Wiki Page, Math 13 - Perez Sandbox : Module 23 - Estimating a Population Mean (2a of 3)

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

What If We Do Not Know the Population Standard Deviation? Can We Still Estimate the Population Mean?

The answer is yes, but we have to make some adjustments to our confidence interval formula.

On the previous page, we used a confidence interval to estimate the population mean, . For this confidence interval, we had to use the population standard deviation, , based on previous studies. It may have occurred to you that if we do not know , it is unlikely that we know . So we now take a different approach. We estimate using the sample standard deviation, .

Note: This is the same type of adjustment we used previously when working with proportions. The standard error of the sampling distribution for proportions involves p, the population proportion. If we knew p, then we wouldn’t need to build a confidence interval. We approximate by the sample proportion, .

Standard error = becomes

Our process for adjusting the confidence interval for estimating is similar. We use the sample standard deviation, , to estimate . The standard error for the sampling distribution becomes . This adjustment affects the margin of error in the confidence interval formula, but this adjustment is not as straightforward as our work with proportions.

This sample estimate for introduces more uncertainty in the process. The problem is worse with smaller samples because the sample standard deviations vary more. For small samples, is a worse approximation for . Unfortunately, this approximation makes the normal model a bad fit for the sampling distribution and inappropriate for determining the margin of error. For this reason, we use what is called a T-model (also known as the T-distribution) for this purpose.

Introduction to the T-Model

Here is the formula for the confidence interval based on the T-model. Note the the formula has the same structure as our previous confidence interval: sample statistic ± margin of error, where the margin of error is the number of standard errors associated with the confidence level.

As before, we can only use this formula when certain conditions are met for use of the T-model.

When Is a T-Model a Good Fit For the Sampling Distribution of Sample Means?

Check these conditions before using the T-model:

Use the T-model if (the population standard deviation) is unknown. If is known, then use the normal model instead of the T-model (if conditions for use of the normal model are met.)

Use the T-model if variable values are normally distributed in the population. If this is not true, then make sure the sample size is large (more than 30).

How Does the T-Model Compare To the Standard Normal Curve?

Up to this point, we have been using a normal curve to model probability distributions, specifically the Standard Normal curve of Z-scores which has a mean of 0 and standard deviation of 1. Unlike the Standard Normal Curve, the distribution of T-scores depends on the sample size, n. There is a different T-model for every n, so the T-model is a family of curves.

Instead of referring to n to specify which T-model to use, statisticians refer to the degrees of freedom, or df for short. The number of degrees of freedom is 1 less than the sample size. That is, df = n – 1.1

Let’s take a look at a few T-model curves (for various df) to see how they compare to the Standard Normal curve for Z-scores.

We can see from the picture that as the sample size increases, the T-curve gets closer to the Standard Normal curve. To say this another way, when degrees of freedom are large, the T-curve is in essence the Standard Normal curve. This makes sense because when samples are large, we expect the sample standard deviation to closely approximate the population standard deviation, so the T-score will be very close to the Z-score.

Similarities between T-model and Standard Normal model:

Symmetric with a central peak, bell-shaped.

Centered at 0.

The larger the degrees of freedom, the closer the T-model is to the Standard Normal model.

Area under the curve represents probability.

Difference between T-model and standard normal model:

For smaller samples, the T-model has more spread than the Standard Normal model.

The T-model has more probability in the tails and less in the center than the Standard Normal model. We can see this in the fatter tails and lower central peak of the T-model.

How Do We Find the T-Score Associated With a Given Confidence Level?

As noted above, the T-model depends on the sample size. Therefore, we need to first use the degrees of freedom to identify the correct T-curve, then we need to determine the middle area based on the confidence level. We will use an applet to find based on the confidence level and degrees of freedom. Eventually, we will just use StatCrunch, or a similar statistical package, that does all of the calculations for us and outputs the lower limit and upper limit for the confidence interval.

EXAMPLE

Suppose that we want to estimate a population mean with a 95% confidence interval. We estimate the standard error using the standard deviation from a random sample of 40 individuals, what is the value of ?

This is a large enough sample for use to use the T-model. Use the T-distribution Inverse Calculator to find the critical T-score, aka .

T-distribution Inverse Calculator

Our answer: is 2.0227.

Set the confidence level (which is the central probability) to 0.95 and the degrees of freedom to 39 (one less than n).

NOTE: For 95% confidence, the Empirical Rule approximates the critical Z-score as 2. The Empirical Rule is based on the Normal model. Using the T-model for df = 39, the critical T-score (2.0227) is very close to 2. This makes sense because for larger df, the closer the T-model is to the Standard Normal model. We will see that the critical T-score differs more from the critical Z-score when the sample sizes are small.

EXAMPLE

Suppose that we want to estimate the mean birth weight of babies born in a town with a 90% confidence interval. Based on past studies, we assume that birth weight is normally distributed in the population. We estimate the standard error using the standard deviation from a random sample of 15 babies, what is the value of ?

We can use the T-model even with this small random sample because we are assuming that birth weights are normally distributed in the population of babies in this town.

Use the T-distribution Inverse Calculator above to find the critical T-score, aka ., where df = 15–1 = 14.

Our answer: = 1.7613. Set the confidence level (central probability) to 0.9 and the degrees of freedom to 14 (one less than n) as shown below.

Note: You may recall that for 90% confidence is 1.645. We see a bigger difference between and when the sample is small.

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