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Problem 21 Hard Difficulty

The College Board reported the following mean scores for the three parts of the Scholas-
tic Aptitude Test (SAT) (The World Almanac, 2009):
$\begin{array}{ll}{\text { Critical Reading }} & {502} \\ {\text { Mathematics }} & {515} \\ {\text { Writing }} & {494}\end{array}$
Assume that the population standard deviation on each part of the test is $\sigma=100$ .
a. What is the probability that a random sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 502 on the Critical Reading part of the test?
b. What is the probability that a random sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 515 on the Mathematics part of the test? Compare this probability to the value computed in part (a).
c. What is the probability that a random sample of 100 test takers will provide a sample mean test score within 10 of the population mean of 494 on the writing part of the test? Comment on the differences between this probability and the values computed in parts (a) and (b).

Answer

a. 0.6578
b. 0.6578
c. 0.6826 , Larger, due to the larger sample size.

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Video Transcript

all right, We're getting some statistics about the population. Means score on the critical reading, math and writing sections of the S A T. And we're also given that the population standard deviation for all three of these categories is 100. Now, for each of these parts were goingto excuse me, we're giving a sample size a section, and then we're supposed to find the probability that our point estimate for the mean will fall under a certain interval. So for part A, our sample size is 90. We're looking at the creek reading section and our error, the error. We're trying to find the probability of its plus or minus 10. Really? Technically, none of this writing and math and critical reading stuff the scores don't matter because we're only looking at the error. So yeah, keep that in mind anyway. So let's start with our standard deviation of the sampling distribution. Ah, I I think it's safe to say that 90 is small enough compared to all the people who take the S a T that we can just assume better. Our population sizes relatively infinite, so we're just going to use that formula, so it's gonna be 100 over the square root of 90. This equals about 10.54 Now we're going to Z score the upper and lower bounds. So negative. 10. Get See. Score to approximately 0.95 on lowered snake. It's positive. 10. Not negative. Over 10.54 Okay. Looking at our table since by central limit their, um this is big enough that we can just assume it's normal. Billy, on the lower end of zero point 1711 Probability on the upper end is 0.82 uh, 89 We find our probability by subtracting the upper limit minus the lower limit. And this equals zero. Oops. 0.6578 All right, party sample sizes still 90. Now, we're looking at the math section and we're still looking for an air of 10. But you probably noticed, in part A. We never used this bit of information here. Like the population mean never came into it, so we don't need to worry about it. Fact, we don't even need to worry about it here or here. It's completely irrelevant to the question anyway, so let's do this. Our standard deviation of the sampling. Distribution is also 10.54 And you might notice this means literally. All the math is the same. So for a gravity sake, I'm not gonna write it out again. This probability is 0.6578 again. All right. We were looking at the writing section. Not that it matters. Our sample size is 100 we're still looking for an error of plus or minus 10. So let's find our standard deviation with the sampling distribution. This time, it will be different. Uh, well, this isn't even approximately. It's ah 100 divided by 10. So this is 10. All right, let's see. Score the upper and lower. But if the standard deviation is 10 and we're looking for an error of plus or minus 10 batches means not ours. E lower is negative one and ours ear upper is one. So this meat So what? Z equals negative one. Our probability is 0.1587 And when Z equals one, our probability is equal to 0.84 13 It's probability lower probability, upper so probably upper minus probability. Lower equals 0.8413 13 There we go, minus 0.1587 This equals 0.68 to 7. And this should make sense because of the 68 95 99.7 role that 68% uh, is congruent with what we know about the set of points between negative one and one standard deviation. That means now the problem asks us to analyze the standard deviation and compare it two parts A and B, and we notice it's larger. And this makes sense because we looked at a larger sample. So yeah, that means we're having more. We have a wider standard deviation here for four hour point estimate for the meat, so they don't lead to a larger probability, and there you have it.