Scores on the math portion of the SAT exam for those taking the exam in 2016 fol

Question

Scores on the math portion of the SAT exam for those taking the exam in 2016 follow a normal distribution with a mean of 475 and a standard deviation of 105. We plan to select a SRS of 400 SAT takers from 2016 and record the math score for each person in the sample. a. What is the population of interest in this problem? b. What variable is recorded on each unit in this population? c. What type of variable is being recorded? d. What are the parameters of interest in the problem? (Include the correct symbol) a. Once a sample is selected, the men of the 400 scores in the sample will be determined. Is this quantity a parameter or statistic? Explain for full credit. b. What is the sampling distribution for the sample mean for sample of size 400. Verify the conditions hold for the CLT for sample means. Suppose you want the standard deviation for the sample means to be 7 or less. How large of a sample size is needed? Show work. Round your answer to the next whole number.

Explanation / Answer

4 a) Population of interest : All persons taking the SAT exam in 2016 form the population of interest in this example b) Maths score for each person taking the SAT exam is the variable recorded for each unit of the population c) Maths score is a discrete variable that is recorded. d) The parameter of interest is i) µ, the average of the Maths scores of persons taking the SAT exam in 2016 ii) , the standard deviation of the Maths scores of persons taking the SAT exam in 2016 5 a) Since it is the mean of the sample of 400 scores, it is a statistic and not a parameter Parameters are associated with populations and statistics with samples b) The sampling distribution for the sample mean for sample size of 400 is the Standard Normal Distribution. Following conditions of the CLT for mean hold good in this example 1. Since the distribution of the population is normal, the distribution of the sample mean is also normal. 2. As per CLT, for large sample sizes , the distribution of sample mean will approach a Normal Distribution. In this example, the sample size is 400 which is large enough. 6 We know, the distribution of the sample is a normal distribution with mean x^ and standard deviation s = where = population standard deviation Given = 105 We have to find n such that s is less than or equal to 7 i.e.

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