Question

A carpet company advertises that it will deliver your carpet within 15 days of purchase. A sample of 49 past customers is taken. The average delivery time in the sample was 16.2 days. Assume the population standard deviation is known to be 5.6 days.
a. State the null and alternative hypotheses.
b. Using a critical value, test the null hypothesis at the 5% level of significance.
c. Using a p-value, test the hypothesis at the 5% level of significance.
d. What type of error may have been committed for this hypothesis test

Answer 1

Answer:

a) $H_0: \mu\leq15\\\\H_1: \mu>15$

b) The z-value (1.5) is smaller than z=1.645 (critical value), so it is in the "acceptance region". It failed to reject the null hypothesis.

c) The p-value (0.07) is greater than the significance level (0.05), so it failed to reject the null hypothesis.

d) In this case, the error we may hae comitted is a Type II error (failed to reject a null hypothesis that is false).

Step-by-step explanation:

We have to perfomr a hypothesis test on the mean, with known standard deviation of the population.

a) The null hypothesis is that the deliver time is 15 days or less.

The null and alternative hypothesis are then:

$H_0: \mu\leq15\\\\H_1: \mu>15$

The significance level is defined as 0.05.

b) The critical value of z for a one-side test (rigth side) and a significance level of 0.05 is z=1.645.

If the z-value for this sample is higher than 1.645, it is in the "rejection region".

Calculating the z-value:

$z=\frac{M-\mu}{\sigma/\sqrt{n}}=\frac{16.2-15}{5.6/\sqrt{49}}=\frac{1.2}{0.8}=1.5$

The z-value (1.5) is smaller than z=1.645 (critical value), so it is in the "acceptance region". It failed to reject the null hypothesis.

c) The p-value for z=1.5 is:

$P(z>1.5)=0.067$

The p-value (0.07) is greater than the significance level (0.05), so it failed to reject the null hypothesis.

d) There are two types of error:

Type I errors: happen when we reject a null hypothesis that is true.

Type II errors: happen when we failed to reject a null hypothesis that is false.

In this case, the error we may hae comitted is a Type II error.