Obj 1: Sampling Distributions: I Can Find Sampling Distributions and Verify Their Properties

Statistics5.4: Sampling Distributions and the Central Limit Theorem

Obj 1: Sampling Distributions: I can find sampling distributions and verify their properties.

In previous sections, we studied the relationship between the mean of a population and values of a random variable. In this section, you will study the relationship between a population mean and the means of ______taken from the ______.

A ______is the probability distribution of a sample statistic that is formed when samples of size n are repeatedly taken from a population. If the sampling statistic is the sample mean (rather than variance or SD), then the distribution is called the ______

*Every sample statistic has a sampling distribution.

* A sample statistics is a numerical value taken from a sample.

Properties of Sampling Distributions of Sample Means

1) The mean of the sample means

2) The SD of the sample means

Read Ex 1, page 271

TIY 1: List all possible samples of n = 3, with replacement, from the population {1, 3, 5, 7}. Calculate the mean, variance, and SD of the sample means by hand. Compare these values with the population parameters and the formulas learned above.

Obj 2: The Central Limit Theorem: I can interpret the Central Limit Theorem

The CLT (Central Limit Theorem) describes the relationship between the sampling distribution of sample means and the population the samples are taken from. We use this theorem to make inferences about a population mean.

Properties of the CLT (Central Limit Theorem)

1) If the sample size is ______, then the sampling distribution of sample means is approximately ______. The greater the sample size, the better the approximation.

2) If the population is given as normally distributed, then the sampling distribution of sample means is ______

* Recall the mean, variance, and SD are

Read Ex 2, pg 273

TIY 2: Suppose random samples of size 100 are drawn from the population in Example 2. Find the mean and standard error of the mean of the sampling distribution. Sketch a graph of the sampling distribution and compare it with the sampling distribution in Example 2.

Read Ex 3, pg 274

TIY 3: The diameters of fully grown white oak trees are normally distributed, with a mean of 3.5 feet and a SD of 0.2 foot. 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 sampling distributions.

5.4: Obj 3—Probability and the Central Limit Theorem

In section 5.2, you learned how to find the probability that a random variable x will fall in a given interval of population values. In a similar manner, you can find the probability that a sample mean will fall in a given interval of the sampling distribution.

To transform into a z-score, you can use the formula:

Read Ex 4, pg 275

Read Ex 5, pg 276

Read Ex 6, pg 277