Lesson 13: Sampling Distribution Central Limit Theorem

LESSON 13: SAMPLING DISTRIBUTION

Outline

• Central Limit Theorem • Sampling Distribution of Mean

Central Limit Theorem: If a random sample is drawn from any population, the sampling distribution of the sample mean is approximately normal for a sufficiently large sample size. The larger the sample size, the more closely the sampling distribution of X will resemble a normal distribution.

1 Sample Size and Mean

0.1

0.08

0.06

0.04

0.02 Relative Frequency 0

1 5 9 13 17 21 25 29 33 37 41 45 49 Class Number

Distribution of random numbers 3

Sample Size and Mean

0.1

0.08

0.06

0.04

0.02 Relative Frequency 0

1 5 9 13 17 21 25 29 33 37 41 45 49 Class Number

Distribution of means of n random numbers, n=4 4

2 Sample Size and Mean

0.1

0.08

0.06

0.04

0.02 Relative Frequency 0

1 5 9 13 17 21 25 29 33 37 41 45 49 Class Number

Distribution of means of n random numbers, n=10 5

SAMPLING DISTRIBUTION OF THE SAMPLE MEAN

• If the sample size increases, the variation of the sample mean decreases.

s 2 s m = m, s 2 = , s = X X n n

• Where, m = Population mean s = Population standard deviation n = Sample size

mX = Mean of the sample means

s X = Standard deviation of the sample means 6

3 SAMPLING DISTRIBUTION OF THE SAMPLE MEAN

• Summary: For any general distribution with mean m and standard deviation s – The distribution of mean of a sample of size n can be approximated by a normal distribution with mean, µ s standard deviation, s = X n

• Exercise: Generate 1000 random numbers uniformly distributed between 0 and 1. Consider 200 samples of size 5 each. Compute the sample means. Check if the histogram of sample means is normally distributed and mean and standard deviation follow the above rules. 7

SAMPLING DISTRIBUTION OF THE SAMPLE MEAN

Example 1: An automatic machine in a manufacturing process requires an important sub-component. The lengths of the sub-component are normally distributed with a mean, m=120 cm and standard deviation, s=5 cm. What does the central limit theorem say about the sampling distribution of the mean if samples of size 4 are drawn from this population?

4 SAMPLING DISTRIBUTION OF THE SAMPLE MEAN

Example 2: An automatic machine in a manufacturing process requires an important sub-component. The lengths of the sub-component are normally distributed with a mean, m=120 cm and standard deviation, s=5 cm. Find the probability that one randomly selected unit has a length greater than 123 cm. )

x s ( f

m 9

SAMPLING DISTRIBUTION OF THE SAMPLE MEAN

Example 3: An automatic machine in a manufacturing process requires an important sub-component. The lengths of the sub-component are normally distributed with a mean, m=120 cm and standard deviation, s=5 cm. Find the probability that, if four units are randomly selected, their mean length exceeds 123 cm. )

x s ( f

m 10

5 SAMPLING DISTRIBUTION OF THE SAMPLE MEAN

Example 4: An automatic machine in a manufacturing process requires an important sub-component. The lengths of the sub-component are normally distributed with a mean, m=120 cm and standard deviation, s=5 cm. Find the probability that, if four units are randomly selected, all four have lengths that exceed 123 cm.

CORRECTION FOR SMALL SAMPLE SIZE

• For a small, finite population N, the formula for the standard deviation of sampling mean is corrected as follows:

s N -n s = X n N -1

6 READING AND EXERCISES

Lesson 13

Reading: Sections 8-1, 8-2, 8-3, pp. 260-276

Exercises: 9-3,9-4, 9-8, 9-17, 9-19