Independent random samples, each containing 600 observations, were selected from
ID: 3218865 • Letter: I
Question
Independent random samples, each containing 600 observations, were selected from two binomial populations. The samples from populations 1 and 2 produced 118 and 530 successes, respectively. Test H_0: (p_1 - p_2) = 0 against H_alpha (p_l - p_2) notequalto 0. Use alpha = 0.05 test statistic = _____________ rejection region |z| > ____________ The final conclusion is There is not sufficient evidence to reject the null hypothesis that (p_l - p_2) = 0. We can reject the null hypothesis that (p_l - p_2) = 0 and support that (p_l - p_2) notequalto 0. Test H_0: (p_l - p_2) lessthanorequalto 0 against H_alpha: (p_l - p_2) > 0. Use alpha = 0.04 test statistic = ______________rejection region z > ____________ The final conclustion is We can reject the null hypothesis that (p_l - p_2) lessthanorequalto 0 and support that (p_l - p_2) > 0. There is not sufficient evidence to reject the null hypothesis that (p_l - p_2) = 0.Explanation / Answer
Solution:-
a)
p1 = 118/600
p1 = 0.1967
p2 = 530/600
p2 = 0.883
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: P1 - P2 = 0
Alternative hypothesis: P1 - P2 0
Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the proportion from population 1 is too big or if it is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a two-proportion z-test.
Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).
p = (p1 * n1 + p2 * n2) / (n1 + n2)
p = 0.54
SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }
S.E = 0.0288
z = (p1 - p2) / SE
z = - 23.84
zcritical = + 1.96
where p1 is the sample proportion in sample 1, where p2 is the sample proportion in sample 2, n1 is the size of sample 1, and n2 is the size of sample 2.
Interpret results. Since z value is less than the critical value , we cannot accept the null hypothesis.
Hence we have sufficient evidence that there is significant difference in peoportion of two groups.
b)
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: P1 - P2< 0
Alternative hypothesis: P1 - P2 > 0
Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the proportion from population 1 is too big or if it is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.04. The test method is a two-proportion z-test.
Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).
p = (p1 * n1 + p2 * n2) / (n1 + n2)
p = 0.54
SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }
S.E = 0.0288
z = (p1 - p2) / SE
z = - 23.84
zcritical = - 2.06
where p1 is the sample proportion in sample 1, where p2 is the sample proportion in sample 2, n1 is the size of sample 1, and n2 is the size of sample 2.
Interpret results. Since z value is less than the critical value , we cannot accept the null hypothesis.