are 90% confident that²lies within this interval.