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+\. first construct the frequency distribution. class: 0 - 4, 5 - 9, 10 - 14, 15 - 19, 20 - 24, 25 - 29, 30+; frequency, f: 6, 10, 23, 19, 14, 11, 5. find an approximation for the sample mean. x = 16.43 (type an integer or decimal rounded to the nearest tenth as needed.)

Explanation:

Step1: Find mid - points of each class

For 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+, $x_7 = 32$ (given).

Step2: Calculate the product of mid - points and frequencies

Let $f_1 = 6,f_2=10,f_3 = 23,f_4=19,f_5=14,f_6=11,f_7 = 5$. Then $f_1x_1=6\times2 = 12$, $f_2x_2=10\times7 = 70$, $f_3x_3=23\times12 = 276$, $f_4x_4=19\times17 = 323$, $f_5x_5=14\times22 = 308$, $f_6x_6=11\times27 = 297$, $f_7x_7=5\times32 = 160$.

Step3: Calculate the sum of frequencies and the sum of products

$\sum f_i=6 + 10+23+19+14+11+5=88$. $\sum f_ix_i=12 + 70+276+323+308+297+160 = 1446$.

Step4: Calculate the sample mean

The sample mean $\bar{x}=\frac{\sum f_ix_i}{\sum f_i}=\frac{1446}{88}\approx16.4$.

Answer:

$16.4$