Answer to Question #341263 in Statistics and Probability for Clouds

Question

A garment factory distributes two brands of jeans. If it is found that 75 out of 250 customers prefer brand A and that 30 out of 150 prefer brand B, can we conclude at 0.05 level of significance that brand A outsells brand B?

a. State the null and alternative hypothesis

b. Determine the significance level

c. Select the test statistic to be used

d. Compute the test statistic value

e. Determine the critical value

f. Sketch the rejection region

g. Draw the conclusion

Answer 1

a.

$H_0: p_1\le p_2$

$H_a:p_1>p_2$

b.The significance level is $\alpha = 0.05.$

c. This corresponds to a right-tailed test, and a z-test for two population proportions will be used.

d. The value of the pooled proportion is computed as

$$\bar{p}=\dfrac{X_1+X_2}{n_1+n_2}=\dfrac{75+30}{250+150}=0.2625$$

The z-statistic is computed as follows:

$$z=\dfrac{\hat{p}_1-\hat{p}_2}{\sqrt{\bar{p}(1-\bar{p})(1/n_1+1/n_2}}$$

$$=\dfrac{75/250+30/105}{\sqrt{0.2625(1-0.2625)(1/250+1/150)}}=2.2006$$

e. Based on the information provided, the significance level is $\alpha = 0.05,$ and the critical value for a right-tailed test is $z_c = 1.6449.$

f. The rejection region for this right-tailed test is $R = \{z: z > 1.6449\}.$

g. Since it is observed that $z = 2.2006 >1.6449= z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is $p=P(Z>2.2006)=0.013882,$ and since $p=0.013882<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population proportion $p_1$ is greater than $p_2,$ at the $\alpha = 0.05$ significance level.