Answer to Question #109867 in Statistics and Probability for julaine

Question

in a bid to determine the effectiveness of their succession planning. Elemenopee ltd is interested in determining the age distribution of its employees. In this regard
the company collected ages from each of their employees and summarized as follows;

AGE FREQUENCY
0 under 20 4
20 under 30 9
30 under 40 18
40 under 50 26
50 under 60 17
60 under 70 6

Using the grouped frequency distribution calculate
1. Mean
2. Median
3. Mode
4.Standard deviation
5. from the calculation in question 3.1 above do you think that the company has sufficient talent to adequately plan its successors? Justify your answer.
6. what is the best measure of central location in this scenario? Justify your answer

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} Class\ limits & f & x & fx & \begin{gathered} Cumulative \\ frequency, F \end{gathered} \\ \hline 0-19 & 4 & 9.5 & 38 & 4 \\ \hline 20-29 & 9 & 24.5 & 220.5 & 13 \\ \hline 30-39 & 18 & 34.5 & 621 & 31 \\ \hline 40-49 & 26 & 44.5 & 1157 & 57 \\ \hline 50-59 & 17 & 54.5 & 926.5 & 74 \\ \hline 60-69 & 6 & 64.5 & 387 & 80 \\ \hdashline Total & 80 & & 3350 \end{array}$$

1.

$$Mean=\bar{x}={\sum fx \over \sum f}={3350 \over 80}=41.875$$

2. $80/2=40$

From the column of cumulative frequency, we find that the 40th observation lies in the class 40-49.

The median class is $39.5-49.5$

$L=39.5, n=80, F=31,f=26, class\ length=c=10$

$$Median=M=L+{{n \over 2}-F\over f}\cdot c=$$

$$=39.5+{{80 \over 2}- 31\over 26}\cdot 10\approx42.9615$$

3. Here class length is not equal for each data. So mode can not be obtained directly.

$$Mode=Z=3M-2\bar{x}=$$

$$=3(42.915)-2(41.875)\approx45.1345$$

4.

$A=44.5, h=1, d={x-A \over h}$

$$Sample\ variance=S^2={\sum fd^2-{(\sum fd)^2 \over n} \over n-1}\cdot h^2$$

$$\sum fd=4(9.5-44.5)+9(24.5-44.5)+18(34.5-44.5)+$$

$$+26(44.5-44.5)+17(54.5-44.5)+6(534.5-44.5)=-210$$

$$\sum fd^2=4(9.5-44.5)^2+9(24.5-44.5)^2+18(34.5-44.5)^2+$$

$$+26(44.5-44.5)^2+17(54.5-44.5)^2+6(534.5-44.5)^2=14400$$

$$S^2={14400-{(-210)^2 \over 80} \over 80-1}\cdot (1)^2\approx175.3006$$

$$Standard\ deviation=S=\sqrt{S^2}=$$

$$=\sqrt{175.3006}\approx13.2401$$

5.

$Mean<Median<Mode.$

The distribution is left-skewed. The company has not sufficient talent to adequately plan its successors.

6. Type of Variable: Interval/Ratio (skewed)

Best measure of central tendency: Median