Question

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 heel 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 10³ miles) and the variance was 7.75 (in 10⁶ miles²). For the cars with brand B, the corresponding statistics were 32.92 (in 10³ miles) and 6.47 (in 10⁶ miles²) respectively. The mileage before heel and toe wear is detectable is assumed to be Normally distributed for both brands.

Calculate the pooled variance s² 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.

Answer 1

The pooled variance for the two brands of tires, taking into account their sample sizes and individual variances, is calculated to be 7.159 x 10¶ miles² to three decimal places.

Step-by-step explanation:

To calculate the pooled variance (s²) for the two brands of tires, Brand A and Brand B, we first need to understand that pooled variance is a weighted average of the variances from two or more groups. Since the groups have different sample sizes, each group's variance contributes proportionally to the pooled variance.

For Brand A, there were 15 cars with a mean mileage of 24.99 x 10³ miles and a variance of 7.75 x 10¶ miles². For Brand B, there were 13 cars with a mean mileage of 32.92 x 10³ miles and a variance of 6.47 x 10¶ miles².

The formula for pooled variance is:

s² = ((n1 - 1)*s1² + (n2 - 1)*s2²) / (n1 + n2 - 2)

where:

n1 and n2 are the sample sizes of the two groups

s1² and s2² are the variances of the two groups

Let's calculate it:

s² = ((15 - 1)*7.75 + (13 - 1)*6.47) / (15 + 13 - 2)

s² = (14*7.75 + 12*6.47) / 26

s² = (108.5 + 77.64) / 26

s² = 186.14 / 26

s² = 7.159

Thus, the pooled variance s² to 3 decimal places is 7.159 x 10¶ miles².