Question

A student complains that the average time students have to wait in line in the school cafeteria is over 5 minutes. To test this claim, she times a sample of 18 students. The mean time required by the people in the sample was 5.1 minutes, with a sample standard deviation of 0.2 minutes. At α = 0.05, test the claim that the average waiting time is more than 5 minutes.a) What kind of test would you use to test this claim?

Answer 1

We have to perform and hypothesis test for the population mean.

In this case, the standard deviation of the population is not known, so the sample standard deviation is needed to perform the test. Then, the appropiate test is the T-test for the mean.

We can only use Z test for means when we know the standard deviation of the population or the sample is big enough that the the T-test is very close to a Z-test.

The claim is that the average time students have to wait in line in the school cafeteria is over 5 minutes.

Then, the null and alternative hypothesis are:

`\begin{gathered} H_0\colon\mu=5 \\ H_a\colon\mu>5 \end{gathered}`

The significance level is 0.05.

The sample has a size n=18.

The sample mean is M=5.1.

As the standard deviation of the population is not known, we estimate it with the sample standard deviation, that has a value of s=0.2.

The estimated standard error of the mean is computed using the formula:

`s_M=(s)/(√(n))=(0.2)/(√(18))=0.0471`

Then, we can calculate the t-statistic as:

`t=(M-\mu)/(s/√(n))=(5.1-5)/(0.0471)=(0.1)/(0.0471)=2.121`

The degrees of freedom for this sample size are:

`df=n-1=18-1=17`

This test is a right-tailed test, with 17 degrees of freedom and t=2.121, so the P-value for this test is calculated as (using a t-table):

`\text{P-value}=P(t>2.121)=0.0244`

As the P-value (0.0244) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

At a significance level of 0.05, there is enough evidence to support the claim that the average time students have to wait in line in the school cafeteria is over 5 minutes.

Answer:

a) T-test for a mean

b) The claim is that the average time students have to wait in line in the school cafeteria is over 5 minutes.

c) H0 is μ=5 and Ha is μ>5 (Note: Ha, the alternative hypothesis, represents the claim).

d) μ = 5, X (with hat, or sample mean) = 5.1, s = 0.2, n = 18.

e) The null hypothesis is rejected.

f) There is enough evidence to support the claim that the average time is over 5 minutes.