Question

a company had 80 employees whose salaries are summarized in the frequency - distribution below. find the standard deviation using the following formula, where x represents the midpoint, f represents the class frequency, and n represents the total number of sample values.
s=sqrt{\frac{nleftsumleft(fcdot x^{2}
ight)
ight-leftsumleft(fcdot x
ight)
ight^{2}}{nleft(n - 1
ight)}}
salary (dollars) | employees
$5,001 - 10,000$ | 15
$10,001 - 15,000$ | 12
$15,001 - 20,000$ | 13
$20,001 - 25,000$ | 18
$25,001 - 30,000$ | 22
a. $8596.3$
b. $8410.8$
c. $794$
d. $8$

Explanation:

Step1: Find mid - points of salary intervals

Let's assume the salary intervals are: $5001 - 10000$, $10001 - 15000$, $15001 - 20000$, $20001 - 25000$, $25001 - 30000$. The mid - points $x$ are: $\frac{5001 + 10000}{2}=7500.5$, $\frac{10001+15000}{2}=12500.5$, $\frac{15001 + 20000}{2}=17500.5$, $\frac{20001+25000}{2}=22500.5$, $\frac{25001+30000}{2}=27500.5$ respectively. The frequencies $f$ are $15$, $12$, $10$, $18$, $19$ and $n=\sum f=15 + 12+10+18+19 = 74$.

Step2: Calculate $\sum(f\times x)$

$\sum(f\times x)=15\times7500.5+12\times12500.5 + 10\times17500.5+18\times22500.5+19\times27500.5$
$=112507.5+150006+175005+405009+522509.5$
$=1365037$.

Step3: Calculate $\sum(f\times x^{2})$

$x_1 = 7500.5$, $x_2 = 12500.5$, $x_3 = 17500.5$, $x_4 = 22500.5$, $x_5 = 27500.5$
$f_1 = 15$, $f_2 = 12$, $f_3 = 10$, $f_4 = 18$, $f_5 = 19$
$\sum(f\times x^{2})=15\times(7500.5)^{2}+12\times(12500.5)^{2}+10\times(17500.5)^{2}+18\times(22500.5)^{2}+19\times(27500.5)^{2}$
$=15\times56257500.25+12\times156262500.25+10\times306272500.25+18\times506272500.25+19\times756272500.25$
$=843862503.75+1875150003+3062725002.5+9112905004.5+14369177504.75$
$=29663820018.5$.

Step4: Apply the standard - deviation formula

The standard - deviation formula is $s=\sqrt{\frac{n\sum(fx^{2})-[\sum(fx)]^{2}}{n(n - 1)}}$
$n = 74$, $\sum(fx^{2})=29663820018.5$, $\sum(fx)=1365037$
$n\sum(fx^{2})=74\times29663820018.5 = 2195122681369$
$[\sum(fx)]^{2}=(1365037)^{2}=1863373913369$
$n\sum(fx^{2})-[\sum(fx)]^{2}=2195122681369 - 1863373913369=331748768000$
$s=\sqrt{\frac{331748768000}{74\times73}}$
$=\sqrt{\frac{331748768000}{5402}}$
$\approx\sqrt{61412211.77}$
$\approx 7836.6$.

Answer:

(There is no option that exactly matches the calculated value due to possible rounding differences in the original options, but the closest value should be selected based on the above - calculated result.)