Question

Jay –Ethan Company is a multi-billion pharmaceutical company that has won a contract by the government to produce hand sanitizers and respiratory gadgets for frontline health workers in Bono East region. As a result, the company has decided to give bonuses to its employees to motivate them for the risky task ahead. A sample of the weekly bonus received by the employees is organized in the table below:
Weekly Bonus ($) 30<35 35<40 40<45 45<50 50<55 55<60 60<56 65<70
Number of employees(f) 8 6 10 5 15 4 9 3
NB: Approximate to 2 decimal places
i) Find the mean amount of bonus given to the employees

ii) Calculate the standard deviation and interpret it.

iii) What is the modal amount paid?

iv) Would you say the bonus distribution follows the standard normal distribution? Why?

Answer 1

Suppose, random variable X denotes amount of bonus given to an employee.

Necessary calculations are as follows.

(i)

Mean amount of bonus given to the employees is given by

Hence, mean amount of bonus is 48.83.

(ii)

Variance of amount of bonus given to the employees is given by

So, standard deviation of amount of bonus given to the employees is given by

Hence, standard deviation of bonus is 10.44.

Interpretation-

Standard deviation is square root of mean of all squared deviations of values from mean. So, square root of mean of all squared deviations of values from mean (48.83) is 10.44.

(iii)

Here, highest frequency corresponds to class 50-55. So, this is the modal class.

We know,

Here,

Lower boundary of modal class

Frequency of modal class

Frequency preceding modal class

Frequency proceeding modal class

Width of modal class

Hence, modal amount of bonus is 52.38.

(iv)

Bar graph of given data is as follows.

We observe that the distribution is approximately normally distributed. So, we may conclude that the bonus distribution follows (approximately) normal distribution.

For standard normal distribution, mean is to be 0 and standard deviation 1. Given data of X does not satisfy either of those properties (However, after applying proper transformation may satisfy those properties). So, we conclude that the bonus distribution does not follow standard normal distribution.