Question

Market research organization selects various car models every year and assesses their fuel efficiency. This year, in the analysis of the two similar subcompact models from two different manufacturers, the average mileage of 12 A-brand cars was 27.2 miles per gallon, and the standard deviation was 3.8 mpg. The 9 B-brand cars a tested were averaging 32.1 mpg with a standard deviation of 4.3 mpg. For a = 0.01, can it be concluded that mark A gives a lower average mileage than mark B? Find the value of p, interpret the result.

Answer 1

(1) Null and Alternative Hypotehsis:

H0: Mark A not gives a lower average mileage than mark B

H1: Mark A gives a lower average mileage than mark B

i.e.

Ho: Mi = 12 Ha: ui <H2 This corresponds to a left-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 = 0.01, and the degrees of freedom are df = 19. 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 left-tailed test is te = -2.539, for a = 0.01 and df = 19. The rejection region for this left-tailed test is R = {t:t < -2.539}. (3) Test Statistics Since it is assumed that the population variances are equal, the t-statistic is computed as follows: X1 - X2 (n-1) +(n -1) ( 1 1 ) V 71-19-224+1) 27.2 -32.1 == -2.766 (12-1)3.8 +(9-14.32 12-9-2** G +) (4) Decision about the null hypothesis Since it is observed that t = -2.766 <te = -2.539, it is then concluded that the null hypothesis is rejected.

(4) Decision about the null hypothesis Since it is observed that t= -2.766 <tc = -2.539, it is then concluded that the null hypothesis is rejected Using the P-value approach: The p-value is p=0.0062, and since p = 0.0062 <0.01, it is concluded that the null hypothesis is rejected. (5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean Mi is less than um, at the 0.01 significance level.

Thus we conclude the there is sufficient evidence that Mark A gives a lower average mileage than mark B