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)?
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
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