H0: MA - MB = 0 (the mean response time of company A does not differ from the mean response time of company B).
Vs.
H1: MA - MB ≠ 0 (the mean response time of company A differs from the mean response time of company B).
For this case study the alternative hypothesis: H1 represents the claim that the mean response time of company A differs from the mean response time of company B.
Decision rule: Reject the null hypothesis; H0 if the z statistics > critical z Value (Conover 2005).
- The hypothesis test is a two tailed test. To compare the mean response time of the two companies, the z test is the correct and appropriate test to use. This is because the random samples of company A and company B are independent (Green 2008). In addition, the sample sizes (n) for the random calls for company A and company B are 50 each, hence, n > 30.
- At the level of significance = 5%, the two tailed test will have the critical values;
Critical z Value: Z0.025 = 1.96 (Lisa 2011). Hence, the critical regions include the region Z > 1.96 or the region Z < -1.96. - The z statistics is for the hypothesis test is given as follows, Z statistic = MA- MB- 0SA2nA+ SB2nB Z statistic = 8.5 – 5.5- 01.8250+ 1.6250 Z statistic = 30.341 Z statistic = 8.81 - Decision: Reject the null hypothesis; H0 since the z statistics > critical z Value (8.81 > 1.96). Thus, the z statistics lies on the critical region.
- At 5% level of significance, reject the null hypothesis; H0 and accept the alternative hypothesis; H1. Hence, the claim that the mean response time of company A differs from the mean response time of company B is TRUE.
- The 95% confidence interval for the difference between the mean response time of company A and the mean response time of company B is given by (Mogull 2004);
d – Z0.025*S (d) < d < d + Z0.025*S (d)
S (d) = standard deviation of the difference MA - MB.
S (d) = 0.341
d = MA - MB = 8.5 – 5.5 = 3
Z0.025 = 1.96
Thus, the 95% confidence interval is as shown follows;
d – Z0.025*S (d) < d < d + Z0.025*S (d)
3 – 1.96 * 0.341 < d < 3 + 1.96 * 0.341
3 – 0.668 < d < 3 + 0.668
2.332 < d < 3.668
The 95% confidence interval for the difference between the mean response time of company A and the mean response time of company B is;
2.332 < d < 3.668
According to the null hypothesis; H0, the difference between the mean response time of company A and the mean response time of company B is d = 0. However, the difference, d = 0, falls outside the 95% confidence interval for the difference; 2.332 < d < 3.668. This, therefore, agrees with the decision to reject the null hypothesis; H0, at significance level, a = 5% and subsequently, conclude that the mean response time of company A differs from the mean response time of company B.
Works Cited
Conover, W.J. Practical Nonparametric Statistics, New York: Wiley & Sons. 2005. Print.
Green, S. B. Using SPSS for Window and Macintosh: Analyzing and Understanding data (5th ed.), Upper Saddle River, NJ: Pearson Prentice Hall. 2008. Print.
Mogull, Robert G. Second-Semester Applied Statistics, Kendall/Hunt Publishing Company. 2004. Print.
. Essentials of Biostatistics in Public Health, Jones & Bartlett Learning. 2011. Print.
