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HW3-Solutions

# HW3-Solutions - STAT 101 Agresti Homework 3 Solutions...

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STAT 101 - Agresti Homework 3 Solutions 9/27/10 Chapter 4 4.27. (a) The sampling distribution of the sample proportion of heads for flipping a balanced coin once is p 0 1 Probabilit y 0.5 0 0.50 (b) The sampling distribution of the sample proportion of heads for flipping a balanced coin twice is p 0 0.5 1 Probabilit y 0.2 5 0.5 0 0.25 (c) The sampling distribution of the sample proportion of heads for flipping a balanced coin three times is p 0 1/3 2/3 1 Probabilit y 0.12 5 0.37 5 0.37 5 0.125 (d) The sampling distribution of the sample proportion of heads for flipping a balanced coin four times is p 0 0.2 5 0.50 0.7 5 1 Probabilit y 0.062 5 0.2 5 0.37 5 0.2 5 0.0625 (e) As the number of flips increases, the sampling distribution of the sample proportion of heads seems to be getting more normal, with the probabilities concentrating more closely around 0.50. 4.29. (a) 0.5 0.0104 2293 y n σ σ = = = . (b) If actually 50% of the population voted for DeWine, it would be surprising to obtain 44% in this exit poll, since 44% is 6% lower than 50%, and the standard error for the sampling distribution is 1.04%; that is, the sample proportion of 0.44 is nearly 6 standard errors below 0.50. (c) Based on the information from the exit poll, I would be willing to predict that Sherrod Brown would win the Senatorial election. 4.33. (a) The probability that PDI is below 90 is ( 29 ( 29 90 100 90 0.67 0.2514 15 P Y P Z P Z - < = < = < - = . (b) The probability that the sample mean PDI is below 90 is ( 29 ( 29 90 100 90 3.33 0.00135 15 25 P Y P Z P Z - < = < = < - < . (c) An individual PDI of 90 is not surprising, since the probability is 0.2514 of that value or lower. However, a sample mean PDI of 90 would be surprising since this value would happen almost never. (d) The sketch of the sampling distribution should be less spread out and have a taller peak and thinner tails than the sketch of the population distribution.

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4.36. (a) The population distribution is skewed to the right with mean 5.2 and standard deviation 3.0. (b) The sample data distribution based on the sample of 36 families and is skewed to the right with mean 4.6 and standard deviation 3.2. (c) The sampling distribution of y is approximately normal with mean 5.2 and standard error 3.0 36 0.5 = . This distribution describes the theoretical distribution for the sample mean. 4.41 (b) Even though the population distribution is not normal (there are only two possible values), the sample proportions for the 1000 samples of size 100 each should have a histogram with an approximately bell shape. 4.42 (a) The population distribution is skewed, but the empirical distribution of sample means probably has a bell shape, reflecting the Central Limit Theorem. (b) The Central Limit Theorem applies to relatively large random samples, but here n = 2 for each sample. 4.46. (a) The sample data distribution tends to resemble the population distribution more closely than the sampling distribution. A random sample of data from a population should be representative of the population, and its distribution should be similar to the population distribution. (b) The sample data distribution is the distribution of data that we actually observe. The sampling distribution of y is the
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