• Create a sampling distribution given [latex]\mu[/latex] and [latex]n[/latex].
• Know and check the conditions of the Central Limit Theorem.
• Use the normal approximation to compute probabilities involving sample means when appropriate.

Sampling Variability

Data collected by the Centers for Disease Control and Prevention show that the average birthweight for babies in the United States is [latex]7.17[/latex] pounds, and the standard deviation of birthweights is [latex]1.30[/latex] pounds^[1]. Assume that birthweights in the United States follow an approximately normal distribution.

Let’s use the statistical tool to simulate random samples of births and examine the mean birthweight for each sample.

Step 1: Select Population Distribution: Bell-Shaped.
Step 2: Enter [latex]7.17[/latex] and [latex]1.30[/latex] for the population mean and standard deviation, respectively. (You will need to select the Enter values for [latex]\mu[/latex] and [latex]\sigma[/latex] option.)
Step 3: Select [latex]n=5[/latex] and draw 1 sample.
Step 4: Under the Data Distribution (Histogram from last generated sample), you can find its [latex]\bar(x)[/latex] and [latex]s[/latex].

[Trouble viewing? Click to open in a new tab.]

Because the birth weights of different babies will vary from sample to sample, the sample mean birth weights will also vary from sample to sample. The tendency of samples to have different statistics (means, proportions) than the population as a whole due to randomness is called sampling variability, and the distribution of these statistics is called a sampling distribution.

Mean and Standard Deviation

In the case of sample means, if we sample from a normal population as the one seen here, the sampling distribution of the sample means will also have a normal distribution.

If the mean and standard deviation of the population are [latex]\mu[/latex] and [latex]\sigma[/latex], respectively, the mean and standard deviation of the sample means for random samples of size [latex]n[/latex] are:

• Mean of the sample means = [latex]\mu[/latex]
• Standard deviation of the sample means = [latex]\dfrac{\sigma}{\sqrt{n}}[/latex]

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