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# Finding z-score for a percentile

AP.STATS:
VAR‑2 (EU)
,
VAR‑2.B (LO)
,
VAR‑2.B.4 (EK)
CCSS.Math:

## Video transcript

the distribution of resting pulse rates of all students at Santa Maria high school was approximately normal with mean of 80 beats per minute and standard deviation of nine beats per minute the school nurse plans to provide additional screening to students whose resting pulse rates are in the top 30 percent of the students who are tested what is the minimum resting pulse rate at that school for students who will receive additional screening round to the nearest whole number if you feel like you know how to tackle this I encourage you to pause this video and try to work it out all right now let's work this out together they're telling us that the distribution of resting pulse rates are approximately normal so we could use a normal distribution and they tell us several things about this normal distribution they tell us that the mean is 80 beats per minute so that is the mean right over there and it tell us that the standard deviation is nine beats per minute so on this normal distribution we have one standard deviation above the mean two standard deviations above the mean so this distance right over here is nine so this would be 89 this one right over here would be 98 and you could also go standard deviations below the mean this right over here would be 71 this would be 62 but what we're concerned about is the top 30% because that is who is going to be tested so there's going to be some value here some threshold let's say it is right over here that if you are in that if you are at that score you have reached the minimum threshold to get a radition 'el screening you are in the top 30% so that means that this area right over here is going to be 30% or 0.3 so what we can do we can use the Z table to say for what z-score is 70% of the distribution less than that and then we can take that z-score and use the mean and the standard deviation to come up with an actual value in previous exam we started with the z-score and we're looking for the percentage this time we're looking for the percentage we want it to be at least 70% and then come up with the corresponding z-score so let's see immediately when we look at this and we are to the right of the mean and so we're going to have a positive z-score so we're starting at 50% here we definitely want to get the 67% 68-69 we're getting close and on our table this is the lowest z-score that gets us across that 70% threshold it's at zero point seven zero one nine so it definitely crosses the threshold and so that is a z-score of zero point five three zero point five two is too little so we need a z-score of zero point five three let's write that down zero point five three right over there and we just now have to figure out what value gives us a z-score of zero point five three well this just means zero point five three standard deviations above the mean so to get the value we would take our mean and we would add zero point five three standard deviations so zero point five three times nine and this will get us zero point five three times nine is equal to four point seven seven plus 80 is equal to 84 point seven seven eighty four point seven seven and they want us to round to the nearest whole number so we will just round to 85 beats per minute so that's the threshold if you have that resting heart beat then the school nurse is going to give you some additional screening you are in the top 30 percent of students who are tested
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