An automotive company is considering two types of batteries for its automobile. Sample information on battery life is collected for 20 batteries of type A and 20 batteries of type B. The summary statistics are:

- [tex]\bar{X}_A = 32.91[/tex]
- [tex]\bar{X}_B = 30.47[/tex]
- [tex]s_A = 1.57[/tex]
- [tex]s_B = 1.74[/tex]

Assume the data on each battery are normally distributed.

Find a 95% confidence interval on [tex]\mu_A - \mu_B[/tex].

To find the 95% confidence interval on the difference in means of two types of automotive batteries, calculate the standard error of the difference, find the critical value, and apply these in the confidence interval formula.

Explanation:

The question asks to find a 95% confidence interval on the difference in means (μA - μB) of two types of automotive batteries, given sample means (Xa=32.91, Xb=30.47) and standard deviations (sa=1.57, sb=1.74) for 20 batteries of each type. Assuming the distributions are normal and equal variances, the step by step calculation involves using the formula for the confidence interval of the difference between two independent means.

Step 1: Calculate the standard error of the difference in means

SE = √((sa²/na) + (sb²/nb)) = √((1.57²/20) + (1.74²/20))

Step 2: Find the critical value

For a 95% confidence interval, with degrees of freedom approximated by the smaller of na-1 and nb-1, assume a critical value (z-value or t-value) accordingly.

Step 3: Calculate the confidence interval

CI = (Xa - Xb) ± (critical value * SE)

Plug in the calculated values and complete the calculation to obtain the 95% confidence interval on the difference in the means of the battery life.