Module 10: Inference for Means
Distribution of Sample Means (1 of 4)
Distribution of Sample Means (1 of 4)
Learning outcomes
- Describe the sampling distribution of sample means.
- Draw conclusions about a population mean from a simulation.
How Sample Means Vary in Random Samples
In Inference for Means, 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 with sample proportions in the module Linking Probability to Statistical Inference to understand the distribution of sample proportions. Ultimately, we develop a probability model based on this sampling distribution. We use the probability model with an actual sample mean to test a claim about population mean or to estimate a population mean. This task is similar to the type of work we did in Inference for One Proportion with proportions when we tested hypotheses and created confidence intervals.
Example
Birth Weights
The World Health Organization (WHO) monitors many variables to assess a population’s overall health. One of these variables is low birth weight. A birth weight under 2,500 grams is a low birth weight. Low birth weight is a categorical variable because the birth weight is either under 2,500 grams or it is not. The WHO collects data from hospitals and other health-care institutions and can use this sample data to find a confidence interval to estimate the proportion of all babies in a country with a low birth weight. This type of inference comes from Inference for One Proportion.
In this module, we work with quantitative variables. In this example, we use birth weight as a quantitative variable. To analyze the quantitative variable birth weight, we use means.
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: µ = 3,500 grams.
- The 3,400 is a statistic from a sample, so we write [latex]\bar{x}[/latex]= 3,400 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 µ = 3,500, and take random samples of 9 babies at a