Finding z-score for a percentile | AP Statistics | Khan Academy

The distribution of resting pulse rates of all students at Santa Maria High School was approximately normal, with a mean of 80 beats per minute and a 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% 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 is approximately normal, so we could use a normal distribution. 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.

They tell us that the standard deviation is 9 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 N9. So this would be 89. This one right over here would be 98.

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 at that score, you have reached the minimum threshold to get additional screening; you are in the top 30%. So that means that this area right over here is going to be 30% or 0.3.

What we can do is use a z-table to say for what z-score is 70% of the distribution less than that. Then we can take that z-score and use the mean and the standard deviation to come up with an actual value.

In previous examples, we started with the z-score and were 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, we are to the right of the mean, and so we're going to have a positive z-score. We're starting at 50% here; we definitely want to get this 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 0.719, so definitely crosses the threshold.

And so that is a z-score of 0.53. 0.52 is too little, so we need a z-score of 0.53. Let's write that down, 0.53 right over there. Now we just have to figure out what value gives us a z-score of 0.53.

Well, this just means 0.53 standard deviations above the mean. So to get the value, we would take our mean and we would add 0.53 standard deviations. So 0.53 times 9 will get us.

0.53 * 9 is equal to 4.77. Plus 80 is equal to 84.77. 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 heartbeat, then the school nurse is going to give you some additional screening; you are in the top 30% of students who are tested.