Question

It is important that face masks used by firefighters be able to withstand high temperatures because firefighters commonly work in temperatures of 200–500°F. In a test of one type of mask, 11 of 55 masks had lenses pop out at 250°. Construct a 90% upper confidence bound for the true proportion of masks of this type whose lenses would pop out at 250°.

(a) Answer the question as given except find the 90% confidence interval estimate, not the 90% upper confidence bound.

(b) If we wanted a 90% confidence interval of width 10%, what sample size is required? (i) Assume we have no idea what the sample proportion will be. (ii) Assume we know the sample proportion will be about 20%.

Answer 1

First we construct the upper bound 90% confidence interval here:

From standard normal tables, we get:

P(Z < 1.282 ) = 0.9

The sample proportion here is computed as: p = 11/55 = 0.2

Therefore the upper bound here is computed as:

a) From standard normal tables, we get:

P(-1.65 < Z < 1.65 ) = 0.9

Therefore the confidence interval here is obtained as:

This is the required confidence interval here.

b) (i) For no prior proportion, we take p = 0.5 to be on the safe side.

The interval width is 10% which means margin of error is 10/2 = 5% that is 0.05

The margin of error is computed as:

Therefore 271 is the minimum sample size required here.

ii) Here, we are given a prior proportion value of p = 0.2, therefore the minimum sample size computed here is:

Therefore 174 is the minimum sample size required here.