Overview 7.2 Central Limit Theorem for Means

Objectives: By the end of this section, I will be able to… 1) Describe the sampling distribution of x for skewed and symmetric populations as the sample size increases. 2) Apply the Central Limit Theorem for Means to solve probability questions about the sample mean.

Skewed and Symmetric Populations For a skewed population, sampling distribution of the mean becomes approximately normal as the sample size approaches 30. For a symmetric distribution, at n=20 the sampling distribution is approximately normal.

Central Limit Theorem for Means Population with mean μ Standard deviation σ The sampling distribution of the sample mean x becomes approximately normal (μ, ) as the sample size gets larger Regardless of the shape of the population.

Rule of Thumb We consider n ≥ 30 as large enough to apply the Central Limit Theorem for any population.

Three Cases for the Sampling Distribution of the Sample Mean x Case 1 The population is normal. Then the sampling distribution of x is normal (Fact 3).

Three Cases for the Sampling Distribution of the Sample Mean x continued Case 2 The population is either non-normal or of unknown distribution and the sample size is at least 30. Central Limit Theorem for Means

Three Cases for the Sampling Distribution of the Sample Mean x continued Case 3 The population is either non-normal or of unknown distribution and the sample size is less than 30. Insufficient information to conclude that the sampling distribution of the sample mean x is either normal or approximately normal

Example Sometimes the solution to a problem lies beyond our available means The U.S. Small Business Administration (SBA) provides information on the number of small businesses for each metropolitan area in the United States. Figure 7.15 shows a histogram of our population for this example, the number of small businesses in each of the 328 cities nationwide. (For example, Austin, Texas, has 22,305 small businesses, while Pensacola, Florida, has 6020.)

Example 7.12 continued The mean is μ = 12,485 and the standard deviation is σ = 21,973. Find the probability that a random sample of size n = 10 cities will have a mean number of small businesses greater than 17,000. Figure 7.15

Example 7.12 continued Solution Try Case 1 Population is not normal, therefore Case 1 does not apply The sample size n=10 is too small to apply Case 2 Default to Case 3

Example 7.12 continued Solution Using the methods in this textbook, we cannot find the probability that a random sample of size n = 10 cities will have a mean number of small businesses greater than 17,000.

Example Application of the Central Limit Theorem for the Mean Suppose we have the same data set as in Example 7.12, but this time we increase our sample size to 36. Now, try again to find the probability that a random sample of size n = 36 cities will have a mean number of small businesses greater than 17,000.

Example Application of the Central Limit Theorem for the Mean Continued Solution Try to apply Case 2. Sample size n = 36 is large enough, the Central Limit Theorem applies. Sampling distribution of the sample mean x is approximately normal. _

Example Application of the Central Limit Theorem for the Mean Continued Find  x and  x Facts 1 and 2 __ _

Example Application of the Central Limit Theorem for the Mean Continued CLT indicates solve a normal probability problem Use Fact 5

Example Application of the Central Limit Theorem for the Mean Continued FIGURE 7.17 FIGURE 7.16

Example Application of the Central Limit Theorem for the Mean Continued Look up Z = 1.23 in the Z table and subtract this table area (0.8907) from 1 to get the desired tail area: P( Z > 1.23) = = There is a 10.93% probability that a random sample of 36 cities will have a mean number of small businesses greater than 17,000.

Summary In this section, we examine the behavior of the sample mean when the population is not normal. The approximate normality of the sampling distribution of the sample mean kicks in much quicker when the original population is symmetric rather than skewed. The Central Limit Theorem is one of the most important results in statistics and is stated as follows:

Summary Given a population with mean μ and standard deviation σ, the sampling distribution of the sample mean x becomes approximately normal(μ, ) as the sample size gets larger, regardless of the shape of the population. This approximation applies for smaller sample sizes when the original distribution is more symmetric.