Question

the stem - and - leaf plot shows the test scores of 16 randomly selected students. construct a 99% confidence interval for the population standard deviation.
5 | 9
6 | 5 8 3
7 | 7 4 4 2 9
8 | 5 8 3 5
9 | 3 1 7
a. (57.97,413.27)
b. (7.89,19.07)
c. (7.61,20.33)
d. (62.18,363.63)

Explanation:

Step1: Calculate sample standard deviation

First, write out the data: \(59,63,65,68,72,74,74,77,79,83,85,88,91,93,97\).
The formula for sample standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\), where \(n = 16\).
\(\bar{x}=\frac{59+63+65+68+72+74+74+77+79+83+85+88+91+93+97}{16}=\frac{1218}{16}=76.125\)
\(\sum_{i = 1}^{16}(x_{i}-\bar{x})^{2}=(59 - 76.125)^{2}+(63 - 76.125)^{2}+\cdots+(97 - 76.125)^{2}=4357.875\)
\(s=\sqrt{\frac{4357.875}{15}}\approx17.0\)

Step2: Find critical values

For a \(99\%\) confidence interval and \(n-1 = 15\) degrees of freedom, \(\alpha=0.01\), \(\frac{\alpha}{2}=0.005\) and \(1-\frac{\alpha}{2}=0.995\)
From the \(\chi^{2}\) - distribution table, \(\chi_{L}^{2}=\chi_{0.005,15}^{2}=32.801\), \(\chi_{R}^{2}=\chi_{0.995,15}^{2}=4.601\)

Step3: Calculate confidence interval

The formula for the confidence interval for the population standard deviation \(\sigma\) is \(\sqrt{\frac{(n - 1)s^{2}}{\chi_{U}^{2}}}<\sigma<\sqrt{\frac{(n - 1)s^{2}}{\chi_{L}^{2}}}\)
Substitute \(n = 16\), \(s\approx17.0\), \(\chi_{L}^{2}=32.801\), \(\chi_{R}^{2}=4.601\)
\(\sqrt{\frac{15\times17^{2}}{32.801}}\approx7.89\), \(\sqrt{\frac{15\times17^{2}}{4.601}}\approx19.07\)

Answer:

B. \((7.89,19.07)\)