Lesson 7.3 The Central Limit Theorem for Sums

Notation and Formulae 

X is a random variable with a distribution that may be known or unknown. Using a subscript that matches the random variable, suppose

• μ_X = the mean of X
• σ_X = the standard deviation of X

If you draw random samples of size n, then as n increases, the random variable

 

of the sample sums tends to be normally distributed as follows:

Notice that the mean of the sample sums is equal to the mean of the original distribution multiplied by the sample size and the standard deviation of the sample sums is equal to the standard deviation of the original distribution multiplied by the square root of the sample size.

 

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CLT Problems for Sums Using TI-83 or TI-84 calculators

Example: In 1940, the United States was still very much a farming nation. The average farm at that time was 174 acres. Suppose the standard deviation was 55 acres. We randomly survey 30 farmers from 1940.

Let X = the size, in acres, of one U.S. farm in 1940.

and follows a normal distribution:

 

where μ_X = 174 and σ_X = 55. The mean of the sums is (30)(174) = 5220.

Below are some typical problems. The answers have been calculated using technology (TI-83 calculator).

What is the probability that the total (sum) number of acres of 30 farms in 1940 is more than 5000 acres?   The probability that the total number of acres of 30 farms in 1940 is more than 5000 acres is 0.7674. These calculations were done using the TI-83 DISTR calculator function 2nd DISTR. Example: Find the 90th percentile for the total (sum) number of acres for 30 farms in 1940.   k = 5606.1 acres (to 1 decimal place). The 90th percentile is 5606.1 acres. These calculations were done using the TI-83 DISTR calculator function 2nd DISTR.

Think About It

Do the Try-It examples in Introductory Statistics.

This is the last section of this lesson. When you have completed the assignment and the quiz for Lesson 7, you are ready to begin Lesson 8 - Confidence Intervals.  

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