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

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2026-01-17T21:15:40-08:00

In our discussion about the structure of confidence intervals, we said choosing a higher level of confidence means that we sacrifice some precision. This is true only if nothing else changes. But there is one way to keep a high level of confidence without sacrificing precision: increase the sample size. We investigate the impact of sample size on the confidence interval next.

EXAMPLE Cable strength revisited

Recall the engineers who are trying to determine the breaking weight of a cable. In that example, we had a random sample of 45 cables with a mean breaking weight of 768.2 lb and a standard deviation of 15.1 lb. From that sample we computed a 95% confidence interval for the mean breaking weight of all such cables. Here are the important numbers we found from that calculation on the previous page:

standard error:

critical T-value: (we used the applet to find this)

margin of error:

confidence interval: or (763.67, 772.73)

Now let’s increase the sample size and investigate the impact on the confidence interval. We calculate the confidence interval for a larger sample of 101 cables (n = 101).

Sample size affects our calculations in two ways:

The sample size (n) appears in our formula for standard error.

The critical T-value depends on degrees of freedom, and df = n - 1.

Finding the standard error:

Note: The standard error is smaller when the sample size is larger. We were expecting this because we know there is less variability in sample means when the samples are larger.

Finding the critical T-value:

To find the critical T-value, we use the applet. We set the df to 100 and the central probability to 0.95. We see that the critical T-value is 1.984.

Here is a picture of our work with the applet.

Note: Increasing the sample size decreased the critical T-value. The T-value went from 2.015 to 1.984 when we increased the sample size. You might also notice that both of the critical T-values for 95% confidence are larger than the critical Z-value for 95% confidence, which is approximately 1.96. This makes sense because the T-models are wider than the the standard normal curve.

Finding the margin of error:

Here is the margin of error calculation.

Finding the confidence interval:

Side-by-side comparison:

Let’s take a look at these two intervals to study the effects of changing the sample size.

Comparison of calculations for different sample sizes

Sample Size | n = 45 | n = 101
Standard Error | 2.25 | 1.50
Critical T-value | 2.015 | 1.984
Margin of Error | 4.53 | 2.98
Confidence Interval | (763.67, 772.73) | (765.22, 771.18)

Increasing the sample size had the following effects on the confidence interval estimate:

Decreased standard error

Decreased critical T-value

Decreased margin of error and hence decreased the interval width

Improved interval precision

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