Question

Do teachers find their work rewarding and satisfying? An article reports the results of a survey of 393 elementary school teachers and 261 high school teachers. Of the elementary school teachers, 221 said they were very satisfied with their jobs, whereas 123 of the high school teachers were very satisfied with their work. Estimate the difference between the proportion of all elementary school teachers who are satisfied and all high school teachers who are satisfied by calculating a 95% CI. (Use p_elementary ? p_{high school}. Round your answers to four decimal places.)

Answer 1

For sample 1, we have that the sample size is N1?=393, the number of favorable cases is X1?=221, so then the sample proportion is ?=X1/N1??=221/393?=0.5623

For sample 2, we have that the sample size is N2?=261, the number of favorable cases is X2?=123, so then the sample proportion is ?=X2/N2 ??=123/261?=0.4713

The value of the pooled proportion is computed as ?=X1?+X2??/N1?+N2? =221+123/?393+261 =0.526

Also, the given significance level is ?=0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:p1?=p2?

Ha:p1??p2?

This corresponds to a two-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

Based on the information provided, the significance level is ?=0.05, and the critical value for a two-tailed test is zc?=1.96.

The rejection region for this two-tailed test is R={z:?z?>1.96}

(3) Test Statistics

The z-statistic is computed as follows:

Z =?

??=0.5623?0.4713?/0.526?(1?0.526)(1/393+1/261)

?=2.284

(4) Decision about the null hypothesis

Since it is observed that ?z?=2.284>zc?=1.96, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: p=0.0224, and since p=0.0224<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion p1? is different than p2?, at the 0.05 significance level.

Confidence Interval

The 95% confidence interval for p1??p2? is: 0.0129<p1??p2?<0.1691 .Since it does not contain zero the confidence interval is significant.