Question

1) A study was conducted to determine the proportion of people who dream in black and white instead of color. Among 323 people over the age of​ 55, 66 dream in black and​ white, and among 281people under the age of​ 25,15 dream in black and white. Use a 0.05 significance level to test the claim that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25. Complete parts​ (a) through​ (c) below.

2) A simple random sample of​ front-seat occupants involved in car crashes is obtained. Among 2728 occupants not wearing seat​ belts, 31 were killed. Among 7791occupants wearing seat​ belts, 17 were killed. Use a 0.01 significance level to test the claim that seat belts are effective in reducing fatalities. Complete parts​ (a) through​ (c) below.

3)Since an instant replay system for tennis was introduced at a major​ tournament, men challenged 1408 referee​ calls, with the result that 430 of the calls were overturned. Women challenged 767 referee​ calls, and 228 of the calls were overturned. Use a 0.01 significance level to test the claim that men and women have equal success in challenging calls. Complete parts​ (a) through​ (c) below.

Answer 1

Answering 1

For sample 1, we have that the sample size is N1​=323, the number of favorable cases is X1​=66, so then the sample proportion is

For sample 2, we have that the sample size is N2​=281, the number of favorable cases is X2​=15, so then the sample proportion is

The value of the pooled proportion is computed as

Also, the given significance level is α=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 right-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 α=0.05, and the critical value for a right-tailed test is zc​=1.64.

The rejection region for this right-tailed test is R={z:z>1.64}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z=5.43>zc​=1.64, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0, and since p=0<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 greater than p2​, at the 0.05 significance level.

Graphically

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