Wiki Page, Math 13 - Perez Sandbox : Module 23 - Estimating a Population Mean (1a of 3)

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2026-01-17T21:15:39-08:00

In this module we learn to create a confidence interval to estimate a population mean based on a sample mean. The ideas are very similar to confidence intervals for proportions. The table below highlights these similarities.

Study this table to make connections between what we have already learned about the confidence interval to estimate a population proportion and what we are learning now about the confidence intervals to estimate a population mean.

Comparison of confidence interval concepts for proportions and means

CONCEPT | FOR PROPORTIONS | FOR MEANS

When we take lots of random samples, we use the standard deviation of the sample statistics to describe the error that is due to random chance. This is called the standard error. The standard error depends on a population parameter and sample size n.

Standard error =

Standard error =

If a normal model is a good fit for the sampling distribution, then 95% of sample statistics estimate the population parameter within 2 standard errors. This describes the margin of error. | 95% Margin of error = | 95% Margin of error =

We create an interval to estimate the population parameter. The interval is sample statistic ± margin of error. We say we are 95% confident that the confidence interval contains the population parameter.

Confidence interval is

Confidence interval is

These confidence intervals require a normal model. So we can only use these formulas when the normality criteria are met. | Both the expected successes (np) and expected failures (n(1–p)) are at least 10. | Variable is normally distributed in the population, OR sample size (n) is at least 30.

Comment

You may remember that we adjusted the standard error by replacing the population proportion p with the sample proportion p-hat. Doing so made sense because the goal of the confidence interval is to estimate p; therefore, it is odd to assume we know p when calculating the standard error.

When we made this adjustment, the margin of error in the confidence interval formula changed. Here is the adjusted formula.

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, . 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 assuming we know .

We can realistically assume that we know 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 .

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 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 year. More specifically, assume that individual SAT math scores consistently have a normal distribution with a standard deviation of 100 in a state regardless of year.

An educational researcher wants to estimate the mean SAT math score ( ) for his state this year. The researcher chooses a random sample of 650 exams in his state. The average score for the sample is 475. Estimate the mean SAT math score in this state for this year.

Checking conditions:

From our previous work we know that a normal model is a good fit for the distribution of sample means from random samples if one of two conditions is met:

The population of individual values is normal (in which case the sample size is not important).

If we do not know if the population of individual values is normal, then we must have a large sample size (more than 30).

Because we assume that the distribution of individual SAT math scores is normal in this example, a normal model is also a good fit for the distribution of sample means.

Finding the margin of error:

Keep in mind that the sample mean, , is only a single-value estimate for the population mean, . Because it comes from a random sample, we expect there to be some error in the estimate. But how much error should we expect?

We know that the sample distribution of means is approximately normal because conditions are met. Recall that in a normal model, 95% of the values fall within 2 standard deviations of the mean, so we use 2 standard errors for our margin of error.

standard error is

margin of error is

Finding the confidence interval:

We are 95% confident that and are within 7.8 points of each other. So we construct a 95% confidence interval from this sample mean by adding and subtracting 7.8 points.

Conclusion:

We are 95% confident that the mean SAT math score in this state this year is between 467.2 and 484.8. Recall from our previous work that being 95% confident says that this method, in the long run, produces an interval that captures the true population mean about 95% of the time.

Summary

If we want to estimate µ, a population mean, we want to calculate a confidence interval. The 95% confidence interval is:

We can use this formula only if a normal model is a good fit for the sampling distribution of sample means. If the sample size is large (n 30), we can use a normal model. If the sample size is not greater than 30, then we can use a normal model only if the variable is normally distributed in the population. As always, we must have a random sample. If the sample is not random, we cannot use it to estimate .

We say we are 95% confident that this interval contains , which means that in the long run, 95% of these confidence intervals contain .

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