Question

4. (16 pts) In a random sample of 200 men, 130 said they used seat belts. In a random sample of 300 women, 63 said they used seat belts. Test the claim that men are more safety-conscious than women, at. (a) State the null and alternative hypotheses and identify the claim to test. (b) Using level of significance , find the test statistic and the critical value, and make your conclusion about the hypothesis test. (c) Find the P-value and show the comparison with , and make your conclusion about the hypothesis test. (d) Find the 98% confidence interval for the differences between the means and explain why it agrees with the previous results in (b) and (c) above.

alpha: 0.01

Answer 1

a) H0: The men are not more safety-conscious than women
H1: The men are more safety-conscious than women

i.e.

Ho: P1 = P2 Ha: p1 > P2

b) Let the los be alpha = 2%,

130 200 - For sample 1, we have that the sample size is N1 200, the number of favorable cases is X1 = 130 , so then the sample proportion is pi 0.65 For sample 2, we have that the sample size is N2 = 300, the number of favorable cases is X2 = 63, so then the sample proportion is P2 = x The value of the pooled proportion is computed as p NIN 0.386 Also, the given significance level is a = 0.02. 36% = 0.21 130-63 200-300

Test Statistic:

The z-statistic is computed as follows: P1 - P2 p(1-7)(1/nı +1/n) 0.65 -0.21 0.386. (1 – 0.386)(1/200 +1/300) 9.901

Since it is observed that z = 9.901 > Ze=2.05, it is then concluded that the null hypothesis is rejected.

Thus we conclude that  the men are more safety-conscious than women

c)

Using the P-value approach: The p-value is p = 0, and since p=0 < 0.02, it is concluded that the null hypothesis is rejected.

Thus we conclude that  the men are more safety-conscious than women

d)

The critical value for a = 0.02 is ze = 21-a/2 = 2.326. The corresponding confidence interval is computed as shown below CI (n- P1 - 2 – 2c1 pi(1 - 1) Ni P2(1 - 2) ,1 - 2 + zc1 N2 P1(1 - i) Ni P2(1 - P2) N 0.21(1 – 0.21) 0.65 -0.21 – 2.326 x 0.65(1 – 0.65) 200 0.21(1-0.21) ,0.65 -0.21 +2.326 x 300 0.65(1 -0.65) 200 300 (0.344, 0.536) Therefore, based on the data provided, the 98% confidence interval for the difference between the population proportions p1 - p2 is 0.344<p<0.536, which indicates that we are 98% confident that the true difference between population proportions is contained by the interval (0.344, 0.536).

The interval not contain zero and also interval has positive range so we conclude that  the men are more safety-conscious than women