Some car tires can develop what is known as "heel and toe" wear if not rotated after a certain mileage. To assess this issue, a consumer group investigated the tire wear on two brands of tire, A and B, say. Fifteen cars were fitted with new brand A tires and thirteen with brand B tires, the cars assigned to brand at random. (Two cars initially assigned to brand B suffered serious tire faults other than heel and toe wear, and were excluded from the study.) The cars were driven in regular driving conditions, and the mileage at which heal and toe wear could be observed was recorded on each car. For the cars with brand A tires, the mean mileage observed was $24.99$ (in $10^3$ miles ) and the variance was $7.75$ (in $10^6$ miles$^2$). For the cars with brand B, the corresponding statistics were $32.92$ (in $10^3$ miles) and $6.47$ (in $10^6$ miles$^2$ ) respectively. The mileage before heal and toe wear is detectable is assumed to be Normally distributed for both brands.

Part a) Calculate the pooled variance $s^2$ to 3 decimal places. During intermediate steps to arrive at the answer, make sure you keep as many decimal places as possible so that you can achieve the precision required in this question. $\times 10^6$ miles $^2$

Part b) Determine a 95% confidence interval for $\mu_A - \mu_B$, the difference in the mean $10^3$ mileages before heal and toe wear for the two brands of tire. Leave your answer to 2 decimal places. ( ,)

Part c) Based on the 95% confidence interval constructed in the previous part, which of the following conclusions can be drawn when we test $H_0 : \mu_A = \mu_B$ vs. $H_a: \mu_A \ne \mu_B$ with $\alpha = 0.05$.

A. Do not reject $H_0$ since 0 is not in the interval found in part (b).
B. Do not reject $H_0$ since $-7.93$ is within the interval found in part (b).
C. Reject $H_0$ since 0 is not within the interval found in part (b).
D. Reject $H_0$ since 0 is in the interval found in part (b).
E. Do not reject $H_0$ since 0 is within the interval found in part (b).