Chapter 7: 7.1 Intro to the Central Limit Theorem

CLT: Mean of a Sampling Distribution of Sample Means

1. The mean of a sampling distribution of sample means, μx̅, equals the mean of the population, μ. μx̅ = μ μx̅: mean of the sampling distribution of sampling means μ: population mean

CLT: The Standard Deviation of a Sampling Distribution of Sample Means

2. The standard deviation of a sampling distribution of sample means, σx̅, equals the standard deviation of the population, σ, divided by the square root of the sample size, √ n. σx̅ = σ/√ n σx̅: standard deviation of sample means σ: population standard deviation n: sample size

CLT: The Shape of a Sampling Distribution of Sample Means

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.

The Central Limit Theorem (CLT)

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.

Example 7.2: Applying the Central Limit Theorem The following histogram represents the population distribution of the weights of 600 horse jockeys. The mean of the population is 116.2 pounds and the standard deviation is 3.9 pounds.

Histogram is skewed to the right (data lies on the left)

Example 7.1: Calculating the Standard Deviation for a Sampling Distribution of Sample Means Using the Central Limit Theorem Suppose that the standard deviation of movie ticket prices in the United States is $0.72. Suppose sample means are calculated for samples of size 52; that is, ticket prices are recorded for different samples of 52 theaters and the sample means are calculated. What would be the standard deviation of the sampling distribution of the sample means, that is, the standard error of the mean?

Take a mean of each sample and those are your data points. σ = $0.72 Use formula: σx̅ = σ/√ n 1. Check: Sample size is at least 30 ✓ We can use CLT for the problem. 2. σx̅ = σ/√ n = 0.72/√52 = 0099846 = 0.100 => σx̅ = $0.10

Sampling Distribution of Sample Means

The distribution of the sample means for all possible samples of a given size, n. Ex: Take however many 30 (n) sample statistics classes and get the average test scores of all classes.

Sampling Distribution

The distribution of the values of a particular sample statistic for all possible samples of a given size, n.

Example 7.3: Applying the Central Limit Theorem (cont.) a. Describe the shape of this distribution. b. Consider the sampling distribution of the sample means created from this population for samples of size 25. Can the Central Limit Theorem be applied in this situation? c .Consider the sampling distribution of the sample means created from this population for samples of size 50. Can the Central Limit Theorem be applied in this situation?

a. Bimodial Distribution (2 clear peaks) b. Check: No; sample size (25) isn't large enough. c. Check: Yes; sample size is at least 30 ✓

Example 7.2: Applying the Central Limit Theorem (cont.) Let's now consider the sampling distribution of the sample means for samples of size n = 45. a. Can the Central Limit Theorem be applied to this sampling distribution? If so, explain how the conditions are met. b. What is the sampling distribution's mean? c. What is the sampling distribution's standard deviation? d. What is the sampling distribution's shape?

a. Check: Yes; sample size is at least 30 ✓ b. By the CLT, μx̅ = μ => μx̅ = 116.2 c. σx̅ = σ/√ n (by the CLT) => = 8.9/√ 45 = 0.58 d. Approximately normal or bell-shaped (by the CLT).

Example 7.3: Applying the Central Limit Theorem (cont.) d. Consider the sampling distribution of the sample means created from this population for samples of size 200. What would you expect the shape of this distribution to be? How would it compare to the sampling distribution described in part c.?

d. Would resemble a normal distribution (by the CLT), even more than the distribution in part c. Since, sample in part d is even larger.