Question

Conduct a test at the a = 0.10 level of significance by determining (a) the null and alternative hypotheses, (b) the test statistic, and (c) the P-value. Assume the samples were obtained independently from a large population using simple random sampling. Test whether 21 > P2. The sample data are x1 = 127, ny = 241, X2 = 131, and n2 = 318. (a) Choose the correct null and alternative hypotheses below. O A. Ho: P1 = P2 versus Hy: P1 P2 OB. Ho: P1 = P2 versus Hy: P1 P2 OC. Ho: P1 = 0 versus Hy: P170 OD. Ho: P1 = P2 versus H: P1 *P2 (b) Determine the test statistic. Zo (Round to two decimal places as needed.) (c) Determine the P-value. The P-value is (Round to three decimal places as needed.) What is the result of this hypothesis test? A. Do not reject the null hypothesis because there is not sufficient evidence to conclude that P1 P2. O B. Do not reject the alternative hypothesis because there is sufficient evidence to conclude that P1 * P2-

Answer 1

(a)Here we want to test $p_1>p_2$ . Hence the appropriate hypothesis from the given options is (B) . It is a right tailed test.

(b)Given that

127 131 241, 12 131, n2 = 318, P1 0.5270,P2 21 = 127, n1 0.4120 241 318

Let $P=\frac{x_1+x_2}{n_1+n_2}=\frac{127+131}{241+318}=0.46154$ .

The test statistic is given by, .

Pi P2 Z= N(0,1) (PQCH + ))

The value of the test statistic is $Z=\frac{0.5270-0.4120}{\sqrt{\left ( 0.46154\times(1-0.46154)(\frac{1}{241}+\frac{1}{318}) \right )}}=2.701049.$ =2.70

(c) Since the given test is right tailed test and the test statistics follows standard normal distribution the P value of the test is given by,

(d) What is the result of this hypothesis test?

Given that the significance level of the test
a = 0.10
, and P value is 0.0035. Since P value is less than the significance level we reject the null hypothesis and hence $p_1>p_2$ . Hence the correct option is (c) Reject the null hypothesis because there is sufficient evidence to conclude that $p_1>p_2$ .