Question

An oil company sends out monthly statements to its customers who purchased gasoline and other items using the company’s credit card. Until now, the company has not included a pre-addressed envelope for returning payments. The mean number of days before payment is received is 9.8. As an experiment to determine whether enclosing pre-addressed envelopes speeds up payment, 150 customers selected at random were sent pre-addressed envelopes with their bills. The sample statistics showed a mean of 9.16 days and a standard deviation of 2.642 days. Do the data provide sufficient evidence at 5% level of significance to establish that enclosure of pre-addressed envelopes improves the average speed of payments?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\ge9.8$

$H_a:\mu<9.8$

This corresponds to a left-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is $\alpha = 0.05,$ $df=n-1=149$ degrees of freedom, and the critical value for a left-tailed test is $t_c =-1.655145.$The rejection region for this left-tailed test is $R = \{t:t<-1.655145\}.$

The t-statistic is computed as follows:

$$t=\dfrac{9.16-9.8}{2.642/\sqrt{150}}=-2.9668$$

Since it is observed that $t =-2.9668<-1.655145=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for left-tailed $df=149$ degrees of freedom, $t=-2.9668$ is $p= 0.001753,$ and since $p= 0.001753<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is less than 9.8, at the $\alpha = 0.05$ significance level.