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# More empirical rule and z-score practice (from ck12.org)

## Video transcript

never hurts to get a bit more practice so this is problem number five from the normal distribution chapter from ck-12 dot orgs ap statistics flex book 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 one point three four they cite some college board stuff here I didn't copy and paste that what is the approximate z-score remember z-scores just how many standard how many standard deviations you are away from the mean what is the approximate z-score that corresponds to an exam score of five so we really just have to figure out this is a pretty straightforward problem we just need to figure out how how many standard deviations is five from the mean well five well it's one you just take five 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 right so the mean is 2.8 so 5 minus 2.8 is equal to two point two 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 we divide by one point three for one point three four divided by one point three four I'll take out the calculator for this so we have two point two divided by one point three four is equal to one point six four so this is this is equal to one point six four 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 five 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 five it's one point six four 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 be it could apply to any distribution because that you can 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 I guess that's a good point of clarification to get out of the way I thought I would do two problems in this video just because that one was pretty short so problem number 6 the heights of fifth-grade boys in the United States is approximately normally distributed that's good to know with the mean height of one hundred and forty three point five centimeters so it's a mean of 140 three point five centimeters and a standard a standard deviation of about seven point one centimeters standard deviation of seven point one centimeters what is the probability that a randomly chosen fifth grade boy would be taller than 150 seven point seven centimeters so let's just draw out this distribution like we've done in a bunch of problems so far there's asking us one question so you 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 one forty three point five they're asking is taller than one fifty-seven point seven so we're going the upwards direction so one standard deviation above the mean will take us right there and we just have to add seven point one to this number right here we're going up by seven point one so one forty three point five plus seven point one is what 150 point six one fifty point two six that's one standard deviation if we were to go another standard deviation we go seven point one more with seven point one plus one fifty point six it's one fifty seven point seven which just happens to be is the exact number they ask for they're asking for Heights 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 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 the area let me pick a different color so that I don't so we know what this we know what this area is the area within two standard deviation the empirical rule tells us or even better the 6895 99.7 rule tells us that this area because it's within two standard deviations is 95% or 0.95 or it's 95 percent 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 five percent so those two combined have to be 5% and these are symmetrical we've done this before this is actually redundant from other problems we've done but if these are added combined five percent and they're the same then each of these are two and a half percent each of these are two and a half percent so two the answer the question what is the probability that a randomly chosen 5th grade boy would be taller than 150 seven point seven 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 and we just figure it out it's 2.5 percent so there's a two and a half percent chance we'd randomly find a fifth-grade boy who's taller than 150 seven point seven centimeters assuming this is the mean the standard deviation and we are dealing with a normal distribution