Wiki Page, Math 13 - Perez Sandbox : Module 22 - Distribution of Sample Means (1a of 4)

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2026-01-17T21:15:35-08:00

How Sample Means Vary in Random Samples

In this unit we work with quantitative variables, so the statistics and parameters will be means instead of proportions.

We begin this module with a discussion of the sampling distribution of sample means. Our goal is to understand how sample means vary when we select random samples from a population with a known mean. We did this same type of thinking previously with sample proportions to understand the distribution of sample proportions.

Ultimately, we will develop a probability model based on the distribution of sample means. We use the probability model with an actual sample mean to test a claim about population mean or to estimate a population mean.

EXAMPLE Birth weights

The World Health Organization (WHO) monitors many variables to assess a population’s overall health. One of these variables is birth weight. Birth weight is a quantitative variable; we can calculate the mean birth weight for a population.

Suppose that babies in a town had a mean birth weight of 3,500 grams in 2005. This year, a random sample of 9 babies has a mean weight of 3,400 grams.

The 3,500 is a parameter from a population. We use the Greek letter µ to represent it: grams.

The 3,400 is a statistic from a sample, so we write grams.

Obviously, this sample weighs less on average than the population of babies in the town. A decrease in the town’s mean birth weight could indicate a decline in overall health of the town. But does this sample give strong evidence that the town’s mean birth weight is less than 3,500 grams this year?

To answer this question, we need to understand how much the means from random samples vary. Would a sample be likely—or unlikely—to have a mean birth weight of 3,400 grams if the mean weight of all the babies is 3,500 grams?

We outline this investigation in the following diagram:

As before, the logic of inference is the same. Begin with a population with , and take random samples of 9 babies at a time.

If a sample mean of 3,400 is likely to occur when sampling from a population with , then this sample could have come from a population with a mean of 3,500. The evidence from the sample therefore is not strong enough to reject the idea that .

If a sample mean of 3,400 is unlikely when sampling from a population with , then the sample provides evidence that the mean weight for all babies in the population is less than 3,500.

Likely or unlikely? It depends on how much the sample means vary. We need to investigate the sampling distribution of sample means.

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