#37

Let x = age in years of a rural Quebec woman at thetime of her first marriage. In the year 1941, the populationvariance of x was approximately ?² =5.1. Suppose a recent study of age at first marriage for a randomsample of 31 women in rural Quebec gave a sample variances² = 2.3. Use a 5% level of significance totest the claim that the current variance is less than 5.1. Find a90% 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: ?² > 5.1

(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 originaldistribution?

We assume a normal population distribution.

We assume a uniform population distribution.  

  We assume a binomial population distribution.

We assume a exponential population distribution.

(c) Find or estimate the P-value of the sample teststatistic.

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 orfail to reject the null hypothesis?

Since the P-value > ?, we fail to rejectthe null hypothesis.Since the P-value > ?, wereject the null hypothesis.

   Since the P-value ? ?, wereject the null hypothesis.Since the P-value ? ?,we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of theapplication.

At the 5% level of significance, there is insufficient evidenceto conclude that the variance of age at first marriage is less than5.1.

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

(f) Find the requested confidence interval for the populationvariance. (Round your answers to two decimal places.)

lower limit
upper limit

Interpret the results in the context of the application.

We are 90% confident that ?² lies below thisinterval.We are 90% confident that ?² lieswithin this interval.  

  We are 90% confident that ?²lies above this interval.We are 90% confident that?² lies outside this interval.