b)
Given:
[tex]\begin{gathered} \bar{x}=10.2 \\ s=2.12 \end{gathered}[/tex]
Hence,
[tex]\begin{gathered} \bar{x}\pm s=10.2\pm2.12 \\ \bar{x}+s=12.32 \\ \bar{x}-s=8.08 \end{gathered}[/tex]
So, the measurements in the data between 8.08 and 12.32 are 11, 9, 12, 10 12, 12 , 12, 9, 9, 9, 11, 11, 12 and 11.
Therefore, the number of measurements in interval x±s is 14.
The percentage of the measurements that fall between the interval x±s is,
[tex]\text{Percent}=\frac{14}{20}\times100=70[/tex]
Therefore, the percentage of the measurements that fall between the interval x±s is 70%.
Now,
[tex]\begin{gathered} \bar{x}\pm2s=10.2\pm2\times2.12 \\ \bar{x}\pm2s=10.2\pm4.24 \\ \bar{x}+2s=14.44 \\ \bar{x}-2s=5.96 \end{gathered}[/tex]
So, all the measurements in the data are between 5.96 and 14.44.Therefore, the number of measurements in interval x±2s is 20.
Therefore, the percentage of the measurements that fall between the interval x±2s is 100%.
Now,
[tex]\begin{gathered} \bar{x}\pm3s=10.2\pm3\times2.12 \\ \bar{x}\pm3s=10.2\pm6.36 \\ \bar{x}+3s=16.56 \\ \bar{x}-3s=3.84 \end{gathered}[/tex]
So, all the measurements in the data are between 3.84 and 16.56.Therefore, the number of measurements in interval x±3s is 20.
Therefore, the percentage of the measurements that fall between the interval x±3s is 100%.
Last part: compare the percentage .
According to empirical rule, approximately 68% of the measurements in a sample will fall within the interval x±s.
From part b, the obtained percentage of measurements that fall within the interval x±s is 70%.
Therefore, percentage of measurements that fall within the interval x±s is greater than the predicted percentage for x±s using the empirical rule.
Option C is correct.
