LESSON 13 SAMPLING DISTRIBUTION Outline Central Limit Theorem

LESSON 13: SAMPLING DISTRIBUTION Outline • Central Limit Theorem • Sampling Distribution of Mean 1

CENTRAL LIMIT THEOREM 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 will resemble a normal distribution. 2

Distribution of random numbers 3

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

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. • Where, = Population mean = Population standard deviation = Sample size = Mean of the sample means = Standard deviation of the sample means 6

SAMPLING DISTRIBUTION OF THE SAMPLE MEAN • Summary: For any general distribution with mean and standard deviation – The distribution of mean of a sample of size can be approximated by a normal distribution with • 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 7 mean and standard deviation follow the above rules.

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, =120 cm and standard deviation, =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? 8

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, =120 cm and standard deviation, =5 cm. Find the probability that one randomly selected unit has a length greater than 123 cm. 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, =120 cm and standard deviation, =5 cm. Find the probability that, if four units are randomly selected, their mean length exceeds 123 cm. 10

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, =120 cm and standard deviation, =5 cm. Find the probability that, if four units are randomly selected, all four have lengths that exceed 123 cm. 11

CORRECTION FOR SMALL SAMPLE SIZE • For a small, finite population N, the formula for the standard deviation of sampling mean is corrected as follows: 12

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 13