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

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

Now let's work through an example of finding the confidence interval for a population mean using the sample standard deviation and the T-model.

EXAMPLE Cable Strength

A group of engineers developed a new design for a steel cable. They need to estimate the amount of weight the cable can hold. The weight limit will be reported on cable packaging.

The engineers take a random sample of 45 cables and apply weights to each of them until they break. The mean breaking weight for the 45 cables is lb. The standard deviation of the breaking weight for the sample is s = 15.1 lb.

What should the engineers report as the mean amount of weight held by this type of cable?

Let’s use these sample statistics to construct a 95% confidence interval for the mean breaking weight of this type of cable.

Checking conditions:

Since we do not know the standard deviation of breaking weights of all of the cables (the population parameter , we use the sample standard deviation as an approximation for . Since we don’t know , we must use the T-distribution to model the sampling distribution of means.

Is the T-model a good fit for the sampling distribution?

Yes, because the conditions are met:

is unknown.

The sample size (n=45) is large enough (greater than 30).

Finding the standard error:

As usual, we start by estimating the standard error. This estimate comes from the formula . However, since we don’t know , we use as an approximation for . So our estimate for the standard error of all sample means is .

Finding the margin of error:

To find the margin of error, we need to find the critical T-score that corresponds to a 95% confidence level. This is just like the critical Z-score when we built confidence intervals for proportions, except that it comes from the T-model instead of the Standard Normal model.

We will use technology to find the critical T-score. Set the confidence level to 95% and the degrees of freedom to 44 (n – 1 = 45 – 1 = 44)

T-distribution Inverse Calculator

We get a T-score of 2.0154 as shown below.

Now we are ready to calculate the margin of error for this confidence interval:

Finding the confidence interval:

We have all the pieces to build the confidence interval. In our example, the confidence interval is

Conclusion:

We are 95% confident that the mean breaking weight for all cables of this type is between 763.7 lb and 772.7 lb.

EXAMPLE Cat's tail length

We want to estimate the mean length of a female cat's tail with 95% confidence. We assume that tail length (like most body measurements) are normally distributed in the population of female cats. We measure the tails of a random sample of 25 cats and find a mean of 9 inches and a standard deviation of 3 inches.

Even though the sample is less than 30 cats, we can use the T-model because we are assuming that tail length is normally distributed in the population of all female cats.

Using the applet, we find the critical T-value for a 95% confidence interval with a df = 24 is 2.064.

Standard error is

Margin of error (ME):

So the confidence interval is . We are 95% confident that the mean tail length of a female cat lies between 7.76 and 10.24 inches.

As before we can envision this confidence interval as an interval on the number line. As we learned previously, 95% of the confidence intervals with this margin of error will accurately estimate the mean tail length of all cats.

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