Chapter 7 The Normal Probability Distribution 7.5 Sampling Distributions; The Central Limit Theorem Illustrating Sampling Distributions Step 1: Obtain a simple random sample of size n. Illustrating Sampling Distributions Step 1: Obtain a simple random sample of size n. Step 2: Compute the sample mean. Illustrating Sampling Distributions Step 1: Obtain a simple random sample of size n. Step 2: Compute the sample mean. Step 3: Assuming we are sampling from a finite population, repeat Steps 1 and 2 until all simple random samples of size n have been obtained. EXAMPLE Illustrating a Sampling Distribution The following data represents the ages of 6 individuals that are members of a weekend golf club. 23, 25, 49, 32, 38, 43 Treat these 7 individuals as a population. (a) Compute the population mean. (b) List all possible samples of size n = 2 and determine the sample mean age. (c) Construct a relative frequency distribution of the sample means. This distribution represents the sampling distribution of the sample mean. EXAMPLE Illustrating a Sampling Distribution The weights of pennies minted after 1982 are approximately normally distributed with mean 2.46 grams and standard deviation 0.02 grams. Approximate the sampling distribution of the sample mean by obtaining 200 simple random samples of size n = 5 from this population. The data on the following slide represent the sample means for the 200 simple random samples of size n = 5. For example, the first sample of n = 5 had the following data: 2.493 2.466 2.473 2.492 2.471 The mean of the sample means is 2.46, the same as the mean of the population. The standard deviation of the sample means is 0.0086, which is smaller than the standard deviation of the population. The next slide shows the histogram of the sample means. EXAMPLE Finding the Area Under a Normal Curve The weights of pennies minted after 1982 are approximately normally distributed with mean 2.46 grams and standard deviation 0.02 grams Approximate the sampling distribution of the sample mean by obtaining 200 simple random samples of size n = 15 from this population of pennies minted after 1982. The mean of the 200 sample means is 2.46 (the mean of the population). The standard deviation of the 200 sample means is 0.0049. Notice that the standard deviation of the sample means is smaller for the larger sample size. EXAMPLE Describing the Distribution of the Sample Mean The weights of pennies minted after 1982 are approximately normally distributed with mean 2.46 grams and standard deviation 0.02 grams. What is the probability that in a simple random sample of 10 pennies minted after 1982, we obtain a sample mean of at least 2.465 grams? EXAMPLE Sampling from a Non-normal Population The following distribution represents the number of people living in a household for all homes in the United States in 2000. Obtain 200 simple random samples of size n = 4; n = 10 and n = 30. Draw the histogram of the sampling distribution of the sample mean. EXAMPLE Using the Central Limit Theorem Suppose that the mean time for an oil change at a “10-minute oil change joint” is 11.4 minutes with a standard deviation of 3.2 minutes. (a) If a random sample of n = 35 oil changes is selected, describe the sampling distribution of the sample mean. (b) If a random sample of n = 35 oil changes is selected, what is the probability the mean oil change time is less than 11 minutes? EXAMPLE Using the Central Limit Theorem (c ) If a random sample of n = 50 oil changes is selected, what is the probability the mean oil change time is less than 11 minutes? (d) What effect did increasing the sample size have on the probability?