The capacities (in ampere-hours) of 258 batteries were measured. The average was

Question

The capacities (in ampere-hours) of 258 batteries were measured. The average was 181.5, and the standard deviation was 15.1. a) Find a 92% confidence interval for the true average capacity of the batteries. Express your answer accurate to three decimal places. b) Suppose that it's desired to have a 92% confidence interval whose margin of error (or half-width) is only 1 ampere-hour. Find the number of additional batteries that have to be tested (in other words, not counting the 258 batteries that have already been tested) to obtain this margin of error.

Explanation / Answer

sample size, n = 258
mean = 181.5
std. dev. = 15.1

(A)
For 92% CI, z-value = 1.75
lower limit = 181.5 - 1.75*15.1/sqrt(258) = 179.8549
upper limit = 181.5 + 1.75*15.1/sqrt(258) = 183.1452

hence 92% CI = (179.8549, 183.1452)

(B)
Margin of error = 1
ME = z*SE
1 = 1.75 * 15.1/sqrt(n)
n = (1.75 * 15.1)^2
n = 698.2806 (698)

Hence 698 - 258 = 440 more batteries are required to be tested.

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