1. The scores of students on the ACT college entrance examination in a recent ye
ID: 1254216 • Letter: 1
Question
1. The scores of students on the ACT college entrance examination in a recent year had a normal distribution with a mean of 18.6 and a standard deviation of 5.9. A simple random sample of 60 students who took the exam is selected for study:a) What is the shape, mean(expected value), and standard deviation of the sampling distribution of the sample mean score for samples of size 60?
b) What is the probability that the sample mean is 20 or higher?
c) What is the probability that the sample mean falls within 2 points of the population mean?
d) What value does the sample mean have be in order for it to be in the top 1% of the sampling distribution?
Explanation / Answer
(a) What is the probability that a randomly chosen student scores 21 or higher? SOLUTION: z = (21 - 18.6)/5.9 = 0.407. The right-tail prob. is .3421. (b) Now take an SRS of 50 students who took the test. What are the mean and std.dev. of the sample mean score ¯x of these 50 students? ANSWER: Std.dev. of the sample mean = / p n = 5.9/ p 50 = 0.834. (c) What is the probability that the mean score of these 50 students is 21 or higher? SOLUTION: z = (21-18.6)/0.834 = 2.88. The right-tail prob. is .002 . About 34% of students (from this population) scored a 21 or higher on the ACT The probability that a single student randomly chosen from this population would have a score of 21 or higher is 0.34 Now take a SRS of 50 students who took the test. What are the mean and standard deviation of the sample mean score x-bar of these 50 students? ¨Mean = 18.6 [same as µ] ¨Standard Deviation = 0.8344 [sigma/sqrt(50)] nAbout 0.2 % of all random samples of size 50 (from this population) would have a mean score x-bar of 21 or higher. n nThe probability of having a mean score x-bar of 21 or higher from a sample of 50 students (from this population) is 0.002.
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