Question

Sample annual salaries​ (in thousands of​ dollars) for employees at a company are listed.

40  33  54  52  28  28  40  33  54  27  52  40  45

  

a. Find the sample mean and sample standard deviation.

​(b) Each employee in the sample is given a ​$22000 raise. Find the sample mean and sample standard deviation for the revised data set.

​(c) Each employee in the sample takes a pay cut of ​$55000 from their original salary. Find the sample mean and the sample standard deviation for the revised data set.

​(d) What can you conclude from the results of​ (a), (b), and​ (c)?

Answer 1

a)

X	(X - X̄)²
40	0.213
33	55.675
54	183.290
52	133.136
28	155.290
28	155.290
40	0.213
33	55.675
54	183.290
27	181.213
52	133.136
40	0.213
45	20.598

	X	(X - X̄)²
total sum	526	1257.230769
n	13	13

mean =    ΣX/n =    40.46153(in thousands of​ dollars)
            
sample std dev =   √ [ Σ(X - X̄)²/(n-1)] =   10.2357(in thousands of​ dollars)

__________________________________________

b)

Each employee in the sample is given a ​$22000 raise i.e $ 22 (in thousands of​ dollars)

so, mean = Σ(x+22)/n = (Σx+22n)/n = Σx/n + 22 = old mean+22 = 40.46153+22 = 62.46154 (in thousands of​ dollars)

std dev = √ [ Σ((X+22) - (X̄+22))²/(n-1)] = √ [ Σ(X - X̄)²/(n-1)] =  10.2357(in thousands of​ dollars)

c) Each employee in the sample takes a pay cut of ​$55000 from their original salary.i.e $ 55(in thousands of​ dollars)

so, mean = Σ(x-55)/n = (Σx-55n)/n = Σx/n - 55 = old mean-55 = 40.46153-55 = -14.5384 (in thousands of​ dollars)

std dev = √ [ Σ((X-55) - (X̄-55))²/(n-1)] = √ [ Σ(X - X̄)²/(n-1)] =  10.2357(in thousands of​ dollars)

d)

we conclude that on increasing/decreasing a partcular value from each sample value, sample mean increases/decreases resp.

but std dev gets remain the same