Question

Conduct the following test at the alphaequals0.01 level of significance by determining ​(a) the null and alternative​ hypotheses, ​(b) the test​ statistic, and​ (c) the​ P-value. Assume that the samples were obtained independently using simple random sampling. Test whether p 1 not equals p 2. Sample data are x 1 equals 28​, n 1 equals 254​, x 2 equals 38​, and n 2 equals 302. ​(a) Determine the null and alternative hypotheses. Choose the correct answer below. A. Upper H 0 : p 1 equals p 2 versus Upper H 1 : p 1 greater than p 2 B. Upper H 0 : p 1 equals 0 versus Upper H 1 : p 1 equals 0 C. Upper H 0 : p 1 equals p 2 versus Upper H 1 : p 1 not equals p 2 D. Upper H 0 : p 1 equals p 2 versus Upper H 1 : p 1 less than p 2 ​(b) The test statistic z 0 is nothing. ​(Round to two decimal places as​ needed.) ​(c) The​ P-value is nothing. ​(Round to three decimal places as​ needed.) Test the null hypothesis. Choose the correct conclusion below. A. Reject the null hypothesis because there is sufficient evidence to conclude that p 1 not equals p 2. B. Do not reject the null hypothesis because there is sufficient evidence to conclude that p 1 greater than p 2. C. Do not reject the null hypothesis because there is not sufficient evidence to conclude that p 1 not equals p 2. D. Reject the null hypothesis because there is not sufficient evidence to conclude that p 1 less than p 2.

Answer 1

a) H₀: P1 = P2

    H₁: P1 P2

= 28/254 = 0.1102

= 38/302 = 0.1258

b) The pooled sample proportion(P) = ( * n1 + * n2)/(n1 + n2)

                                                         = (0.1102 * 254 + 0.1258 * 302)/(254 + 302)

                                                         = 0.1187

SE = sqrt(P(1 - P)(1/n1 + 1/n2))

     = sqrt(0.1187 * (1 - 0.1187) * (1/254 + 1/302))

    = 0.0275

The test statistic z = ()/SE

                            = (0.1102 - 0.1258)/0.0275 = -0.57

c) P-value = 2 * P(Z < -0.57)

                = 2 * 0.284 = 0.568

D) Since the P-value is greater than the significance level(0.568 > 0.01), so we should not reject the null hypothesis.

Option - C) Do not reject the null hypothesis because there is not sufficient evidence to conclude that P1 not equals to P2.