Question

The manager of the service department of a local car dealership has noted that the service times of a sample of 15 new automobiles has a sample standard deviation of S = 4 hours. We are interested to test the following hypotheses using α = 0.05 level of significance:

  H₀ : σ² ≤ 14

  H_a : σ² > 14

1. Assuming that the population of service times follows an approximately normal distribution, what is the proper test statistic?

A.Chi-square distribution with 4 degrees of freedom

B.Chi-square distribution with 14 degrees of freedom

C.Chi-square distribution with 15 degrees of freedom

D.Chi-square distribution with 16 degrees of freedom

2. Assuming that the population of service times follows an approximately normal distribution, what is the value of the test statistic?

A.4

B.14

C.15

D.16

3. Assuming that the population of service times follows an approximately normal distribution, what is the p-value?

A.

More than 0.10

B.

Between 0.05 and 0.10  

C.

Less than 0.01

D.

between 0.025 and 0.05

4.Assuming that the population of service times follows an approximately normal distribution, what is the conclusion?

A.	Reject the null hypothesis at  α = 0.05 . The sample supports the alternative hypothesis that the population variance of the service times exceeds 14.

B.	We cannot reject the null hypothesis at  α = 0.05 . The sample does not support the alternative hypothesis that the population variance of the service times exceeds 14.
	C.	We can reject the null hypothesis only if the level of significance, α, is reduced from 0.05 to0.01.
	D.	The test is inconclusive.

Answer 1

Degrees of freedom is (sample size-1)=(15-114
Chi-square distribution with 14 degrees of freedom

where N is the sample size and s is the sample standard deviation. The key element of this formula is the ratio
s/σ0 which compares the ratio of the sample standard deviation to the target standard deviation. The more this ratio deviates from 1, the more likely we are to reject the null hypothesis.
Here s=4 and

Valu of test statistics is 16

p-value is greater than 0.1

We cannot reject the null hypothesis at  α = 0.05 . The sample does not support the alternative hypothesis that the population variance of the service times exceeds 14.
Since it is observed that it is then concluded that the null hypothesis is not rejected.