Question

the distribution of the number of hours that a random sample of people spend doing chores per week is shown in the pie chart. use 32 as the midpoint for \30+ hours.\ make a frequency distribution for the data. then use the table to estimate the sample mean and the sample standard deviation of the data set. click the icon to view the pie chart.
first construct the frequency distribution.
find an approximation for the sample mean.
find an approximation for the sample standard deviation.

Explanation:

Step1: Calculate midpoints

For class \(0 - 4\), midpoint \(x_1=\frac{0 + 4}{2}=2\); for \(5 - 9\), \(x_2=\frac{5+9}{2}=7\); for \(10 - 14\), \(x_3=\frac{10 + 14}{2}=12\); for \(15 - 19\), \(x_4=\frac{15+19}{2}=17\); for \(20 - 24\), \(x_5=\frac{20+24}{2}=22\); for \(25 - 29\), \(x_6=\frac{25+29}{2}=27\); for \(30+\), \(x_7 = 32\).

Step2: Calculate \(\sum f\)

\(\sum f=6 + 9+23+18+16+13+5=90\)

Step3: Calculate \(\sum(f\cdot x)\)

\(\sum(f\cdot x)=6\times2+9\times7 + 23\times12+18\times17+16\times22+13\times27+5\times32\)
\(=12+63+276+306+352+351+160\)
\(=1520\)

Step4: Calculate sample mean (already given as \(\bar{x}=16.9\))

Step5: Calculate \(\sum(f\cdot x^{2})\)

\(\sum(f\cdot x^{2})=6\times2^{2}+9\times7^{2}+23\times12^{2}+18\times17^{2}+16\times22^{2}+13\times27^{2}+5\times32^{2}\)
\(=6\times4+9\times49+23\times144+18\times289+16\times484+13\times729+5\times1024\)
\(=24+441+3312+5202+7744+9477+5120\)
\(=31320\)

Step6: Calculate sample standard deviation formula

\(s=\sqrt{\frac{\sum(f\cdot x^{2})}{n - 1}-\frac{(\sum(f\cdot x))^{2}}{(n - 1)n}}\), where \(n=\sum f = 90\)
\(s=\sqrt{\frac{31320}{89}-\frac{1520^{2}}{89\times90}}\)
\(=\sqrt{\frac{31320\times90-1520^{2}}{89\times90}}\)
\(=\sqrt{\frac{2818800 - 2310400}{8010}}\)
\(=\sqrt{\frac{508400}{8010}}\approx\sqrt{63.47}\approx7.97\approx8.0\)

Answer:

\(s = 8.0\)