How much does width grow from 90% to 95%?

Question

A quality team samples n = 64 parts with mean \bar{x} = 250 and known standard deviation \sigma = 16. Construct the 90% and 95% intervals for the mean, and report by how much the width increases going from 90% to 95%.

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Setup. The shared standard error is \sigma/\sqrt{n} = 16/8 = 2. Critical values: 1.645 at 90% and 1.96 at 95%. Compute the intervals. 90\%:\ 250 \pm 1.645(2) = [246.71,\ 253.29], \quad 	ext{width } 6.58. 95\%:\ 250 \pm 1.96(2) = [246.08,\ 253.92], \quad 	ext{width } 7.84. The increase. The width grows by 7.84 - 6.58 = 1.26. Interpret. Reported as a ratio this is 7.84/6.58 \approx 1.19, matching the 1.96/1.645 rule; reported additively it is 1.26 units of width. Takeaway. You can describe the cost of extra confidence either as an additive width increase or a multiplicative ratio. The ratio i…

How much does width grow from 90% to 95%?

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