Answer to Question #226056 in Statistics and Probability for Carrie

Question

P Q Printers is evaluating the delivery time of two courier delivery services in Johannesburg. Their

initial belief is that there is no difference between the average delivery times of the two courier

services.

To examine this view, P Q Printers used both courier services daily on a random basis over a period

of three months for deliveries to similar destinations. A dispatch clerk in the marketing department

recorded delivery times. Courier A was used 60 times over this period and a sample mean delivery

was 42 minutes. Courier B was used 48 times over the same period and their sample mean delivery

time was 38 minutes.

Consider the population standard deviation of the delivery times for courier A is 14 minutes, and

for courier B is 10 minutes. Assume the delivery times are normally distributed.

Which one of the following statements is correct?

1. The hypotheses are H0 : 1 D 2 vs H2 : 1 > 2

:

2. The rejection region at 5% significance level is t > 1:96 or t > 

Answer 1

1.The following null and alternative hypotheses need to be tested:

$H_0: \mu_1=\mu_2$

$H_1: \mu_1>\mu_2$

Or

$H_0: \mu_1\leq\mu_2$

$H_1: \mu_1>\mu_2$

The given statement is correct.

2. This corresponds to a right-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

The degrees of freedom are computed as follows, assuming that the population variances are equal:

$$df_{total}=df_1+df_2=60-1+48-1=106$$

The critical value for this right-tailed test is $t_c= 1.659356,$ for $\alpha=0.05$ and $df=106.$

The rejection region for this right-tailed test is $R=\{t:t>1.66\}$

The degrees of freedom are computed as follows, assuming that the population variances are unequal:

$$df_{total}=\dfrac{(\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2})^2}{\dfrac{(s_1^2/n_1)^2}{n_1-1}+\dfrac{(s_2^2/n_2)^2}{n_2-1}}$$

$$=104.76272802168$$

The critical value for this right-tailed test is $t_c= 1.66,$ for $\alpha=0.05$ and $df=104.76272802168.$

The rejection region for this right-tailed test is $R=\{t:t>1.66\}.$

The given statement is uncorrect.