Lesson 13 sampling distribution
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LESSON 13: SAMPLING DISTRIBUTION. Outline Central Limit Theorem Sampling Distribution of Mean. CENTRAL LIMIT THEOREM.

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LESSON 13: SAMPLING DISTRIBUTION

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Lesson 13 sampling distribution

LESSON 13: SAMPLING DISTRIBUTION

Outline

  • Central Limit Theorem

  • Sampling Distribution of Mean


Lesson 13 sampling distribution

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.


Lesson 13 sampling distribution

Distribution of random numbers


Lesson 13 sampling distribution

Distribution of means of n random numbers, n=4


Lesson 13 sampling distribution

Distribution of means of n random numbers, n=10


Lesson 13 sampling distribution

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


Lesson 13 sampling distribution

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 mean and standard deviation follow the above rules.


Lesson 13 sampling distribution

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?


Lesson 13 sampling distribution

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.


Lesson 13 sampling distribution

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.


Lesson 13 sampling distribution

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.


Lesson 13 sampling distribution

CORRECTION FOR SMALL SAMPLE SIZE

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


Lesson 13 sampling distribution

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


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