lab3 - STAT 350 – Fall 2008 Lab #3 – SOLUTION Sampling...

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Unformatted text preview: STAT 350 – Fall 2008 Lab #3 – SOLUTION Sampling Distributions 1. Side-by-side Histograms a. Give the side-by-side histograms for xN , 25 and xN ,100 . Histogram of Normal25 mean, Normal100 mean 48 Normal25 mean 40 49 50 51 52 53 Normal100 mean Frequency 30 20 10 0 (i) (ii) (iii) (iv) (v) (vi) 48 49 50 51 52 53 What would you have expected the statistic to be (what value)? I would expect the sample mean to be close to the population mean. The samples came from a Normal(μ = 50, σ = 5) distribution which has mean 50, so I would expect the statistic to be 50. What is the shape of each distribution? Each distribution is approximately symmetric and bell-shaped What is the center of each distribution (the mean of the distribution, see Step 8 in the directions)? Answers will vary because the data was simulated independently by each student, but answers should be close to 50. For the data graphed above, the centers were 49.962 and 50.012 respectively. Does the center appear to be affected by increasing the sample size. If so, how? No – the center was not [significantly] affected by increasing sample size. What is the standard deviation of each distribution (see Step 8 in the directions)? Answers will vary because the data was simulated independently by each student, but 5 answers should be close to (1 and 0.5). For the data graphed above, the standard n deviations were 1.017 and 0.507 respectively. Does the standard deviation appear to be affected by increasing the sample size. If so, how? Yes, standard deviation was approximately cut in half when the sample size was quadrupled Lab #3 – Solution Page 1 of 1 b. Give the side-by-side histograms for sN ,25 and sN ,100 . Histogram of Normal25 stdev, Normal100 stdev 2.4 Normal25 stdev 40 3.2 4.0 4.8 5.6 6.4 Normal100 stdev Frequency 30 20 10 0 (i) (ii) (iii) (iv) (v) (vi) 2.4 3.2 4.0 4.8 5.6 6.4 What would you have expected the statistic to be (what value)? I would expect the sample standard deviation to be close to the population standard deviation. The samples came from a Normal(μ = 50, σ = 5) distribution which has standard deviation of 5, so I would expect the statistic to be 5. What is the shape of each distribution? Each distribution is approximately symmetric and bell-shaped What is the center of each distribution (the mean of the distribution, see Step 8 in the directions)? Answers will vary because the data was simulated independently by each student, but answers should be close to 5. For the data graphed above, the centers were 4.8757 and 5.0124 respectively. Does the center appear to be affected by increasing the sample size. If so, how? No – the center was not [significantly] affected by increasing sample size. What is the standard deviation of each distribution (see Step 8 in the directions)? Answers will vary because the data was simulated independently by each student. For the data graphed above, the standard deviations were 0.7121 and 0.3821 respectively. Does the standard deviation appear to be affected by increasing the sample size. If so, how? Yes, standard deviation was approximately cut in half when the sample size was quadrupled Lab #3 – Solution Page 2 of 2 c. Give the side-by-side histograms for xP , 25 and xP ,100 . Histogram of Poisson25 mean, Poisson100 mean 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 Poisson25 mean 30 Poisson100 mean Frequency 25 20 15 10 5 0 (i) (ii) (iii) (iv) (v) (vi) 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 What would you have expected the statistic to be (what value)? I would expect the sample mean to be close to the population mean. The samples came from a Poisson(2) distribution which has mean 2, so I would expect the statistic to be 2. What is the shape of each distribution? Each distribution is approximately symmetric and bell-shaped What is the center of each distribution (the mean of the distribution, see Step 8 in the directions)? Answers will vary because the data was simulated independently by each student, but answers should be close to 2. For the data graphed above, the centers were 2.0428 and 1.9917 respectively. Does the center appear to be affected by increasing the sample size. If so, how? No – the center was not [significantly] affected by increasing sample size. What is the standard deviation of each distribution (see Step 8 in the directions)? Answers will vary because the data was simulated independently by each student, but 2 (0.2828 and 0.1414). For the data graphed above, the answers should be close to n standard deviations were 0.2959 and 0.1397 respectively. Does the standard deviation appear to be affected by increasing the sample size. If so, how? Yes, standard deviation was approximately cut in half when the sample size was quadrupled Lab #3 – Solution Page 3 of 3 d. Give the side-by-side histograms for sP ,25 and sP ,100 . Histogram of Poisson25 stdev, Poisson100 stdev 1.0 Poisson25 stdev 30 1.2 1.4 1.6 1.8 Poisson100 stdev Frequency 25 20 15 10 5 0 (i) (ii) (iii) (iv) (v) (vi) 1.0 1.2 1.4 1.6 1.8 What would you have expected the statistic to be (what value)? I would expect the sample standard deviation to be close to the population standard deviation. The samples came from a Poisson(2) distribution which has standard deviation of 2 ≈ 1.414, so I would expect the statistic to be 1.414. What is the shape of each distribution? Each distribution is approximately symmetric and bell-shaped What is the center of each distribution (the mean of the distribution, see Step 8 in the directions)? Answers will vary because the data was simulated independently by each student, but answers should be close to 1.414. For the data graphed above, the centers were 1.4120 and 1.4081 respectively. Does the center appear to be affected by increasing the sample size. If so, how? No – the center was not [significantly] affected by increasing sample size. What is the standard deviation of each distribution (see Step 8 in the directions)? Answers will vary because the data was simulated independently by each student. For the data graphed above, the standard deviations were 0.2097 and 0.1074 respectively. Does the standard deviation appear to be affected by increasing the sample size. If so, how? Yes, standard deviation was approximately cut in half when the sample size was quadrupled Lab #3 – Solution Page 4 of 4 2. Variable Normal25 mean Normal25 stdev Normal100 mean Normal100 stdev Poisson25 mean Poisson25 stdev Poisson100 mean Poisson100 stdev Lab #3 – Solution Total Count 100 100 100 100 100 100 100 100 Mean 49.962 4.8757 50.012 5.0124 2.0428 1.4120 1.9917 1.4081 StDev 1.017 0.7121 0.507 0.3821 0.2959 0.2097 0.1397 0.1074 Page 5 of 5 ...
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This note was uploaded on 02/16/2010 for the course STAT 350 taught by Professor Staff during the Spring '08 term at Purdue University-West Lafayette.

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