0.8 Ch. 8: confidence intervals

You want a confidence interval for a single mean where $\sigma$ is not known. If you use the TI-83/84 series, enter the data into a list and then use the function TInterval, data option. C-level is 95. The confidence interval is (296.4, 334.2). This function also calculates the sample mean (315.3) and sample standard deviation (20.4). TInterval is found in STAT TESTS.

If you want the students to use the formulas for a normal or for the Student-t confidence interval, you will need to use a table for the z-score or the t-score. The book does not have the tables but the Internet has several. Do a search on "z-score table" and "Student-t table."

First, you need to calculate the sample mean and the sample standard deviation.

• $\stackrel{-}{\mathrm{x}}=315.29$
• $s=20.40$

The confidence interval has the pattern : $(\stackrel{-}{x}-\text{EBM},\stackrel{-}{x}+\text{EBM})$

The error bound formula is : $\text{EBM}=t_{\frac{\alpha }{2}}\cdot \left(\frac{s}{\sqrt{n}}\right)$

$\text{CL}=0.95$ so $\alpha =0.05$ . Therefore, $\frac{\alpha }{2}=0.025$ .

Using the Student-t table with $\mathrm{df}=7-1=6$ , $t_{.025}=2.45$ .

$\text{EBM}=t_{\frac{\alpha }{2}}\cdot \left(\frac{s}{\sqrt{n}}\right)=t_{0.25}\cdot \frac{20.40}{\sqrt{7}}=2.45\cdot \frac{20.40}{\sqrt{7}}=18.89$

The confidence interval is $(\stackrel{-}{x}-\text{EBM},\stackrel{-}{x}+\text{EBM})$ = (315.29 - 18.89, 315.29 +18.89) = (296.4, 334.2)

We are 95% confident that the true average number of calories in a 4 ounce serving of french fries is between 196.4 and 334.2 calories.

You want a confidence interval for a single proportion. If you use the TI-83/84 series, use the function 1-PropZinterval. $x=102$ , $n=450$ , $\text{C−level}=80$ . The confidence interval is (.2077, .2590)

If you want to use the formulas, first, you need to calculate the estimated proportion.

$\mathrm{p\text{'}}=\frac{x}{n}=\frac{102}{450}=0.23$

The confidence interval has the pattern $(\mathrm{p\text{'}}-\text{EBP},\mathrm{p\text{'}}+\text{EBP})$ .

The error bound formula is $\text{EBP}=z_{\frac{\alpha }{2}}\cdot \sqrt{\frac{p\text{'}q\text{'}}{n}}$ where $q=1-p\text{'}$

$\text{CL}=0.80$ so $\alpha =0.20$ . Therefore, $\frac{\alpha }{2}=0.10$ .

Using the normal table (find one on the Internet), $z_{.10}=1.28$ . (Remind students that 0.10 is the area to the right. The area to the left is 0.90.)

$\text{EBP}=z_{\frac{\alpha }{2}}\cdot \sqrt{\frac{p\text{'}q\text{'}}{n}}=z_{.10}\cdot \sqrt{\frac{p\text{'}q\text{'}}{n}}=z_{.10}\cdot \sqrt{\frac{.23\cdot .77}{n}}=1.28\cdot \sqrt{\frac{.23\cdot .77}{450}}=0.03$

The confidence interval is : $(\mathrm{p\text{'}}-\text{EBP},\mathrm{p\text{'}}+\text{EBP})$ = (0.23 - 0.03, 0.23 + 0.03) = (0.20, 0.26)

We are 80% confident that the true proportion of families that have children on the swim team in any year is between 0.20 and 0.26.