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Problem 68 Hard Difficulty

The diameter of each cork, as described in Applied Example $6.13,$ is measured in several places and an average diameter is reported for the cork. The average diameter has a normal distribution with a mean of $24.0 \mathrm{mm}$ and standard deviation of $0.13 \mathrm{mm}.$
a. The specs for this variable, given in Applied Example $6.13,$ were $"24 \mathrm{mm}+0.6 \mathrm{mm} /-0.4 \mathrm{mm}"$ Express these specs as an interval.
b. What percentage of the corks is expected to fall within the specs?
c. What percentage of the tested corks will have an average diameter of more than 24.5 millimeters?
d. What percentage of the tested corks will have an average diameter within 0.35 millimeter of 24?

Answer

a. from $23.6 \mathrm{~mm}$ to $24.6 \mathrm{~mm}$
b. $99.90 \%$
c. $100 \%$
d. $99.29 \%$

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Video Transcript

So in this problem, we're talking about the diameter of corks, and it is recorded that the average diameter is 24.0 millimeters on the standard deviation is 0.13 millimeters. It's also reported in the problem that we are dealing with a normal distribution. So we do know at some point we're going to use our bell shaped curve. So that's two problem or part A. In part, A. They give you some notation 24 millimeters plus 0.6 millimeters, and then they use the division sign negative 0.4 millimeters. What this notation means is that you're going to find an upper limit and a lower limit of these corks. So what we're going to do is we're gonna take that 24 we're gonna add the 240.6 and get 24.6, and we're going to take that 24. We're going to subtract the point for getting us to a 23.6. So the interval that the average cork diameter could be is anywhere from 23.6 up to 24.6. They're often times we show intervals in a parenthetical notation, and that would look like this. 23.6 comma, 24.6. So let's do part B in part B. You are asked to find the percentage of corks that fall within the specs from Part A. So, in other words, we're going to find the probability that the corks are greater than 23.6, but less than 24.6. So we are going to want to set up our bell and in the center of that bell, we're going to put the average of 24. So the 23.6 is going to be over here on the left, and the 24.6 is going to be over here on the right. Now we will need to find a Z score for each of those values. So let's refresh our memories on what the Z score formula looks like. Z equals X minus mu over sigma, So the Z score associated with 23.6 will be 23.6, minus 24 divided by the standard deviation of 0.13 and that's going to lead us to a negative 3.8 So that's the left side here. And then we also want to do the Z score for the 24.6. So we're gonna do 24.6, minus 24 defied by the standard deviation of 0.13 So the Z score for the right side is going to be a 4.62 So when the problem is asking us to find the probability that the quirk diameter is between the 23.6 on the 24.6, we can also state that as the probability that disease score is between negative 3.8 and 4.62 and we will need to first find the probability that Z is less than 4.62 and then we're going to subtract from it the probability that Z is less than negative 3.8 So we are going to need the standard normal table in the back of your textbook. And when you look at 4.62 it's off the chart. Um, if you look at the last number in the chart, you see that it's really close to one. So this is a number close 21 so you can kind of think of it as 0.99999 and then we're subtracting from that a 0.10 So what we're going to do when we subtract that we're going to get a number just slightly less than 0.999? So therefore, the probability that the corks fall within these specs will be slightly smaller than 0.999 Let's go to part C of this problem imports. See of this problem? Um, actually, let's go back to Part B. One moment. Um, that's the probability and the percentage with the question asked for the percentage so the percentage would be 99.9% were just slightly less than that. Now we'll go to part C in part C. It's asking us what percentage of tested corks will average, um, mawr than 24.5 millimeters. So this is asking us for the probability that X is greater than 24.5. So again, we're going to start with our bell shaped curve and 24 is in the center. So 24.5 is going to be to the right of that. So we'll find our Z score by doing 24.5 minus 24. Divide by the standard deviation, which was 0.13 So in this part of the problem, our Z score turns out to be 3.85 So when we're talking about being a diameter greater than 24.5, we can also say that that's the same thing as a Z score being greater than 3.85 And since our distribution table in the back of the book always takes the area into the left tail will have to do one minus the probability that Z is less than 3.85 And using that book 3.85 um, is going to be again off the charts, so it's going to be really close toe one. So let's put a number there. Let's say it was 10.99999 So that means the result would be under 0.1 Or we would say what percentage of the tested corks have a diameter of more than 24.5. It would end up being less than 0.1% And then finally, for Part D in part D of this problem, it was to know what percentage of the tested corks will have an average diameter within 0.35 millimeters of 24. So here's 24. So it's saying we can go 240.35 millimeters in either direction. So if I did 24 points four plus 40.35 on this end, I'd be a 24.35 and on the other end I would be at 23.65 So the question is basically asking me, What's the probability that X falls between 23.65 and 24.35 so we will use our standard curve again. We're going to put the 24 in this center, and this time we're going to the left to 23.65 and to the right to 24.35 We will need Z scores for both. So 23.65 minus 24 over the standard deviation of 0.13 gets a Z score of negative 2.69 and the other Z score for 24.35 we will subtract 24 and divide by the standard deviation, and we will get a C score of positive 2.69 So when we're talking about having a cork diameter somewhere between 23.65 and 24.65 it's also like we're saying, the probability that the Z score will be between negative 2.69 and positive 2.6 times. So we're going to first calculate the probability that Z is less than 2.69 And from that we're going to subtract the probability that Z is less than negative 2.69 So you can you will rely on the table in the back of your book. The standard normal table tells us that the probability that Z is less than 2.69 is 0.9964 and the probability that Z is less than negative 2.69 is 0.36 So the probability that the cork diameter falls in that range is 0.99 to 8, which says that 99 0.28% of the tested corks will have an average diameter within 0.35 millimeters of 24