A set of solar batteries is used in a research satellite. The satellite can run on only one battery, but it runs best if more than one battery is used. The variance²of lifetimes of these batteries affects the useful lifetime of the satellite before it goes dead. If the variance is too small, all the batteries will tend to die at once. Why? If the variance is too large, the batteries are simply not dependable. Why? Engineers have determined that a variance of²= 23 months (squared) is most desirable for these batteries. A random sample of26batteries gave a sample variance of14.4months (squared). Using a 0.05 level of significance, test the claim that²= 23 against the claim that²is different from 23.

(a) What is the level of significance?

State the null and alternate hypotheses.

H_o:²= 23;H₁:²< 23

H_o:²= 23;H₁:²> 23

H_o:²> 23;H₁:²= 23

H_o:²= 23;H₁:²23

(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 normal population distribution.

We assume a binomial population distribution.

We assume a uniform population distribution.

We assume a exponential 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 battery life is different from 23.

At the 5% level of significance, there is sufficient evidence to conclude that the variance of battery life is different from 23.

(f) Find a 90% confidence interval for the population variance. (Round your answers to two decimal places.)lower limitupper limit

(g) Find a 90% confidence interval for the population standard deviation. (Round your answers to two decimal places.)lower limitmonthsupper limitmonths