A new kind of typhoid shot is being developed by a medical research team. The old typhoid shot was known to protect the population for a mean time of 36 months, with a standard deviation of 3 months. To test the time variability of the new shot, a random sample of 23 people were given the new shot. Regular blood tests showed that the sample standard deviation of protection times was 1.9 months. Using a 0.05 level of signifi­cance, test the claim that the new typhoid shot has a smaller variance of protection times. Find a 90% confidence interval for the population standard deviation.

Please provide the following information.

(a) What is the level of signi­ficance? State the null and alternate hypotheses.

(b) Find the value of the chi-square statistic for the sample. What are the degrees of freedom? What assumptions are you making about the original distribution?

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

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

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

(f) Find the requested confi­dence interval for the population variance or population standard deviation. Interpret the results in the context of the application.