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

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- [Tutor] A set of philosophy exam scores are normally distributed with a mean of 40 points and a standard deviation of three points. Ludwig got a score of 47.5 points on the exam. What proportion of exam scores are higher than Ludwig's score? Give your answer correct to four 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 three points, so this could be one standard deviation above the mean, that would be one standard deviation below the mean. And once again, 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 47.5 points on the exam. So, Ludwig's score is going to be someplace around here. So, Ludwig got a 47.5 on the exam. And they're saying, what proportion of exam scores are higher than Ludwig's score? So, what we need to do is figure out what is the area under the normal distribution curve that is above 47.5. So, the way we will tackle this is we will figure out the z score for 47.5. 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 can take one minus that to figure out the proportion that is above. Remember, the entire area under the curve is one, so if we can figure out this orange area and take one minus that, we're gonna get the red area. So, let's do that. So, first of all, let's figure out the z score for 47.5. So, let's see. We would take 47.5 and we would subtract the mean. So, this is his score. We'll subtract the mean, minus 40. We know what that's gonna be, that's 7.5. That's how much more above the mean. But how many standard deviations is that? Well, each standard deviation is three, so what's 7.5 divided by three? This just means the previous answer divided by three. So, he is 2.5 standard deviations above the mean. So, the z score here, z score here is a positive 2.5. If he was 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 one. 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 hundredths place. So, what we want to do is find 2.5 right over here on the left, and it's actually gonna be 2.50. There's zero hundredths here. So, we want to look up 2.50. Let me scroll my z table. So, I'm gonna go down to 2.5. Alright, 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 the hundredths place and this zero hundredths. 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 Ludwig, but you'd be wrong. This is the proportion that scores lower than Ludwig. So, what we wanna do is take one minus this value. So, let me get my calculator out again. So, what I'm going to do is I'm going to take one minus this. One minus 0.9938 is equal to, now this is, so this is the proportion that scores less than Ludwig. One minus that is gonna be the proportion that scores more than him. The reason why we have 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, 0.0062. So, that's the proportion. If you thought of it in percent, it would be 0.62% scores higher than Ludwig. Now, that makes sense 'cause Ludwig scored over two standard deviations, two and a half standard deviations above the mean. So, our answer is 0.0062. So, this is going to be 0.0062. That's the proportion of exam scores higher than Ludwig's score.
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