166 Estimating a Population Mean (1 of 3)
Learning Objectives
- Construct a confidence interval to estimate a population mean when conditions are met. Interpret the confidence interval in context.
- Interpret the meaning of a confidence level associated with a confidence interval.
In “Estimating a Population Mean,” we focus on how to use a sample mean to estimate a population mean. This is the type of thinking we did in Modules 7 and 8 when we used a sample proportion to estimate a population proportion. Let’s take a moment to review what we learned in the modules Linking Probability to Statistical Inference and Inference for One Proportion, and then we’ll see how it relates to the current module.
- In Linking Probability to Statistical Inference, we noted that random samples vary, so we expect to see variability in sample proportions. In the section “Distribution of Sample Means” in that module, we made the same observations about sample means. In both cases, a normal model is a good fit for the sampling distribution when appropriate conditions are met.
- We also noted in that module that a sample proportion is an estimate for the population proportion. We do not expect the sample proportion to equal the population proportion, so there is some error. The error is due to random chance. Likewise, a sample mean is an estimate for the population mean, but there will be some error due to random chance.
Comment
Recall that, in Inference for One Proportion, we adjusted the standard error by replacing p with the sample proportion. Doing so made sense because the goal of the confidence interval is to estimate p. So the margin of error in the confidence interval formula changed. Here is the adjusted formula.
[latex]\stackrel{ˆ}{p}±2\text{}\sqrt{\frac{\stackrel{ˆ}{p}(1-\stackrel{ˆ}{p})}{n}}[/latex]
This adjustment changed the normality conditions. We use this adjusted confidence interval to estimate p when the successes and failures in the actual sample are at least 10.
We will eventually have to adjust the standard error for the sampling distribution of sample means, too. It makes sense because in many situations we will not know the population standard deviation, σ. This adjustment is more complicated than the adjustment to standard error for sample proportions, so before we do it, let’s practice finding the confidence interval for µ assuming we know σ.
Assuming we know σ is realistic when a lot of previous research has been done. For example, when we are estimating height, weight, or scores on a standardized test, previous research gives us reliable values for σ.
Example
Estimating Mean SAT Math Score
The SAT is the most widely used college admission exam. (Most community colleges do not require students to take this exam.) The mean SAT math score varies by state and by year, so the value of µ depends on the state and the year. But let’s assume that the shape and spread of the distribution of individual SAT math scores in each state is the same each yea