AGENDA: DG 8 --- 15 minutes Begin Part 2 Unit 1 Lesson 11

Lesson #11 Central Limit Theorem Accel Precalc Unit #1 Data Analysis Lesson #11 Central Limit Theorem EQ: How do the mean and standard deviation of a sampling distribution compare to the mean and standard deviation of a population?

Recall: Suppose a teacher gave an 8-point quiz to a small class of four students. The results of the quiz were 2, 6, 4, and 8. Assume that these students constitute a population. Create a histogram of the original quiz scores. 1 .75 .5 .25 Describe the shape of the distribution. P(X) uniform 4 6 8 Quiz Score

 = ________________________ Find the mean and the standard deviation of the population quiz scores.  = ________________________ (2)(.25) + (4)(.25) + (6)(.25) + (8)(.25) = 5  = ____________________________

Now choose 2 quiz scores (with replacement) Now choose 2 quiz scores (with replacement). Create a table listing all possible samples of size 2 then calculate the mean for each sample. Samples of size n = 2 Mean 2 , 2 (2 + 2) / 2 = 2 2 , 4 (2 + 4) / 2 = 3 . 8 , 8 (8 + 8) / 2 = 8

Create a histogram of the sample means. 2 3 4 5 6 7 8 Sample Means Describe the shape of the distribution. Symmetric, bell-shaped

New Notation: mean of the sample means standard deviation of the sample means

Calculate the mean and standard deviation of the sample means from your table above. “mu x-bar” “standard error of the mean”

They are equal.

Three Properties of a Sampling Distribution: . The mean of the sample means will always equal the mean of the population. 2. The standard deviation will always be greater than the standard error of the means.

3. Central Limit Theorem ---As the sample size of a sampling distribution gets larger, the shape of the distribution will become approximately NORMAL regardless of the shape of the population distribution.

Normal Normal Not Normal Original Distribution Sampling Distribution if n > 30, sampling dist becomes approx normal Not Normal

Sample Mean Recall: Standardizing Values z = obs – mean standard deviation Individual Data Sample Mean z = obs – mean standard error

Ex 1. A.C. Nielsen reported that children between the ages of 2 and 5 watch an average of 25 hours of TV 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 number of hours they watch TV will be greater than 26.3 hours. 1.94

Ex 2. The average age of a vehicle registered in the United States is 8 years or 96 months. Assume the standard deviation is 16 months. If a random sample of 36 vehicles is selected, find the probability that the mean of their age is between 90 and 100 months. -2.25 1.5

Ex 3. The average number of pounds of meat that a person consumes a year is 218.4 pounds. Assume that the standard deviation is 25 pounds and the distribution is approximately normal. a. Find the probability that a person selected at random consumes less than 224 pounds per year. 0.22

Ex 3. The average number of pounds of meat that a person consumes a year is 218.4 pounds. Assume that the standard deviation if 25 pounds and the distribution is approximately normal. b. If a sample of 40 individuals is selected, find the probability that the mean of the sample will be less than 224 pounds per year. 1.42

Assignment: Practice Worksheet: Central Limit Theorem