Changing the Confidence Level or Sample Size

Learning Outcomes

  • Explain how the margin of error changes when changing the confidence level or sample size

Recall: Interval Width

To calculate the width of an interval, take the larger number and subtract the smaller number, this will tell you how many numbers are in the interval, which is how wide the interval is. For example, the width of the interval [latex](2,10)[/latex] is [latex]10-2=8[/latex].

Changing the Confidence Level or Sample Size

Example 4

Suppose we change the original problem in Example 2 by using a 95% confidence level. Find a 95% confidence interval for the true (population) mean statistics exam score.

Interpretation

We estimate with 95% confidence that the true population mean for all statistics exam scores is between 67.02 and 68.98.

Explanation of 95% Confidence Level

Ninety-five percent of all confidence intervals constructed in this way contain the true value of the population mean statistics exam score.

Comparing the Results

The 90% confidence interval is (67.18, 68.82). The 95% confidence interval is (67.02, 68.98). The 95% confidence interval is wider. If you look at the graphs, because the area 0.95 is larger than the area 0.90, it makes sense that the 95% confidence interval is wider. To be more confident that the confidence interval actually does contain the true value of the population mean for all statistics exam scores, the confidence interval necessarily needs to be wider.

Graphs of changing the confidence interval from 09% to 95%

Summary: Effect of Changing the Confidence Level

  • Increasing the confidence level increases the error bound, making the confidence interval wider.
  • Decreasing the confidence level decreases the error bound, making the confidence interval narrower.

try it 4

Refer to the pizza-delivery Try It 2 exercise. The population standard deviation is six minutes and the sample mean deliver time is 36 minutes. Use a sample size of 20. Find a 95% confidence interval estimate for the true mean pizza delivery time.

Example 5

Suppose we change the original problem in Example 2 to see what happens to the error bound if the sample size is changed.

Leave everything the same except the sample size. Use the original 90% confidence level. What happens to the error bound and the confidence interval if we increase the sample size and use n = 100 instead of n = 36? What happens if we decrease the sample size to n = 25 instead of n = 36?

  • [latex]\overline{{x}}[/latex] = 68
  • EBM = [latex]{\left ( z_\frac{a}{2} \right )}{\left ( \frac{\sigma}{\sqrt n} \right )}[/latex]
  • σ = 3
  • The confidence level is 90% (CL=0.90)
  • [latex]{z_\frac{a}{2}}[/latex] = z0.05 = 1.645.

Try It 5

Refer to the pizza-delivery Try It 2 exercise. The mean delivery time is 36 minutes and the population standard deviation is six minutes. Assume the sample size is changed to 50 restaurants with the same sample mean. Find a 90% confidence interval estimate for the population mean delivery time.