A manufacturer knows that the number of items produced per...

A manufacturer knows that the number of items produced per hour by machine A is normally distributed with a standard deviation of 8.4 items for machine A. The mean hourly amount produced by machine A for a random sample of 40 hours was 130 units. (a) Construct 90% confidence interval per hour by this machine. (b) What is the margin of error per hour? (c) How large a sample is needed if the desired margin of error is 0.25 for the confidence interval of 90%?

Transcript

00:01 First of all, let me take notes the given values here.

00:03 So we're given the population standard division, which was given as 8 .4, and the mean value given here for the sample, this is 130 units, and the sample size given here, which is 40 hours, so the sample size is 40.

00:18 So in the first part of the question, we have to find the confidence interval for the population mean.

00:23 Remember the formula, the confidence interval for the population mean, this is the sample mean, plus or minus, because we know the population standard division, use the z distribution here, and times population standard division divided by root n.

00:37 Let me get the z value first.

00:38 So the alpha is 1 minus confidence level, but i need alpha over 2.

00:42 That means 1 minus, so by the way, the confidence level here given as 90%, we can write as 0 .90.

00:50 So this is 0 .90 divided by 2, which would be 0 .05.

00:54 To get the z value, so i'm going to use the graphing display calculator application inverse norm, so the area, and the mean and the standard division.

01:03 Press second variance and the inverse norm, this is 0 .05, and the mean and the standard division.

01:09 So the z value would be negative 1 .645.

01:13 Let me put these findings on the formula.

01:15 So the sample mean, 130 plus or minus 1 .645, and times the standard division, 8 .4, and divided by square root of the sample size.

01:25 This is 130 plus 1 .645, and this is times 8 .4 divided by the square root of 40.

01:34 So the answer, the upper boundary would be 132 .18.

01:39 Let me get the lower boundary.

01:40 For the lower boundary, so i'm going to use the minus sign here.

01:44 So the answer would be 127 .82...