00:02
We're first asked to construct a 99 % confidence interval to estimate the proportion of all components that are defective.
00:10
So we're going to use this formula where p hat is our sample proportion of 17 out of 200, which equates to 0 .085 as a decimal.
00:22
So i'm going to plug in 0 .085 for all of these p hats in the formula.
00:27
We also know our sample size is n is 200, and at 99 % confidence z, the critical value is 2 .576.
00:45
So now i'm going to grab my calculator and compute the margin of error for this interval.
00:50
The right side of the plus or minus works out to be 0 .0508.
00:58
And now to get the end points of our interval, i'm going to do .085 minus .0508, which is .0342 at the low end, and .085 plus .0508 is .1358 at the high end.
01:18
So there's our 99 % confidence interval.
01:22
Now our margin of error there was the .0508.
01:26
In part b, we want to get a sample size that will make the margin of error be 0 .04.
01:33
So using that part of our formula, the margin of error needs to be 0 .04, and how we get to that margin of error is multiplying our z score, our z critical value, 2 .576, by the square root of our sample proportion, 0 .085 times 1 minus 0 .085, over some sample size, n, which is what we need to figure out.
01:55
So the first thing to do to solve this equation for n is divide both sides by 2 .576.
02:03
So on the left, that's going to make 0 .01552795...