8.1 A confidence interval for a single population mean using the  (Page 4/7)

To find the confidence interval, you need the sample mean, $\stackrel{-}{x}$ , and the EBM .

• $\stackrel{-}{x}$ = 68
• EBM = $\left(Z_{\frac{\alpha }{2}}\right)\left(\frac{\sigma }{\sqrt{n}}\right)$
• σ = 3; n = 36; The confidence level is 95% ( CL = 0.95).

CL = 0.95 so α = 1 – CL = 1 – 0.95 = 0.05

$\frac{\alpha }{2}=0.025Z_{\frac{\alpha }{2}}=Z_{0.025}$

The area to the right of Z _0.025 is 0.025 and the area to the left of Z _0.025 is 1 – 0.025 = 0.975.

$Z_{\frac{\alpha }{2}}=Z_{0.025}=1.96$

EBM = (1.96) $\left(\frac{3}{\sqrt{36}}\right)$ = 0.98

$\stackrel{-}{x}$ – EBM = 68 – 0.98 = 67.02

$\stackrel{-}{x}$ + EBM = 68 + 0.98 = 68.98

Notice that the EBM is larger for a 95% confidence level in the original problem.

Interpretation

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

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. This demonstrates a very important principle of confidence intervals. There is a trade off between the level of confidence and the width of the interval. Our desire is to have a narrow confidence interval, huge wide intervals provide little information that is useful. But we would also like to have a high level of confidence in our interval. This demonstrates that we cannot have both.

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.