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Statistics
5.4: 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 (from chapter 1).
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 x will fall in a given interval of the sampling distribution.
To transform x into a z-score, you can use the formula:
Read Ex 4, pg 275
Read Ex 5, pg 276
Read Ex 6, pg 277