Normal distributions and the empirical rule
More empirical rule and z-score practice (from ck12.org)
It never hurts to get a bit more practice. So this is problem number five from the normal distribution chapter from ck12.org's AP statistics FlexBook. So they're saying, the 2007 AP statistics examination scores were not normally distributed with a mean of 2.8 and a standard deviation of 1.34. They cite some College Board stuff here. I didn't copy and paste that. What is the approximate z-score? Remember, z-score is just how many standard deviations you are away from the mean. What is the approximate z-score that corresponds to an exam score of 5? So we really just have to figure out-- this is a pretty straightforward problem. We just need to figure out how many standard deviations is 5 from the mean? Well, you just take 5 minus 2.8, right? The mean is 2.8. Let me be very clear, mean is 2.8. They give us that. Didn't even have to calculate it. So the mean is 2.8. So 5 minus 2.8 is equal to 2.2. So we're 2.2 above the mean. And if we want that in terms of standard deviations, we just divide by our standard deviation. You divide by 1.34. Divide by 1.34. I'll take out the calculator for this. So we have 2.2 divided by 1.34 is equal to 1.64. So this is equal to 1.64. And that's choice C. So this was actually very straightforward. We just have to see how far away we are from the mean if we get a score of 5-- which hopefully you will get if you're taking the AP statistics exam after watching these videos. And then you divide by the standard deviation to say, how many standard deviations away from the mean is the score of 5? It's 1.64. I think the only tricky thing here might have been, you might have been tempted to pick choice E, which says, the z-score cannot be calculated because the distribution is not normal. And I think the reason why you might have had that temptation is because we've been using z-scores within the context of a normal distribution. But a z-score literally just means how many standard deviations you are away from the mean. It could apply to any distribution that you could calculate a mean and a standard deviation for. So E is not the correct answer. A z-score can apply to a non-normal distribution. So the answer is C. And I guess that's a good point of clarification to get out of the way. And I thought I would do two problems in this video, just because that one was pretty short. So problem number six. The height of fifth grade boys in the United States is approximately normally distributed-- that's good to know-- with a mean height of 143.5 centimeters. So it's a mean of 143.5 centimeters and a standard deviation of about 7.1 centimeters. What is the probability that a randomly chosen fifth grade boy would be taller than 157.7 centimeters? So let's just draw out this distribution like we've done in a bunch of problems so far. They're just asking us one question, so we can mark this distribution up a good bit. Let's say that's our distribution. And the mean here, the mean they told us is 143.5. They're asking us taller than 157.7. So we're going in the upwards direction. So one standard deviation above the mean will take us right there. And we just have to add 7.1 to this number right here. We're going up by 7.1. So 143.5 plus 7.1 is what? 150.6. That's one standard deviation. If we were to go another standard deviation, we'd go 7.1 more. What's 7.1 plus 150.6? It's 157.7, which just happens to be the exact number they ask for. They're asking for the probability of getting a height higher than that. So they want to know, what's the probability that we fall under this area right here? Or essentially more than two standard deviations from the mean. Or above two standard deviations. We can't count this left tail right there. So we can use the empirical rule. If we do our standard deviations to the left, that's one standard deviation, two standard deviations. We know what this whole area is. Let me pick a different color so that I don't. So we know what this area is, the area within two standard deviations. The empirical rule tells us. Or even better, the 68, 95, 99.7 rule tells us that this area-- because it's within two standard deviations-- is 95%, or 0.95. Or it's 95% of the area under the normal distribution. Which tells us that what's left over-- this tail that we care about and this left tail right here-- has to make up the rest of it, or 5%. So those two combined have to be 5%. And these are symmetrical. We've done this before. This is actually a little redundant from other problems we've done. But if these are added, combined 5%, and they're the same, then each of these are 2.5%. Each of these are 2.5%. So the answer to the question, what is the probability that a randomly chosen fifth grade boy would be taller then 157.7 centimeters. Well, that's literally just the area under this right green part. Maybe I'll do it in a different color. This magenta part that I'm coloring right now. That's just that area. We just figured out it's 2.5%. So there's a 2.5% chance we'd randomly find a fifth grade boy who's taller than 157.7 centimeters, assuming this is the mean, the standard deviation, and we are dealing with a normal distribution.