Questions: 377 of the 627 freshmen psychology majors from a recent study changed their major before they graduated and 416 of the 611 freshmen business majors changed their major before they graduated. What can be concluded at the alpha=0.05 level of significance? If the calculator asks, be sure to use the "Pooled" data option. For this study, we should use z-test for the difference between two population proportions The results are statistically significant at alpha=0.05, so there is sufficient evidence to conclude that the proportion of the 627 freshmen psychology majors who changed their major is less than the proportion of the 611 freshmen business majors who change their major.

377 of the 627 freshmen psychology majors from a recent study changed their major before they graduated and 416 of the 611 freshmen business majors changed their major before they graduated. What can be concluded at the alpha=0.05 level of significance? If the calculator asks, be sure to use the "Pooled" data option.

For this study, we should use z-test for the difference between two population proportions

The results are statistically significant at alpha=0.05, so there is sufficient evidence to conclude that the proportion of the 627 freshmen psychology majors who changed their major is less than the proportion of the 611 freshmen business majors who change their major.

Transcript text: 377 of the 627 freshmen psychology majors from a recent study changed their major before they graduated and 416 of the 611 freshmen business majors changed their major before they graduated. What can be concluded at the $\alpha=0.05$ level of significance? If the calculator asks, be sure to use the "Pooled" data option. For this study, we should use z-test for the difference between two population proportions The results are statistically significant at $\alpha=0.05$, so there is sufficient evidence to conclude that the proportion of the 627 freshmen psychology majors who changed their major is less than the proportion of the 611 freshmen business majors who change their major.

Solution

Solution Steps

Step 1: Define Hypotheses

We set up the null and alternative hypotheses as follows:

Null Hypothesis $ H_0 $: $ P_1 = P_2 $ (The proportion of freshmen psychology majors who changed their major is equal to that of freshmen business majors.)
Alternative Hypothesis $ H_1 $: $ P_1 < P_2 $ (The proportion of freshmen psychology majors who changed their major is less than that of freshmen business majors.)

Step 2: Calculate Sample Proportions

The sample proportions are calculated as: \[ \bar{x}_1 = \frac{377}{627} \approx 0.6013 \] \[ \bar{x}_2 = \frac{416}{611} \approx 0.6809 \]

Step 3: Calculate Standard Error

The standard error $ SE $ is calculated using the formula: \[ SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} = \sqrt{\frac{0.0004}{627} + \frac{0.0004}{611}} \approx 0.0011 \]

Step 4: Calculate Test Statistic

The test statistic $ z $ is calculated as: \[ z = \frac{\bar{x}_1 - \bar{x}_2}{SE} = \frac{0.6013 - 0.6809}{0.0011} \approx -72.5685 \]

Step 5: Calculate P-value

The p-value is calculated using the formula: \[ P = 2 \times (1 - Z(|z|)) \approx 0.0 \]

Step 6: Decision Rule

At the significance level $ \alpha = 0.05 $:

Since $ P < 0.05 $, we reject the null hypothesis.

Step 7: Conclusion

The results are statistically significant at $ \alpha = 0.05 $, so there is sufficient evidence to conclude that the proportion of freshmen psychology majors who changed their major is less than the proportion of freshmen business majors who changed their major.