Standard normal table for proportion between values | AP Statistics | Khan Academy

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 dollars?

So let's think about what they are asking. 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. 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 seven hundred and fifty dollars. They also tell us that we have a standard deviation of sixty dollars. So that means one standard deviation above the mean would be roughly right over here, and that would be 750 plus sixty, so that would be eight hundred and ten dollars.

One standard deviation below the mean would put us right about there, and that would be seven hundred and fifty minus sixty dollars, which would be six hundred and ninety dollars. Then they tell us what proportion of laptop prices are between six hundred twenty-four dollars and seven hundred and sixty-eight dollars.

So the lower bound, 624 dollars, 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 there. And once again, this is just a hand-drawn sketch, but that is 768.

What proportion are between those two values? We want to 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.

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.

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 6 over, let's see, if we divide the numerator and the denominator by 3, 6 twentieths.

This is the same thing as zero point three zero, 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 want to get 0.30.

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 want to 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.

This is the proportion that is less than 768 dollars: 0.6179. So, 0.6179. Now let's do the same exercise but do it for the proportion that's below 624 dollars.

The z-score for 624 is going to be equal to 624 minus the mean of 750, all of that over 60. What is that going to be? I'll get my calculator out for this one; don't want to 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. To figure out the proportion that is less than that—this red area right over here—we go back to our z-table.

We'd actually go to the first part of the z-table. Same idea, but this starts at a z-score of negative 3.4, 3.4 standard deviations below the mean. But just like we saw before, this is our zero, hundredths, one hundredth, two hundredth, so on and so forth. We want to go to negative two point one, we could say negative two point one zero just to be precise.

So this is going to get us, let's see, negative 2.1—there we go. We are negative 2.1—it's negative 2.10. So we have zero hundredths, so we're going to be right here on our table. So we see the proportion that is less than 624 is 0.0179 or 0.0179.

So if we want to 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. We're going to subtract out what we have in red: minus 0.0179. So we're going to subtract this out to get 0.6000.

If we want to give our answer to four decimal places, it would be 0.6000, or another way to think about it is exactly 60 percent is between 624 and 768.