Question

first construct the frequency distribution.
class frequency, f
0 - 4 6
5 - 9 10
10 - 14 23
15 - 19 19
20 - 24 14
25 - 29 11
30+ 5
find an approximation for the sample mean.
x̄ = 16.4 (type an integer or decimal rounded to the nearest tenth as needed.)
find an approximation for the sample standard deviation.
s = (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+, $x_7 = 32$ (assumed mid - point).

Step2: Calculate $\sum_{i = 1}^{n}f_ix_i$ and $\sum_{i=1}^{n}f_i$

$\sum_{i = 1}^{7}f_i=6 + 10+23+19+14+11+5=88$.
$\sum_{i = 1}^{7}f_ix_i=6\times2+10\times7+23\times12+19\times17+14\times22+11\times27+5\times32$
$=12 + 70+276+323+308+297+160$
$=1446$.
The sample mean $\bar{x}=\frac{\sum_{i = 1}^{n}f_ix_i}{\sum_{i = 1}^{n}f_i}=\frac{1446}{88}\approx16.4$ (already given).

Step3: Calculate $\sum_{i = 1}^{n}f_i(x_i-\bar{x})^2$

$(x_1-\bar{x})^2=(2 - 16.4)^2=(- 14.4)^2 = 207.36$, $f_1(x_1-\bar{x})^2=6\times207.36 = 1244.16$;
$(x_2-\bar{x})^2=(7 - 16.4)^2=(-9.4)^2 = 88.36$, $f_2(x_2-\bar{x})^2=10\times88.36 = 883.6$;
$(x_3-\bar{x})^2=(12 - 16.4)^2=(-4.4)^2 = 19.36$, $f_3(x_3-\bar{x})^2=23\times19.36 = 445.28$;
$(x_4-\bar{x})^2=(17 - 16.4)^2=(0.6)^2 = 0.36$, $f_4(x_4-\bar{x})^2=19\times0.36 = 6.84$;
$(x_5-\bar{x})^2=(22 - 16.4)^2=(5.6)^2 = 31.36$, $f_5(x_5-\bar{x})^2=14\times31.36 = 439.04$;
$(x_6-\bar{x})^2=(27 - 16.4)^2=(10.6)^2 = 112.36$, $f_6(x_6-\bar{x})^2=11\times112.36 = 1235.96$;
$(x_7-\bar{x})^2=(32 - 16.4)^2=(15.6)^2 = 243.36$, $f_7(x_7-\bar{x})^2=5\times243.36 = 1216.8$.
$\sum_{i = 1}^{7}f_i(x_i-\bar{x})^2=1244.16+883.6+445.28+6.84+439.04+1235.96+1216.8 = 5471.72$.

Step4: Calculate the sample standard deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}f_i(x_i-\bar{x})^2}{n - 1}}$, where $n=\sum_{i = 1}^{n}f_i = 88$.
$s=\sqrt{\frac{5471.72}{88 - 1}}=\sqrt{\frac{5471.72}{87}}\approx\sqrt{62.8933}\approx7.9$

Answer:

$7.9$