Question

first construct the frequency distribution.
class frequency, f
0 - 4 4
5 - 9 12
10 - 14 25
15 - 19 19
20 - 24 18
25 - 29 14
30+ 4
find an approximation for the sample mean.
x̄ = 16.8 (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+, assume mid - point $x_7 = 32$.

Step2: Calculate $\sum_{i = 1}^{7}f_ix_i$

$f_1 = 4,f_2=12,f_3 = 25,f_4=19,f_5=18,f_6=14,f_7 = 4$.
$\sum_{i = 1}^{7}f_ix_i=4\times2+12\times7 + 25\times12+19\times17+18\times22+14\times27+4\times32$
$=8+84 + 300+323+396+378+128$
$=1617$.

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

$\sum_{i = 1}^{7}f_i=4 + 12+25+19+18+14+4=96$.

Step4: Calculate sample variance formula part 1

The sample mean $\bar{x}=\frac{\sum_{i = 1}^{7}f_ix_i}{\sum_{i = 1}^{7}f_i}=\frac{1617}{96}\approx16.8$ (already given).

Step5: Calculate $\sum_{i = 1}^{7}f_i(x_i-\bar{x})^2$

$(x_1-\bar{x})^2=(2 - 16.8)^2=(- 14.8)^2 = 219.04$, $f_1(x_1-\bar{x})^2=4\times219.04 = 876.16$;
$(x_2-\bar{x})^2=(7 - 16.8)^2=(-9.8)^2 = 96.04$, $f_2(x_2-\bar{x})^2=12\times96.04 = 1152.48$;
$(x_3-\bar{x})^2=(12 - 16.8)^2=(-4.8)^2 = 23.04$, $f_3(x_3-\bar{x})^2=25\times23.04 = 576$;
$(x_4-\bar{x})^2=(17 - 16.8)^2=(0.2)^2 = 0.04$, $f_4(x_4-\bar{x})^2=19\times0.04 = 0.76$;
$(x_5-\bar{x})^2=(22 - 16.8)^2=(5.2)^2 = 27.04$, $f_5(x_5-\bar{x})^2=18\times27.04 = 486.72$;
$(x_6-\bar{x})^2=(27 - 16.8)^2=(10.2)^2 = 104.04$, $f_6(x_6-\bar{x})^2=14\times104.04 = 1456.56$;
$(x_7-\bar{x})^2=(32 - 16.8)^2=(15.2)^2 = 231.04$, $f_7(x_7-\bar{x})^2=4\times231.04 = 924.16$.
$\sum_{i = 1}^{7}f_i(x_i-\bar{x})^2=876.16+1152.48+576+0.76+486.72+1456.56+924.16=5472.8$.

Step6: 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 $n=\sum_{i = 1}^{7}f_i = 96$.
$s^2=\frac{5472.8}{96 - 1}=\frac{5472.8}{95}\approx57.61$.

Step7: Calculate sample standard deviation $s$

$s=\sqrt{s^2}=\sqrt{57.61}\approx7.6$.

Answer:

$7.6$