Name Date Chiek Math 12

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1 Section 6.3: The Central Limit Theorem Definition: 1. A sampling distribution of sample means is a distribution using the means computed from all possible random samples of a specific size taken from a population. In other words, suppose a researcher selects a sample of 30 adults males and finds the mean of the measure of he triglyceride levels for the sample subjects to be 187 milligrams/deciliter. Then suppose a second sample is selected, and the mean of that sample is found to be 192 milligrams/deciliter. What happens then is that the mean becomes a random variable, and the sample means 187, 192, 184,..., 196 constitute a sampling distribution of sample means. If the samples are randomly selected with replacement, the sample mean, for the most part, will be somewhat different from the population mean. These differences are caused by sampling error. 2. Sampling error is the difference between the sample measure and the corresponding population measure due to the fact that the sample is not a perfect representation of the population. Properties of the Distribution of Sample Means: 1. The mean of the sample means will be the same as the population mean. 2. The standard deviation of the sample means will be smaller than the standard deviation of the population, and it will be equal to the population standard deviation divided by the square root of the sample size. Exercise 1. Suppose a professor gave an 8-point quiz to a small class of four students. The results of the quiz were 2, 4, 6, and 8. Assume that the four students constitute the population. a) Find the population mean, µ. 1

2 b) Find the standard deviation for the population, σ. c) Create a frequency distribution and draw a histogram. d) Find all samples of size 2 (with replacement) and their mean. e) Construct a frequency distribution of sample means, find the mean of the sample mean, µ X, and draw a histogram. 2

3 f) What do you notice about µ X and µ? g) Find the standard deviation of sample means, σ X, and compare it to the population standard deviation divided by 2. In summary, if all possible samples of size n are taken with replacement from the same population, the mean of the sample means, denoted by µ X equals the population mean µ ; and the standard deviation of the sample means, denoted by σ X, equals σ / n. 3. The standard deviation of the sample means, σ X = σ, is called the n standard error of the mean. 4. The Central Limit Theorem states for any given population with mean µ and standard deviation σ, a sampling distribution of sample means will have the following three characteristics if either the sample size n, is at least 30 or the population is normally distributed. Property 1: The mean of the sampling distributions of sample means will be the same as the population mean. Property 2: The standard deviation of a sampling distribution of sample means will be smaller than the standard deviation of the population, and it will be equal to the population standard deviation divided by the square root of the sample size. Property 3: The shape of a sampling distribution of sample means will approach that of a normal distribution, regardless of the shape of the population distribution. The larger the sample size, the better the normal distribution approximation will be. (Note: If the population is normally distributed in the first place, then the sample distribution is normal for any value of n.) 3

4 Note: To gain information about an individual data value obtained from the population, use formula z = X µ. To gain information about a sample mean, σ use formula z = X µ σ / n. Exercise 2. A.C. Neilsen reported that children between the ages of 2 and 5 watch an average of 25 hours of television per week. Assume the variable is normally distributed and the standard deviation is 3 hours. If 20 children between the ages of 2 and 5 are randomly selected, find the probability that the mean of the number of hours they watch television will be greater than 26.3 hours. Exercise 3. The average drive to work is 9.6 miles. Assume the standard deviation is 1.8 miles. If a random sample of 36 employed people who drive to work are selected, find the probability that the mean of the sample miles driven to work is between 9 and 10 miles. 4

5 Exercise 4. The average time spent by construction workers who work on weekends is 7.93 hours (over 2 days). Assume the distribution is approximately normal and has a standard deviation of 0.8 hour. a) Find the probability that an individual who works at that trade works fewer than 8 hours on the weekend. b) If a sample of 40 construction workers is randomly selected, find the probability that the mean of the sample will be less than 8 hours. 5

Section 7.2 Homework Answers

Section 7.2 Homework Answers 25.5 30 Sample Mean P 0.1226 sum n b. The two z-scores are z 25 20(1.7) n 1.0 20 sum n 2.012 and z 30 20(1.7) n 1.0 0.894, 20 so the probability is approximately 0.1635 (0.1645 using Table A). P14. a.