Central Limit Theorem

Monday 14th March 2023

Unit 7 - The Central Limit Theorem.

7.2 - The Central Limit Theorem for Sums

Check that you have achieved the above at the end of the lesson.

State the central limit theorem for sums.

Calculate probabilities involving sums after large enough samples are taken from any distribution.

Use the central limit theorem to determine parameters for the distribution of the sums.

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Lesson outcomes

After discussing, click on the picture on the side for the answer. Remember to make sure you know how to arrive at the answers.

In your notebook, discuss with your groupmates and work out the answers to the following;

Central Limit theorem for means

Central limit theorem for sums

Definition

The central limit theorem for sums says that if you repeatedly draw samples of a given size (such as repeatedly rolling ten dice) and calculate the sum of each sample, these sums tend to follow a normal distribution. As sample sizes increase, the distribution of means more closely follows the normal distribution. The normal distribution has a mean equal to the original mean multiplied by the sample size and a standard deviation equal to the original standard deviation multiplied by the square root of the sample size.

b)

a)

Solutions

Central limit theorem for sums

Example:

An unknown distribution has a mean of 90 and a standard deviation of 15. A sample of size 80 is drawn randomly from the population.a) Find the probability that the sum of the 80 values (or the total of the 80 values) is more than 7,500. b) Find the sum that is 1.5 standard deviations above the mean of the sums.

c)

b)

a)

Solutions

Central limit theorem for sums

Example:

In a recent study reported Oct. 29, 2012 on the Flurry Blog, the mean age of tablet users is 34 years. Suppose the standard deviation is 15 years. The sample of size is 50.a) What are the mean and standard deviation for the sum of the ages of tablet users? What is the distribution? b) Find the probability that the sum of the ages is between 1,500 and 1,800 years. c) Find the 80th percentile for the sum of the 50 ages.

Check that you have achieved the above at the end of the lesson.

State the central limit theorem for sums.

Calculate probabilities involving sums after large enough samples are taken from any distribution.

Use the central limit theorem to determine parameters for the distribution of the sums.

02

03

01

Check that you are able to do the following.

Using your device, joint the kahoot using the game pin on the side for a formative assessment.

Practice quiz.

Looks like this!

Homework/Assignment

For your practice, refer to your workbook and attempt work on Central limit theorem for sums

Try your best to complete your work on time.