5.4 Sampling Distributions and the Central Limit

Theorem•Find sampling distributions and verify their properties•Interpret the Central Limit Theorem•Apply the Central Limit Theorem to find the probability of a sample mean

Sampling distributionA sampling distribution is the probability distribution of a sample statistic that is formed when samples of size n are repeated taken from a population.

Sampling distribution of sample means

If the sample statistic is the sample mean, then the distribution is the sampling distribution of sample means. Every sample statistic has a sampling distribution.

Properties of sampling distributions of sample

means1. The mean of the sample

means is equal to the population mean

2. The standard deviation of the sample means is equal to the population standard deviation divided by the square root of the sample size

Standard error of the mean

The standard deviation of the sampling distribution of the sample means is called the standard error of the mean.

A Sampling Distribution of Sample MeansYou write the population values {1, 3, 5, 7} on slips

of paper and put them in a box. Then you randomly choose two slips of paper, with replacement. List all possible samples of size n = 2 and calculate the mean of each. These means form the sample distribution of the sample means. Find the mean, variance, and standard deviation of the sample means. Compare your results with the mean μ = 4, variance σ² = 5, and standard deviation σ = √5 ≈ 2.236 of the population.

Example 1

Sample

Sample mean

Sample

Sample Mean

1, 1 1 5, 1 31, 3 2 5, 3 41, 5 3 5, 5 51, 7 4 5, 7 63, 1 2 7, 1 43, 3 3 7, 3 53, 5 4 7, 5 63, 7 5 7, 7 7

Example 1Mean: 4

Variance:2.5

Standard deviation:1.851

The Central Limit Theorem

1. If samples of size n, where n>30, are drawn from any population with a mean μ and a standard deviation σ, then the sampling distribution of sample means approximates a normal distribution. The greater the sample size, the better the approximation.

The Central Limit Theorem

2. If the population itself is normally distributed, then the sampling distribution of sample means is normally distributed for any sample size n.

The Central Limit TheoremIn either case, the sampling distribution of sample means has a mean equal to the population mean.

The sampling distribution of sample mean has a variance equal to 1/n times the variance of the population and a standard deviation equal to the population standard deviation divided by the square root of n.

The Central Limit Theorem

Interpreting the Central Limit TheoremCellular phone bills for residents of a city have a

mean of $63 and a standard deviation of $11, as shown in the following graph. Random samples of 64 are drawn from this population and the mean of each sample is determined. Find the mean and standard error of the mean of the sampling distribution. Then sketch the graph of the sampling distribution.

Try it yourself 2

Interpreting the Central Limit TheoremThe diameters of fully grown white oak trees are

normally distributed, with a mean of 3.5 feet and a standard deviation of 0.2 foot, as shown in the graph below. Random samples of size 16 are drawn from this population, and the mean of each sample is determined. Find the mean and standard error of the mean of the sample distribution. Then sketch a graph of the sampling distribution.

Try it yourself 3

Try it yourself 3Mean: 63

Standard deviation:1.4

Finding Probabilities for Sampling Distributions

The graph below shows the lengths of time people spend driving each day. You randomly select 100 drivers ages 15 to 19. What is the probability that the mean time they spend driving each day is between 24.7 and 25.5 minutes? Use μ=25 and σ=1.5 minutes.

Try it yourself 4

15-19

20-24

25-54

55-64

65+

0 10 20 30 40 50 60 70

Time behind the wheel

The average time spent driving each day, by age group

0.9768

Finding Probabilities for Sampling Distributions

The average sales price of a single-family house in the United States is $290,600. You randomly select 12 single-family houses. What is the probability that the mean sales price is more than $265,000? Assume that the sales prices are normally distributed with a standard deviation of $36,000.

Try it yourself 5

0.9931

Finding Probabilities for x andA consumer price analyst claims that prices

for liquid crystal display (LCD) computer monitors are normally distributed, with a mean of $190 and a standard deviation of $48.

1. What is the probability that a randomly selected LCD computer monitor costs less than $200?

Try it yourself 6

58%

2. You randomly select 10 LCD computer monitors. What is the probability that their mean cost is less than $200?

3. Compare these two probabilities.

Try it yourself 6

75%