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AP Stats: VAR‑2 (EU), VAR‑2.B (LO), VAR‑2.B.3 (EK)

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- [Instructor] A set of middle school students' heights are normally distributed with a mean of 150 centimesters and a standard deviation of 20 centimeters. Darnell is a middle school school student with a height of 161.4 centimeters. What proportion of student heights are lower than Darnell's height? So, let's think about what they are asking. So, they're saying that heights are normally distributed. So, it would have a shape that looks something like that. That's my hand-drawn version of it. There's a mean of 150 centimeters, so right over here. That would be 150 centimeters. They tell us that there's a standard deviation of 20 centimeters and Darnell has a height of 161.4 centimeters. So, Darnell is above the mean, so let's say he is right over here, and I'm not drawing it exactly, but you get the idea. That is 161.4 centimeters. And we wanna figure out what proportion of students' heights are lower than Darnell's height. So, we wanna figure out what is the area under the normal curve right over here. That will give us the proportion that are lower than Darnell's height. And I'll give you a hint on how to do this. We need to think about how many standard deviations above the mean is Darnell, and we can do that because they tell us what the standard deviation is and we know the difference between Darnell's height and the mean height, and then once we know how many standard deviations he is above the mean, that's our z score, we can look at a z table to tell us what proportion is less than that amount in a normal distribution. So let's do that. So, I have my TI-84 Emulator right over here and let's see. Darnell is 161.4 centimeters, 161.4. Now, the mean is 150, minus 150 is equal to, we coulda done that in our head, 11.4 centimeters. Now, how many standard deviations is that above the mean? Well, they tell us that the standard deviation in this case for this distribution is 20 centimeters, so we'll take 11.4 divided by 20. So, we will just take our previous answer, so this just means our previous answer divided by 20 centimeters and that gets us 0.57. So we can say that this is 0.57 standard deviations, deviations above the mean. Now, why is that useful? Well, you could take this z score right over here and look at a z table to figure out what proportion is less than 0.57 standard deviations above the mean. So, let's get a z table over here. So, what we're going to do is we're gonna look up this z score on this table, and the way that you do it, this first column, each rows tells us our z score up until the tenths place and then each of these columns after that tell us which hundredths we're in. So, 0.57, the tenths place is right over here, so we're going to be in this row. And then our hundredths place is this seven, so we'll look right over here. So, 0.57, this tells us the proportion that is lower than 0.57 standard deviations above the mean, and so it is 0.7157 or another way to think about is if the heights are truly normally distributed, 71.57% of the students would have a height less than Darnell's. But the answer to this question, what proportion of students' heights are lower than Darnell's height? Well, that would be 0.7157, and they want our answer to four decimal places, which is exactly what we have done.