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Section liZscores (8_poluts)Haley has taken an aptitude test given by the career center. Scores on two professions are tallied, Scores for each subtest are normally...

Question

Section liZscores (8_poluts)Haley has taken an aptitude test given by the career center. Scores on two professions are tallied, Scores for each subtest are normally distributed with the following mean and standard deviations.ProfessionMeanStandard DeviationHaley's Raw ScoreSocial Worker 120 20 Psychologist 260 So Calculate Haley' 5 Z-score On1 each subtest,105 290Calculate the percentile rank for each of Haley'$ scores.Which career would you advise Haley to pursue? Why?What percen

Section liZscores (8_poluts) Haley has taken an aptitude test given by the career center. Scores on two professions are tallied, Scores for each subtest are normally distributed with the following mean and standard deviations. Profession Mean Standard Deviation Haley's Raw Score Social Worker 120 20 Psychologist 260 So Calculate Haley' 5 Z-score On1 each subtest, 105 290 Calculate the percentile rank for each of Haley'$ scores. Which career would you advise Haley to pursue? Why? What percentage of scores are above a Z-score of + 1.752 What percentage of scores are below & Z-score of -. 652



Answers

Using the standard normal curve and $z$ : a. Find the minimum score needed to receive an A if the instructor in Example 6.11 said the top $15 \%$ were to get A's. b. Find the 25 th percentile for IQ scores in Example $6.10 .$ c. If SAT scores are normally distributed with a mean of 500 and a standard deviation of $100,$ what score does a student need to at least be considered by a college that takes only students with scores within the top $7 \% ?$

For this question, we're told that scores for an exam are normally distributed and have a mean score of 525 and a standard deviation of 80. For Part A were asked what percentage of of testers score less than 350 on the exam. We're looking for the probability that X is less than 350 and to convert his head scores were using that is equal to X minus mu over Sigma. So this is equal to the probability that said is less than 350 minus 525 over 80 just equal to the probability that said it is less than minus 2.188 equals 0.144 So the probability of scoring less than 350 is 0.1 for four or, in other words, 0.144 of the students to take this exam score less than 350 for Part B, whereas what score is needed to make the top 12%. So we're looking for a score K, such that scoring higher than it is equal to 12% or a 0.12 probability of scoring higher than K is equal to 0.12 Another way to state this is to say the probability of Zed being less than K minus 525 over 80 is equal to 0.12 So you can look in the standard normal table or use a calculator or software to find the said value that has a cumulative area of 0.12 And that gives us value a zed score of 1.175 So we can say that K minus 525 over 80 is equal to 1.175 and therefore K is equal to 619 So you must score higher than 619 in order to be in the top. 12% of testers report see, were asked what the inter quartile range is. So the inter courthouse range is the range from the first quartile to the third, so the range is the difference of these court tiles. I remember the probability that of scoring less than the first quartile is equal to 0.25 So we can say that the probability of Zed being less than Q one minus 525 over 80 is equity 0.25 So we have Q one minus 525 over 80 is equal to 1.175 So this 1.175 is this ed score that has a cumulative area of 0.25 So therefore, yes is equal to you. This which were showing in this equation here. So then isolating Q one we get Q one is equal to 471 0.8 and then we can do the same thing for the beard quartile. But the probability that said is less than here three minus 525 over 80 is equal to 0.75 which tells us that Q three minus 525 over 80 is equal to zero point 67 four and then isolating que gives us 578 0.92 So those were the bounds on our intercourse. How range so to find the size of the range, we just subtract those two values where we subtract Q one from Q three and we get 107 0.84 So this is the answer now for Part D. Were asked, What grade do you need so that only one out of 500 score above it? The one out of 500 is equal to 0.2 So we want to find the probability of scoring greater than a value K being only equal to 0.2 So what value of K such that the probability of scoring higher than K is equal to 0.2 So we can say the probability of Zed being less than K minus 525 over 80 is equal to 0.2 This should be three zeros in here. Well, sorry. No, it's it's 200.0 to Okay, we're good to go. So we can say that K minus 525 over 80 is equal to you. 2.878 And that gives us K is equal to you. 755.2. So you must score higher than 755.2, such that only one out of 500 score that high

And this problem, we have six test scores only are asked to find the percentile that corresponds to each you'll be using this formula. P equals number of numbers lower plus 0.5 over end. Where n is the total number of numbers Times 100 to convert 12%. So we need to identify how maney numbers we have. And we said this is six test scores. So for this problem and equal six, all right. So to calculate the first percentile for the corresponds to test score five. How Maney data values are lower than five and that answer is zero. So we ads. You're a 0.5 divide by six and multiplied by 100. That turns out to be 8.33333 etcetera. So 8.3 repeating, and that working around them corresponds to about the eighth percentile. So for the data value of 12 we have one data value lower than 12. So we're gonna have one plus 10.5, divide by six and multiply by 100. So 1.5 divided by six uh, and then multiply by 100 is 25. So that corresponds then to 25th percentile. Say I'm going to repeat that same pattern. And for 15 we have two values lower. So 2.5 divided by six times 100 and that gives us 41.6, which is going around then to the 42nd percentile. And we're going to continue in that same fashion for the last three numbers. So we've got three numbers less than 16 and when you compute, that comes out to be 58.3, which rounds to the 58th percentile and then four plus 40.5. I'm just gonna go ahead and write 4.5, divided by six and multiply by 100 and that gives us the 75th percentile. And then our last number. We have five numbers less than that. So five plus 50.5 is 5.5, divided by six multiplied by 100 and that corresponds to or that equals 91.7, which is the 92nd percentile. So, um, you may notice that there's a little bit of a pattern that these air evenly spaced, and that makes sense because if we look at 100% split six ways, we've got six data values. It turns out to be 16 0.6 repeating percent. So if you keep adding 16.6% you'll make your way through that. And that's the pattern. It's in that table, All right. For the last part of the problem, we are asked to find the 33rd percentile. So we need to use the other formula. C equals end times p over 100 where n is equal to the number of numbers and P is the percentile. So in this case, 33 make that a multiplication sign there, um, divided by 100 and that comes out to be 1.98 So that means we're looking for the number in the 1.898 position. Um, since that's not a specific position, we round up when we always round up. So that means we're looking for the number that's in the second position. So if we go back and look at our data, then number 12 is right here. That's in our second position. Our second number when we look at the data organized from smallest to largest. So the 33rd percentile Oops, um, is what was that? 12

So we're gonna answer question number two in your textbook today about I. Q. Scores I'm gonna use stop but to help us get a visual picture of this. So it says that what white slur adult intelligence scale is an I. Q. Score obtained through a test. And the scores are normally distributed with a mean of 100 and a standard deviation of 15, 17 as 100. The standard deviation is 15. A bell shaped graph is drawn to represent this distribution. So here's our bell shaped graph. Part A says for the bell shaped graph, what is the area under the curve? So for any normal distribution which is said in this problem that it is normal, the area under the curve is equal to one Or 100% of the total area under this curve is equal to one. So that's part a Part B says. What is the value of a median of the median? So in a symmetric distribution or in a normal distribution the median will equal the mean. So here the mean is 100. So the median is also 100 For part C. What is the value of the mode again in a symmetric distribution or a normal distribution, the mode will also equal the mean. So the mode of this distribution is 100 and what is the value of the variance? Well, we know the standard deviation is 15, the variance is just the standard deviation squared. So 15 squared is 2 25. So the value of the variance is 225


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