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Video transcript

- [Instructor] A set of laptop prices are normally distributed with a mean of $750 and a standard deviation of $60. What proportion of laptop prices are between $624 and $768? So let's think about what they are asking. So we have a normal distribution for the prices. So it would look something like this. This is just my hand-drawn sketch of a normal distribution. So it would look something like this. It should be symmetric, so I'm making it as symmetric as I can hand draw it. And we have the mean right in the center. So the mean would be right there. And that is $750. They also tell us that we have a standard deviation of $60. So that means one standard deviation above the mean would be roughly right over here, and that would be 750 plus 60. So that would be $810. One standard deviation below the mean would put us right about there, and that would be 750 minus $60, which would be $690. And then they tell us, what proportion of laptop prices are between $624 and $768. So the lower bound, $624, that's going to actually be more than another standard deviation less, so that's going to be right around here. So that is $624. And 768 would put us right at about right at about there, and once again this is just a hand-drawn sketch. But that is 768. And so what proportion are between those two values? So we wanna find essentially the area under this distribution between these two values. The way we are going to approach it, we're going to figure out the z-score for 768, it's going to be positive because it's above the mean, and then we're going to use a z-table to figure out, what proportion is below 768. So essentially we're going to figure out this entire area. We're even going to figure out the stuff that's below 624. That's what that z-table will give us. Then we'll figure out the z-score for 624. That will be negative 2 point something, and we will use the z-table again to figure out the proportion that is less than that. And so then we can subtract this red area from the proportion that is less than 768 to get this area in between. So let's do that. Let's figure out first the z-score for 768. And then we'll do it for 624. The z-score for 768, I'll write it like that, is going to be 768 minus 750 over the standard deviation, over 60. So this is going to be equal to 18 over 60, which is the same thing as six over, let's see if we divide the numerator and the denominator by three, 6/20, and this is the same thing as 0.30. So that is the z-score for this upper bound. Let's figure out what proportion is less than that. For that, we take out a z-table, get our z-table. And let's see we wanna get 0.30. And so this is 0.3, this first column, and we've done this in other videos, this goes up until the tenths place for our z-score, and then if we wanna go to our hundredths place, that's what these other columns give us. But we're at 0.3, so we're going to be in this row, and our hundredths place is right over here, it's a zero, so this is the proportion that is less than $768. So .6179. So 0.61, 0.6179. So now let's do the same exercise but do it for the proportion that's below $624. The z-score for 624 is going to be equal to 624 minus the mean of 750, all of that over 60. And so what is that going to be? I'll get my calculator out for this one, don't wanna make a careless error. 624 minus 750 is equal to, and then divide by 60, is equal to negative 2.1. So that lower bound is 2.1 standard deviations below the mean, or you could say it has a z-score of negative 2.1. Is equal to negative 2.1. And so to figure out the proportion that is less than that, this red area right over here, we go back to our z-table. And we actually go to the first part of the z-table. So same idea but this starts at negative, a z-score of negative 3.4, 3.4 standard deviations below the mean. And just like we saw before, this is our zero hundredths, one hundredth, two hundredths, so on and so forth. And we wanna go to negative 2.1. We could say negative 2.10 just to be precise. So this is going to get us, let's see negative 2.1, there we go. And so we are negative 2.1, and it's negative 2.10, so we have zero hundredths, so we're gonna be right here on our table. So we see the proportion that is less than 624 is .0179 or 0.0179. So 0.0179. And so if we wanna figure out the proportion that's in between the two, we just subtract this red area from this entire area, the entire proportion that's less than 768 to get what's in between. 0.6179, once again I know I keep repeating it, that's this entire area right over here, and we're gonna subtract out what we have in red, minus 0.0179, so we're gonna subtract this out, to get 0.6. So if we wanna give our answer to four decimal places, it would be 0.6000, or another way to think about it is exactly 60% is between 624 and 768.
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