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Standard normal table for proportion above

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

Video transcript

a set of philosophy exam scores are normally distributed with the mean of 40 points and a standard deviation of 3 points Ludwig got a score of 40 7.5 points on the exam what proportion of exam scores are higher than Ludwig's score give your answer correct to 4 decimal places so let's just visualize what's going on here so the scores are normally distributed so it would look like this so the distribution would look something like that trying to make that pretty symmetric looking the mean is 40 points so that would be 40 points right over there standard deviation is 3 points so this could be one standard deviation above the mean that would be one standard deviation below the mean and once again on this is just very rough and so this would be 43 this would be 37 right over here and they say Ludwig got a score of 40 7.5 points on the exam so Ludwig's score is going to be someplace around here so Ludwig got a 40 7.5 on the on the exam and they're saying what proportion of exam scores are higher than Ludwig's score so we need to do is figure out what is the area under the normal distribution curve that is above 40 7.5 so the way we will tackle this is we will figure out the z-score for 40 7.5 how many standard deviations above the mean is that then we will look at a Z table to figure out what proportion is below that because that's what Z tables give us they give us the proportion that is below a certain z-score and then we could take one minus that to figure out the proportion that is above remember the entire area under the curve is 1 so if we can figure out this orange area and take one minus that we're going to get the red area so let's do that so first of all let's figure out the z-score for forty seven point five so let's see we would take 47 point five and we would subtract the mean so this is his score we'll subtract the mean minus 40 we know what that's going to be at seven point five so that's how much more above the mean but how many standard deviations is that well each standard deviation is three so what's 7 point 5 divided by 3 this just means the previous answer divided by 3 so he is 2.5 standard deviations above the mean so the z-score here z-score here is a positive 2.5 if you is below the mean it would be a negative so now we can look at a Z table to figure out what proportion is less than 2.5 standard deviations above the mean so that'll give us that Orange and then we'll subtract that from 1 so let's get our Z table so here we go and we've already done this in previous videos but what's going on here is this left column gives us our z-score up to the tenths place and then these other columns give us the hundreds place so what we want to do is find 2 point 5 right over here on the left and it's actually gonna be 2 point 5 0 there's no there's 0 hundredths here so we're going to we're want to look up to point 5 0 so let me scroll my Z table so I'm going to go down to 2 point 5 all right I think I am there so what I have here so I have 2.5 so I am going to be in this row and it's now scrolled off but this first column we saw this is 2 this is the hundreds place and this is 0 hundreds and so 2.50 puts us right over here now you might be tempted to say okay that's the proportion that scores higher than ludwick but you'd be wrong this is the proportion that scores lower than Ludwik so what we want to do is take 1 minus this value so let me get my calculator out again so what I'm going to do is I'm going to take 1 minus this 1 minus 0.99 3/8 is equal to now this is so this is the proportion that scores less than Ludwig one minus that is going to be the proportion that scores more than him the reason why we had to do this is because the z-table gives us the proportion less than a certain z-score so this gives us right over here zero point zero zero six - so that's the proportion if you thought of in percent it would be 0.6 two percent scores higher than Ludwick and that makes sense because Ludwik scored over two standard deviations on two and a half standard deviations above the mean so our answer here zero point zero zero six two so this is going to be zero point zero zero six two that's the proportion of exam scores higher than Ludwig's score
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