7.1 A single population mean using the normal distribution -- rrc  (Page 4/19)

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

• $\overline{x}$ = 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.

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

This can be found using appropriate commands on calculators, using a computer, or using a probability table for the standard normal distribution.

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

$\overline{x}$ – EBM = 68 – 0.98 = 67.02

$\overline{x}$ + 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.

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.

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.

Solution b

If we decrease the sample size n to 25, we increase the error bound.

When n = 25: EBM = $\left(z_{\frac{\alpha }{2}}\right)\left(\frac{\sigma }{\sqrt{n}}\right)$ = (1.645) $\left(\frac{3}{\sqrt{25}}\right)$ = 0.987.