Answer to Question #343683 in Statistics and Probability for Nipuli

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.