Consider the hypothesis statement shown below using alpha = 0.05 and the data to
ID: 3227628 • Letter: C
Question
Consider the hypothesis statement shown below using alpha = 0.05 and the data to the right from two independent samples. H_0: mu_1 - mu_2 = 0 H_1: mu_1 - mu_2 notequalto 0 x bar_1 = 241 x bar_2 = 212 sigma_1 = 49 sigma_2 = 60 n_1 = 43 n_2 = 35 a) Calculate the appropriate test statistic and interpret the result. b) Calculate the p-value and interpret the result. a) The test statistic is. Determine the appropriate critical value(s). Select the correct choice below and fill in the answer box to complete your answer. A. The critical values are plusminus B. The critical value is Since the test statistic in the region, H_0. There is evidence to conclude that the mean of population 1 is different from the mean of population 2. b) The p-value is Since the p-value is H_0. There is evidence to conclude that the mean of population 1 is different from the mean of population 2.Explanation / Answer
Solution:-
The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: 1 - 2 = 0
Alternative hypothesis: 1 - 2 0
Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.
Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).
SE = sqrt[(s12/n1) + (s22/n2)]
SE = sqrt[(492/43) + (602/35)] = 12.597
DF = (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 - 1) ] + [ (s22 / n2)2 / (n2 - 1) ] }
DF = (492/43 + 602/35)2 / { [ (492 / 43)2 / (42) ] + [ (602 / 35)2 / (34) ] }
DF = 25183.897 / { 74.233189 + 311.164465786 } = 65.345, i.e., 65
t = [ (x1 - x2) - d ] / SE = [ (241 - 212) - 0 ] / 12.597 = 2.302
where s1 is the standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2, x1 is the mean of sample 1, x2 is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.
Since we have a two-tailed test, the P-value is
The P-Value is 0.024531.
The result is significant at p < 0.05
Interpret results. Since the P-value (0.024531) is less than the significance level (0.05), we cannot accept the null hypothesis.
a) The test statistic is = 2.30
b) Critical value for t at 65 degree of freedom = + 1.997
Since the t statistic does not fall in the rejection region, do not reject Ho. Insufficient evidence to conclude that the mean of population 1 is different from the mean of population 2.
c) The p-value is 0.023
Since the p-value is less than alpha, do not reject Ho. There is insufficient evidence to conclude that the mean of population 1 is different from the mean of population 2.