Question

A study was conducted to determine if 2 types of car batteries lasted the same amount of time. A sample of 50 brand x batteries lasted and average of 3.5 years, with a standard deviation of 10 years. A sample of 40 brand y batteries lasted 3.2 years with a standard deviation of 1.5 years. Test at the 90% confidence level to determine whether or not the brands last the same amount of time. Use a two tailed test of hypothesis, and the critical value method, do not find the p-value. (20)

Answer 1

(n-1)+(12-12 (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: Mi = 12 Ha: Mi 7 H2 This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used. (2) Rejection Region Based on the information provided, the significance level is a = 0.10, and the degrees of freedom are df = 88. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal: Hence, it is found that the critical value for this two-tailed test is te = 1.662, for a = 0.10 and df = 88. The rejection region for this two-tailed test is R= {t: t > 1.662}. (3) Test Statistics Since it is assumed that the population variances are equal, the t- statistic is computed as follows: X - X2 t m) ni+n2 - 2 + 3.5 - 3.2 (50-1)12+(40-1)1.52 1 50+40-2 (50 + = 1.134 10) (4) Decision about the null hypothesis Since it is observed that t = 1.134 <te = 1.662, it is then concluded that the null hypothesis is not rejected. Using the P-value approach: The p-value is p = 0.2597, and since p = 0.2597 > 0.10, it is concluded that the null hypothesis is not rejected. (5) Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean Mi is different than M2, at the 0.10 significance level. Confidence Interval The 90% confidence interval is -0.14 <Mi - M2 < 0.74.

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