1. The Graduate Record Examination (GRE) is a test required for admission to man
ID: 3052034 • Letter: 1
Question
1. The Graduate Record Examination (GRE) is a test required for admission to many U.S. graduate schools. Students' scores on the quantitative portion of the GRE follow a normal distribution with mean 150 and standard deviation 8.8. In addition to other qualifications, score of at least 160 is required for admission to a particular graduate school. For parts k-m, suppose n 16 randomly selected students take the GRE on the same day. k. Describe the sampling distribution of the sample mean for the quantitative GRE Scores for the 16 students. (give the shape, mean, and SD) I. What is the probability that a random sample of 16 students has a mean score on the quantitative portion of the GRE that is less than 147? Would this be an unusual outcome? m. What is the probability that a random sample of 16 students has a mean score on the quantitative portion of the GRE is greater than 147Explanation / Answer
K. Going by the central limit theorem, the distribution of the sample mean will also be normal. Its mean will be same as the population mean, which is, 150. It's SD (standard deviation) will be equal to the population SD divided by the root of sample size, which is, 8.8 / ((16)^0.5) = 2.2
L. The cumulative probability that the random sample of students has a mean score less than 147 can be calculated using the z - score. First we calculate z-score using z = (value - sample mean) / (sample SD) = -1.364. Then from z score table, we can find how much area of the normal distribution lies to the left of the z-score of -1.364, which comes out to be 0.0864 or 8.64 %. As you can see for such a low probability, this is a rare and unusual outcome.
M. We have already calculated z -score pertaining to the test score of 147 (-1.364), and so we just have to find how much area of the normal distribution lies to the right of the z-score of -1.364. Since, the entire area under a standard normal distribution is equal to 1, we can just subtract 0.0864 from 1 to get the answer, which is 0.9137 or 91.37 %
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