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(2 points) The scores of students on the practicesubject at a certain high school had a normal distributionwith mean ?=549.9 and standarddeviation ?=26.1?=26.1.(a) ...

Question

(2 points) The scores of students on the practicesubject at a certain high school had a normal distributionwith mean ?=549.9 and standarddeviation ?=26.1?=26.1.(a) What is the probability that a single student randomlychosen from all those taking the subject scores 553 orhigher?ANSWER: For parts (b) through (d), consider a simple random sample (SRS) of35 students who took the test.(b) What are the mean and standard deviation of the sample meanscore x¯, of 35 students?The mean of the sampling dis

(2 points) The scores of students on the practice subject at a certain high school had a normal distribution with mean ?=549.9 and standard deviation ?=26.1?=26.1. (a) What is the probability that a single student randomly chosen from all those taking the subject scores 553 or higher? ANSWER: For parts (b) through (d), consider a simple random sample (SRS) of 35 students who took the test. (b) What are the mean and standard deviation of the sample mean score x¯, of 35 students? The mean of the sampling distribution for x¯ is: The standard deviation of the sampling distribution for x¯ is: (c) What z-score corresponds to the mean score x¯ of 553? ANSWER: (d) What is the probability that the mean score x¯x¯ of these students is 553 or higher? ANSWER:



Answers

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).

The S A T is an exam used by colleges and universities to evaluate undergraduate applicants. The test goes on our distributed, the mean was 21.1. I have immune as 21.1 and the standard deviation was 5.3. My Sigma is equal to 5.3 this course of four students selected at random for 15 22 9 and 35. So then mean turns out to be 20.25 This means my expert is equal to 20.25 What is the formula for Z score? Z score is X bar minus meal, which is 20.25 minus 21.1, which is minus 0.85 upon Sigma sigma is 5.3. So minus 0.85 divided by 5.3. This turns out to be minus 0.16 And if I show this on a normal distribution, this is my normal distribution. Z minus 0.16 will be somewhere around here. So we see that this is not at all unusual. Minus 0.16 If I find the area to the left of this which is nothing but a P value. This is going to be the values 0.43 So this region has the area's off approximately 0.430 point 43 So no, this is not at all unusual. This is my sister. Distinct. And if I want the area, there is a value. This is the area.

So we have information about the 2005 math s A. T. So it is called Meth S a T. And then I'll subscript at math and we're told that it has a mean of 520 a standard deviation of 115 and that it is basically a normal distribution. It's not exactly normal, but it's very close to one. And in part, A. We want to know what is the Z score. If somebody scores a 7 20 what is their Z score? Well, there is these scores what they got, minus the mean divided by 1 15. And so we see that that difference is 200. So let's take 200 divided by 115. And we find out that that is 1.7 roughly four. And that means that the person scored 1.74 standard deviations above the mean mhm. On the other hand, let's go the other way that we know that someone has a A Z score of 1.5 or the person scored 1.5 standard deviations above the mean and with them we want to know what was the person score. So remember the Z score is how many standard deviations above or below the mean So we don't know this score. But we know that this is true. And so with this will be multiplying 1.5 this value times 115 and then adding to that by 220 and we find out that the score would be 692.5. Now you can't get a 0.5 on the test, so it's either going to be a 6 92 or a 6 93. And Park Si is doing a comparison and they're dealing with the 2012 exam. And we know that the, uh, the S A T, assuming that it was math again, had a mean for that year of 514 and the standard deviation of 117 very similar to the previous when we looked at and the A C t, on the other hand, as a much lower point total. But the mean that year was 21 with a standard deviation of 5.3, and we have one person who scored on the S a t so we'll do that in blue on the S A T scored a score of 700 and on the other hand, the person on the scored a score of 30 and we want to know, relatively speaking, did which one did better? So let's find out. So let's go through and figure out what this s a T Z value is how many standard deviations above the mean so 700 minus 5 14. Divided by 117. And let me get that left front to see 700 minus 5 14. Divided by 1 17, the C score is 1.59 So this person scored 1.59 Standard deviations above the me. Now let's look at what happens here. What's the Z score here? And we have 30 minus 21 divided by 5.3. And once again, I'll go left front to see 21 books. I'm sorry. 30 right. Minus 21 which I already knew it was nine. I don't know why I don't just put nine divided by 5.3 and I can't do that. 5.3 division in my head and I get this. See, value is 1.698 So this is approximately 1.70 Standard deviations higher. So this person, relatively speaking, did a better job compared to others. Because this person has a score that is 1.7 standard deviations higher than me. And this one is only about 1.6 higher than the main. So this one is the winner.


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