PROBLEM SET-5 Sampling Distributions - Solutions

# PROBLEM SET-5 Sampling Distributions - Solutions -...

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STATISTICS FOR SOCIAL SCIENCES BUS 152 ___________________________________________________________________________ PROBLEM SET – 5 SOLUTIONS SAMPLING DISTRIBUTIONS Problem 1 (7.1) Given a normal distribution with µ = 100 and σ = 10, if you select a sample of n = 25, what is the probability that X is a. less than 95? b. between 95 and 97.5? c. above 102.2? d. There is a 65% chance that X is above what value? SOLUTIONS: a) P( X  < 95) = P(Z < – 2.50) = 0.0062 b) P(95 <  X  < 97.5) = P(– 2.50 < Z < – 1.25) = 0.1056 – 0.0062 = 0.0994 c) P( X  > 102.2) = P(Z > 1.10) = 1.0 – 0.8643 = 0.1357 d) P( X  > A) = P(Z > – 0.39) = 0.65 X  = 100 – 0.39( 10 25 ) = 99.22 _________________________________________________________________________________ Problem 2 (7.4) The following data represent the number of days absent per year in a population of six employees of a small company: 1 3 6 7 9 10 a. Assuming that you sample without replacement, select all possible samples of n = 2 and construct the sampling distribution of the mean. Compute the mean of all the sample means and also compute the population mean. Are they equal? What is this property called? b. Do (a) for all possible samples of n = 3. c. Compare the shape of the sampling distribution of the mean in (a) and (b). Which sampling distribution has less variability? Why? d. Assuming that you sample with replacement, do (a) through (c) and compare the results. Which sampling distributions have the latest variability, those in (a) or (b)? Why? SOLUTIONS: a) Sampling Distribution of the Mean for n = 2 (without replacement) Sample Number                         Outcomes        Sample Means  i X 1 1, 3 1 X  = 2 2 1, 6 2 X  = 3.5 3 1, 7 3 X  = 4 4 1, 9 4 X  = 5 5 1, 10 5 X  = 5.5 6 3, 6 6 X  = 4.5 7 3, 7 7 X  = 5 8 3, 9 8 X  = 6

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9 3, 10 9 X  = 6.5 10 6, 7 10 X  = 6.5 11 6, 9 11 X  = 7.5 12 6, 10 12 X  = 8 13 7, 9 13 X  = 8 14 7, 10 14 X  = 8.5 15 9, 10 15 X  = 9.5 Mean of All Possible Sample Means: Mean of All Population Elements:        μ X = 90 15 = 6               1 3 6 7 9 10 6 6 + + + + + = = Both means are equal to 6.  This property is called unbiasedness. b)
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PROBLEM SET-5 Sampling Distributions - Solutions -...

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