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. class frequency, f 0 - 4 6 5 - 9 12 10 - 14 23 15 - 19 18 20 - 24 16 25 - 29 13 30+ 4 find an approximation for the sample mean. $\bar{x}=16.4$ (type an integer or decimal rounded to the nearest tenth as needed.) find an approximation for the sample standard deviation. $s=square$ (type an integer or decimal rounded to the nearest tenth as needed.)

Explanation:

Step1: Calculate mid - points

For class $0 - 4$, mid - point $x_1=\frac{0 + 4}{2}=2$; 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 $\sum_{i = 1}^{7}f_ix_i$

$f_1 = 6,x_1=2$; $f_2 = 12,x_2=7$; $f_3 = 23,x_3=12$; $f_4 = 18,x_4=17$; $f_5 = 16,x_5=22$; $f_6 = 13,x_6=27$; $f_7 = 4,x_7=32$.
$\sum_{i = 1}^{7}f_ix_i=6\times2+12\times7 + 23\times12+18\times17+16\times22+13\times27+4\times32$
$=12 + 84+276+306+352+351+128$
$=1809$.

Step3: Calculate $\sum_{i = 1}^{7}f_i$

$\sum_{i = 1}^{7}f_i=6 + 12+23+18+16+13+4=92$.

Step4: Calculate sample variance $s^2$

The formula for sample variance $s^2=\frac{\sum_{i = 1}^{n}f_i(x_i-\bar{x})^2}{n - 1}$, where $\bar{x}=\frac{\sum_{i = 1}^{n}f_ix_i}{\sum_{i = 1}^{n}f_i}=16.4$ (already given).
$(x_1-\bar{x})^2=(2 - 16.4)^2=(- 14.4)^2 = 207.36$; $(x_2-\bar{x})^2=(7 - 16.4)^2=(-9.4)^2 = 88.36$; $(x_3-\bar{x})^2=(12 - 16.4)^2=(-4.4)^2 = 19.36$; $(x_4-\bar{x})^2=(17 - 16.4)^2=(0.6)^2 = 0.36$; $(x_5-\bar{x})^2=(22 - 16.4)^2=(5.6)^2 = 31.36$; $(x_6-\bar{x})^2=(27 - 16.4)^2=(10.6)^2 = 112.36$; $(x_7-\bar{x})^2=(32 - 16.4)^2=(15.6)^2 = 243.36$.
$\sum_{i = 1}^{7}f_i(x_i-\bar{x})^2=6\times207.36+12\times88.36+23\times19.36+18\times0.36+16\times31.36+13\times112.36+4\times243.36$
$=1244.16+1060.32+445.28+6.48+501.76+1460.68+973.44$
$=5692.12$.
$s^2=\frac{5692.12}{92 - 1}=\frac{5692.12}{91}\approx62.55$.

Step5: Calculate sample standard deviation $s$

$s=\sqrt{s^2}=\sqrt{62.55}\approx7.9$.

Answer:

$7.9$