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1. The SAT is a standardized test comprised of two parts: Verbaland Math. A college is interested in the SAT scores of theirincoming class, so they randomly sample...

Question

1. The SAT is a standardized test comprised of two parts: Verbaland Math. A college is interested in the SAT scores of theirincoming class, so they randomly sample 34 scores from admittedstudents, yielding the following statistics: the mean SAT Verbalscore was 585.7 with standard deviation 94.8, and the mean SAT Mathscore was 622.1 with standard deviation 134.2. One departmentwants to know if the average SAT Math score of incoming students isstatistically significantly less than 650 points, so

1. The SAT is a standardized test comprised of two parts: Verbal and Math. A college is interested in the SAT scores of their incoming class, so they randomly sample 34 scores from admitted students, yielding the following statistics: the mean SAT Verbal score was 585.7 with standard deviation 94.8, and the mean SAT Math score was 622.1 with standard deviation 134.2. One department wants to know if the average SAT Math score of incoming students is statistically significantly less than 650 points, so they test this hypothesis. What is the value of their test statistic? 2. Helga knows that her quiz performance for class varies based on how well she sleeps. In general, she estimates that if she sleeps well, there is a 91% chance she will pass a quiz; but if she does not sleep well, there is a 50% chance she will pass a quiz. She gets a good night's sleep 76% of the time. Today Helga passed her class quiz. (Yay!) What is the probability she slept well last night?



Answers

Use the following information to answer the next two exercises: The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days.
In 2005, 1,475,623 students heading to college took the SAT. The distribution of scores in the math section of the SAT follows a normal distribution with mean ? = 520 and standard deviation ? = 115.
a. Calculate the z-score for an SAT score of 720. Interpret it using a complete sentence.
b. What math SAT score is 1.5 standard deviations above the mean? What can you say about this SAT score?
c. For 2012, the SAT math test had a mean of 514 and standard deviation 117. The ACT math test is an alternate to the SAT and is approximately normally distributed with mean 21 and standard deviation 5.3. If one person took the SAT math test and scored 700 and a second person took the ACT math test and scored 30, who did better with respect to the test they took?

Problem. 17. We have a table that shows the number of students would take the set exam or the set test specific number of times, for example. We'll take it one time has are these numbers would take it twice are about 600 students, and so on for birthday will let X be a random variable X, the random variable that represents the number of students. The number of times I still don't think that I said this which has the values 12 34 and five For this part of the problem. We want to show the probability distribution for this random. Very. We can make another table by the random variable X, and it will take these values these values one, 234 fight And we can calculate the probability for X or f of X whatever to make the probability distribution by taking age number and divided by the total, it's a get the total. The dotted equals one million 500 and 18,000 and 859. Then the probability of X equals the number of students at X divided by the total number of students. For example, be of one equals 700 21,000 and 769 divided by the total. This gives 4.475 Then we bought here all 0.475 doing the same for the other values. We take this value and avoid by the tooth this value, then divide it by the throat and so on. We have 4.3 96 then all 0.1 one or then all point, all one five. And finally we have Oh, point oh four. Let's check on the total for the probability the total must be equals. Zero equals one. We have 4.475 plus over 396 That's a 0.11 that's open to 15 US Open two oo four. It equals one then our calculations are correct for Barnaby. What is the probability that the student takes a set more than one time? We want to get the probability for the random variable to be more than one time. We can make the compliment event one minus. The probability for X B equals one. The relative or X equals one is given here. Then it equals one minus 0.4 75 Then the answer equals 4.5 to 5 for birth. C. On. To get the probability for X to be greater than three or more X to be three or more again, we will use the compliment event one minus X equals one minus X equals two. It equals the previous probability minus X equals two is given here 4.396 Then the answer is all boy into 1 to 9 for Borden. We want to get the expected value of the number of the time of time they said is taken. The expected value for the random variable equals some mission for X multiplied by its probability for X equals one 25 which means we will get each value of X, then multiply it by its probability and some the products it equals. One multiplied by 0.475 plus two multiplied by 0.396 plus three multiplied by 0.11 last four multiplied by oh point no. 15 plus five multiplied by 4.4. This gives 1.677 which means it's expected or, on average, students think that's it. About 1.677 times or it's expected for a student entering a state exam to take it from 1 to 2 times for party. What is the variant and the Standard Division? Before getting the variance, we should get the expected value of X squared, which equals the submission for X Square. Deployed by B of X from X equals one 25 It's the same here, but we will square the X values one to avoid one square that obviously plus two squared plus three squared last four squared plus five squared equals three point 389 Then we can get the variance of X equals the expected value of X squared minus expected value of X. All the square equals 3.389 minus 1.677 squared equals Oh boy, 56 Oh, boy. In 577 And now we can get the standard division by taking the square root for the variants equals square root of all boy in 577 which gives a 0.7 five My

In this question we want to test the null hypothesis and U equals 516 against the alternative hypothesis, mute is not equal 516. For example, data with and equals 20 X four equals 522. With a standard deviation of the population segment equal 114. Known at alpha equals 1140.1 confidence of this question is to ask your ability to perform a hypothesis test for population we knew where sigma is. No. The first question to answer is a why is it necessary the population normal? It's necessarily population be normal because you end less than 30. Since we don't have a sufficiently large sample, we must meet the requirement of normality for this. Has to be okay to complete. So proceeding on the tests, we first counted at the test net. This is a non equals experimentation unit which is 5 16 over segment we were talking in the numbers about gives 160.235 Next we can compute the critical value for alpha equals 0.5 from a Z. Table. This gives critical Z score plus or minus 1.645 That's we can include. Indeed, we reject a shot we fail to reject H not because, you know, it's not in the critical region. 0.235 is not less than negative 1.645 or greater than 1.645 Therefore it's not in the critical region.

In this question we want to test the null hypothesis and U equals 516 against the alternative hypothesis, mute is not equal 516. For example, data with and equals 20 X four equals 522. With a standard deviation of the population segment equal 114. Known at alpha equals 1140.1 confidence of this question is to ask your ability to perform a hypothesis test for population we knew where sigma is. No. The first question to answer is a why is it necessary the population normal? It's necessarily population be normal because you end less than 30. Since we don't have a sufficiently large sample, we must meet the requirement of normality for this. Has to be okay to complete. So proceeding on the tests, we first counted at the test net. This is a non equals experimentation unit which is 5 16 over segment we were talking in the numbers about gives 160.235 Next we can compute the critical value for alpha equals 0.5 from a Z. Table. This gives critical Z score plus or minus 1.645 That's we can include. Indeed, we reject a shot we fail to reject H not because, you know, it's not in the critical region. 0.235 is not less than negative 1.645 or greater than 1.645 Therefore it's not in the critical region.

So the school administer administrator thinks that students who have not taken the, um so this is, uh, not having English as their first language that there is worse for their verbal score on the S A. T than those who have English as their first language. Now we would assume that they actually are equal to that group. So we would say that people who are not having first language English has as the first language we would assume that their score is no different. It's just 5 15 now. Alternately, we think it's lower. And that's what the administrator thinks. Uh, not for style Prank? Not first. So it's their second language, Possibly. But it's not their first, and we stick with 5. 15. Remember, these two are never different, and I always have to emphasize with my students, these are always parameters. You will never, never, never, never have an X bar here or a P hat. These are always parameters, which is, um, you or I mean, um, now let me draw a little picture of the distribution. We're assuming to be true, and we are going to assume that the mean of that group is 5 15. And we know that when we have the sample of 20 tests that we get an X bar here, that is 458. Now it's lower. But is it significantly lower for this sampling distribution? And our sampling distribution is from a sample size of 20. Yeah, which means the normal distribution or the distribution of X bars here is going to end up being that value they gave us for the test score, which is for individuals divided by the square root of the sample size. So that's how we think this distribution varies, and this is a sampling distribution now. The first question says, Why is it important that the original distribution is normal? Well, it needs to be normal because we have a small sample size. And with that small sample size for the distribution to be approximately normal, we need to have a relatively large sample size and that usually they'll say something about the sample size being 30. But we want normality because of our small sample size, so now we can end up finding How likely is it if this is our sampling distribution for people who don't have any English as their first language. How likely is it to get a score of 4 58 or lower? And I believe our alpha level is 10% were using a 10% significance level. So how likely is it to get an X bar from a population that looks like this being less than or equal to 458? Now we need to change it to a test statistic, which is a Z value, because we know our population standard deviation and we take that 4 58 minus the 5 15. And then we divide it by 1 12, divided by the square root of 20. And let's see what we get here. I have left parentheses E 4 58 minus the 5 15 divided by the square root of And we have, uh, 1 12 divided by whips. I just made a mistake. I mean, back this up here and I put the square root where I didn't want one. So let me get rid of that. So I need a left parenthesis e and then 1 12 divided by the square root of 20. I started with the square root to begin with. It was good, but ready to mess up. All right. And when we do that, I get my test statistic, which again is the C value is negative. 2.27 would come out to be approximately six. And now I'm going to use my normal CDF button, normal CDF second and distribution. I'm on a t I 84 normal CDF, and I'm gonna put my low limit as negative 1000 and then my upper limit. I'm gonna put as this number and I can do second answer for that since that's stored and my calculator and let's find out what that is, which is not very big. 0.114 So there's my P value now that p value is less than our 10% significance level. Therefore, we have evidence to reject the null. Yeah, Mhm. Yeah. Yeah, you are. Yeah. So we would think that the alternative hypothesis is true and that their score is less than 5 15. And so the administrator does seem to be correct. Seems office Frank two p correct. They do seem to have a lower score on the verbal for the S. A. T.


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