Chapter 1, Problem 23P

a.

To determine

To find: The value of $\sum X^2$.

Answer to Problem 23P

The value of $\sum X^2$ is 50.

Explanation of Solution

Given info:

X represents the scores. And the scores are 1,6,2 and 3.

Calculation:

X represents the scores Y represents the values when square of each score is done, that is:

$Y=X^2$

Then,

$\sum Y=\sum X^2$

Software procedure:

Step-by-step procedure to obtain the $\sum X^2$ using the SPSS software:

• Go to Variable View> Enter the name of the variable as X and second variable as Y.
• Go to Data View>Enter the values of X.
• Go to Transform>Choose Compute variable>Enter Y in Target Variable
• Choose X >Move X to Numeric Expression.
• Choose (^) sign > Enter 2.
• Choose OK.
• Choose Analyze > Descriptive Statistics> choose Descriptive.
• Select Y and move it under variable(s)> Choose Options> Select Sum.
• Choose Continue.
• Choose OK.

Output using the SPSS software is given below:

From the SPSS output, the value of $\sum Y=\sum X^2$ is 50.

Thus, the value of $\sum X^2$ is 50.

b.

To determine

To Find: The value of ${\left(\sum X\right)}^2$.

Answer to Problem 23P

The value of ${\left(\sum X\right)}^2$ is 144.

Explanation of Solution

Calculation:

X represents the scores.

$\sum X$ represent the sum of all scores.

Software procedure:

Step-by-step procedure to obtain the $\sum X$ using the SPSS software:

• Go to Variable View> Enter the name of the variable as X.
• Go to Data View>Enter the values of X.
• Choose Analyze > Descriptive Statistics> choose Descriptive.
• Select X and move it under variable(s)> Choose Options> Select Sum.
• Choose Continue
• Choose OK.

Output using the SPSS software is given below:

From the SPSS output, the value of $\sum X$ is 12.

$\left(\sum X\right)=12$

The value of ${\left(\sum X\right)}^2$ is calculated as:

$\begin{array}{c}{\left(\sum X\right)}^2=\left(\sum X\right)\times \left(\sum X\right)\\ =12\times 12\\ =144\end{array}$

Thus, the value of ${\left(\sum X\right)}^2$ is 144.

c.

To determine

To Find: The value of $\sum (X-3)$.

Answer to Problem 23P

The value of $\sum (X-3)$ is 0.

Explanation of Solution

Calculation:

X represents the scores.

Y represents the values when 3 is subtracted from each score, that is:

$Y=X-3$

Then,

$\sum Y=\sum (X-3)$

Software procedure:

Step-by-step procedure to obtain the $\sum (X-3)$ using the SPSS software:

• Go to Variable View> Enter the name of the variable as X and second variable as Y.
• Go to Data View>Enter the values of X.
• Go to Transform>Choose Compute variable>Enter Y in Target Variable> choose X >Move X to Numeric Expression.
• Choose (–) sign > Enter 3.
• Choose OK.
• Choose Analyze > Descriptive Statistics> choose Descriptive.
• Select Y and move it under variable(s)> Choose Options> Select Sum.
• Choose Continue> choose OK.

Output using the SPSS software is given below:

From the SPSS output, the value of $\sum Y=\sum (X-3)$ is 0.

Thus, the value of $\sum (X-3)$ is 0.

d.

To determine

To Find: The value of $\sum {(X-3)}^2$.

Answer to Problem 23P

The value of $\sum {(X-3)}^2$ is 14.

Explanation of Solution

Calculation:

X represents the scores.

Y represents the values when 3 is subtracted from each score and then squaring it that is:

$Y={(X-3)}^2$

Then,

$\sum Y=\sum {(X-3)}^2$

Software procedure:

Step-by-step procedure to obtain the $\sum {(X-3)}^2$ using the SPSS software:

• Go to Variable View> Enter the name of the variable as X and second variable as Y.
• Go to Data View>Enter the values of X.
• Go to Transform>Choose Compute variable>Enter Y in Target Variable.
• Write $\left(X-3\right)\times \left(X-3\right)$ in Numeric Expression.
• Choose OK.
• Choose Analyze > Descriptive Statistics> choose Descriptive.
• Select Y and move it under variable(s)> Choose Options> Select Sum.
• Choose Continue> choose OK.

Output using the SPSS software is given below:

From the SPSS output, the value of $\sum Y=\sum {(X-3)}^2$ is 14.

Thus, the value of $\sum {(X-3)}^2$ is 14.