Chapter 5, Problem 24P

A population consists of the following N – 6 scores: 2, 4, 1, 2, 7, 2

1. a. Compute μ and σ for the population.
2. b. Find the z-score for each score in the population.
3. c. Transform the original population into a new population of N = 5 scores with a mean of μ = 50 and a standard deviation of σ = 10.

To determine

To Find: The mean and standard deviation for the population scores.

Answer to Problem 24P

The mean and standard deviation for the population scores is 3 and 2.

Explanation of Solution

Given info:

The population consists of the score is, 2, 4, 1, 2, 7, and 2.

Calculation:

The formula for calculating the population mean is,

$\mu =\frac{{\displaystyle \sum x_i}}{N}$

The formula for calculating the population standard deviation is,

$\sigma =\sqrt{\frac{{\displaystyle \sum {(x-\mu )}^2}}{N}}$

The population mean is,

$\begin{array}{c}\mu =\frac{{\displaystyle \sum x_i}}{N}\\ =\frac{2+4+1+2+7+2}{6}\\ =\frac{18}{6}\\ =3\end{array}$

The population standard deviation is,

$\begin{array}{c}\sigma =\sqrt{\frac{{\displaystyle \sum {(x-\mu )}^2}}{N}}\\ =\sqrt{\frac{{(2-3)}^2+{(4-3)}^2+{(1-3)}^2+{(2-3)}^2+{(7-3)}^2+{(2-3)}^2}{6}}\\ =\sqrt{4}\\ =2\end{array}$

Conclusion:

Therefore, the required population mean and standard deviation are 3 and 2.

To determine

To Find: The z-score for each score in the population.

Answer to Problem 24P

The z-score for each score in the population is,

X- score	z -score
2	–0.5
4	0.5
1	–1
2	–0.5
7	2
2	–0.5

Explanation of Solution

Given Info:

The population has mean $\mu =3$ and standard deviation $\sigma =2.$

Calculation:

The formula for calculating the z-score is,

$z=\frac{X-\mu }{\sigma }$

The z-score corresponding to X score 2 is,

$\begin{array}{c}z=\frac{X-\mu }{\sigma }\\ =\frac{2-3}{2}\\ =-0.50\end{array}$

The z-score corresponding to X score 4 is,

$\begin{array}{c}z=\frac{X-\mu }{\sigma }\\ =\frac{4-3}{2}\\ =0.50\end{array}$

The z-score corresponding to X score 1 is,

$\begin{array}{c}z=\frac{X-\mu }{\sigma }\\ =\frac{1-3}{2}\\ =-1.00\end{array}$

The z-score corresponding X score 2 is,

$\begin{array}{c}z=\frac{X-\mu }{\sigma }\\ =\frac{2-3}{2}\\ =-0.50\end{array}$

The z-score corresponding X score 7 is,

$\begin{array}{c}z=\frac{X-\mu }{\sigma }\\ =\frac{7-3}{2}\\ =2.00\end{array}$

The z-score corresponding X score 2 is,

$\begin{array}{c}z=\frac{X-\mu }{\sigma }\\ =\frac{2-3}{2}\\ =-0.50\end{array}$

Conclusion:

The required z-score for each score in the population is,

X- score	z-score
2	–0.5
4	0.5
1	–1
2	–0.5
7	2
2	–0.5

To determine

To find: Transform original population in to new population with mean $\mu =50$ and standard deviation $\sigma =10.$

Answer to Problem 24P

The transformed original population in to new population is,

Original X-score	z-score	Transformed X
2	–0.5	45
4	0.5	55
1	–1	40
2	–0.5	45
7	2	70
2	–0.5	45

Explanation of Solution

Given Info:

The sample has mean $\mu =50$ and standard deviation $\sigma =10.$

Calculation:

The formula for transform original population in to new population is,

$X=\mu +z\times \sigma$

The original population value 2 is transfer into a new population is,

$\begin{array}{c}X=\mu +z\times \sigma \\ =50+(-0.50)\times 10\\ =45\end{array}$

The original population value 4 is transfer into a new population is,

$\begin{array}{c}X=\mu +z\times \sigma \\ =50+(0.50)\times 10\\ =55\end{array}$

The original population value 1 is transfer into a new population is,

$\begin{array}{c}X=\mu +z\times \sigma \\ =50+(-1.00)\times 10\\ =40\end{array}$

The original population value 2 is transfer into a new population is,

$\begin{array}{c}X=\mu +z\times \sigma \\ =50+(-0.50)\times 10\\ =45\end{array}$

The original population value 7 is transfer into a new population is,

$\begin{array}{c}X=\mu +z\times \sigma \\ =50+(2.00)\times 10\\ =70\end{array}$

The original population value 2 is transfer into a new population is,

$\begin{array}{c}X=\mu +z\times \sigma \\ =50+(-0.50)\times 10\\ =45\end{array}$

Conclusion:

Thus, the below table shows the transformed original population in to new population:

Original X-score	z-score	Transformed X
2	–0.5	45
4	0.5	55
1	–1	40
2	–0.5	45
7	2	70
2	–0.5	45