Question

A large automobile insurance company wants to test the null hypothesis that the mean age in its population of policyholders is 50, against the alternative hypothesis that it is different from 50. In a random sample of 361 policyholders, the average age is 47.2 years, and the variance is 121 (squared years). The significance level is 5%.

1. What is the population?

a. Bernoulli. Xi , i = 1,…,n are i.i.d. Bernoulli random variables with success probability p

b. Normal.

c. General (arbitrary)

2. Give the value of the critical value (cv) if the rejection region has one critical value or the value of the smaller critical value (cv1) if the rejection region has two critical values. One decimal.

3. Give the value of the larger critical value (cv2) if the rejection region has two critical values. If the rejection region has only one critical value, answer NA. One decimal.

4. What is the conclusion of the test?

a. Fail to reject H0 (accept H0)

b. Reject H0 in favor of HA

Answer 1

Claim: The mean age of the policyholder is 50

The null and alternative hypothesis are,

1. The population is assumed to be Normal.

2.  The population variance is unknown so t test will use.

The alternative hypothesis contains the not equal to sign so the test is two-tailed test.

There are two critical values for the critical region both are same just opposite in sign.

Sample size = n = 361 and significance level = alpha = 5% = 0.05

Degrees of freedom = n - 1 = 360

Using excel function =tinv(probability, degrees of freedom) = tinv(0.05, 360) gives 1.967

Rounded to the 1 decimal place as 2.0

Smaller critical value is negative

cv1 = -2.0

3.  Larger critical value for the critical region is the positive value

cv2 = 2.0

4. Conclusion of the test

To write the conclusion test statistics is needed

The formula of t test statistics,

Where

Decision rule: If the test statistics falls between smaller and larger critical values that is between -2.0 to 2.0 then fail to reject the H0 otherwise reject the H0.

Test statistics -4.8 does not fall between -2.0 to 2.0 so reject the null hypothesis.

Reject H0, in favor of HA

Reject the null hypothesis means we can conclude that the population mean is different than 50.