Question

Your foreman claims that tree planting is a job for young people. He further claims that 3/4 of tree planters are below the age of 21. You think he's exaggerating and the proportion of tree planters under 21 is not nearly that high. You gain access to a page from the personnel files that has the birthdates for 50 tree planters and count 32 who are under 21. Do you have enough evidence (at ?=.05α=.05) to conclude that your foreman is wrong and that the proportion of tree planters under 21 is less than 3/4?

Answer 1

The following information is provided: The sample size is N = 50 , the number of favorable cases is X = 32 , and the sample proportion is , and the significance level is α=0.05

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p = 0.75

The proportion is exactly 3/4

Ha: p < 0.75

The proportion is less than 3/4

This corresponds to a left-tailed test, for which a z-test for one population proportion needs to be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a left-tailed test is z_c = -1.64

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z = -1.796< z_c = -1.64 , it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0362 , and since p = 0.0362 < 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 the population proportion p is less than p_0 at the α=0.05 significance level.