OverviewOverview

7.2 Central Limit Theorem for Means

7.2 Central Limit Theorem for 7.2 Central Limit Theorem for MeansMeansObjectives:

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 PopulationsSkewed 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 MeansCentral 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.

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Rule of ThumbRule of Thumb

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

Three Cases for the Sampling Three Cases for the Sampling Distribution of the Sample Distribution of the Sample Mean xMean x

Case 1

The population is normal.

Then the sampling distribution of x is normal (Fact 3).

Three Cases for the Three Cases for the Sampling Distribution of the Sampling Distribution of the Sample Mean x continuedSample Mean x continuedCase 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 Three Cases for the Sampling Distribution of the Sample Distribution of the Sample Mean x continuedMean 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 7.12 - Sometimes the Example 7.12 - Sometimes the solution to a problem lies solution to a problem lies beyond our available meansbeyond our available meansThe 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 continuedExample 7.12 continuedThe 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 continuedExample 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 continuedExample 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 7.13 - Application of Example 7.13 - Application of the Central Limit Theorem for the Central Limit Theorem for the Meanthe Mean

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

Example 7.13 - Application of Example 7.13 - Application of the Central Limit Theorem for the Central Limit Theorem for the Mean Continuedthe 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 7.13 - Application of Example 7.13 - Application of the Central Limit Theorem for the Central Limit Theorem for the Mean Continuedthe Mean Continued

Find x and x

Facts 1 and 2

_ __

21,9733662.1667

36x

n

12,485x

Example 7.13 - Application of Example 7.13 - Application of the Central Limit Theorem for the Central Limit Theorem for the Mean Continuedthe Mean ContinuedCLT indicates solve a normal probability problem

Use Fact 5

17,000P x

17,000 17,000 12,4851.2329 1.23

3662.1667x

x

Z

Example 7.13 - Application of Example 7.13 - Application of the Central Limit Theorem for the Central Limit Theorem for the Mean Continuedthe Mean Continued

17,000 1.23P x P Z

FIGURE 7.17FIGURE 7.16

Example 7.13 - Application of Example 7.13 - Application of the Central Limit Theorem for the Central Limit Theorem for the Mean Continuedthe 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) = 1 - 0.8907 = 0.1093

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.

SummarySummary

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:

SummarySummary

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.

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