Chapter 1, Problem 19P

For the following set of scores, find the value of each expression:

X

1

2

4

1

3

a. ΣX²

b. (ΣX)²

c. Σ(X + 1)

d. Σ(X + 1)²

To determine

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

Answer to Problem 19P

The value of $\sum X^2$ is 31.

Explanation of Solution

Given info:

X represents the scores. The scores are 1, 2, 4, 1, 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:

Descriptive Statistics
	N	Sum
Y	5	31.00
Valid N (listwise)	5

From the SPSS output, the value of $\sum Y=\sum X^2$ is 31.

Conclusion:

The value of $\sum X^2$ is 31.

To determine

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

Answer to Problem 19P

The value of ${\left(\sum X\right)}^2$ is 121.

Explanation of Solution

Given info:

X represents the scores.

The scores are 1, 2, 4, 1 and 3.

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:

Descriptive Statistics
	N	Sum
X	5	11.00
Valid N ( list wise )	5

From the SPSS output, the value of $\sum X$ is 11.

$\left(\sum X\right)=11$

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

$\begin{array}{c}{\left(\sum X\right)}^2=\left(\sum X\right)\times \left(\sum X\right)\\ =11\times 11\\ =121\end{array}$

Conclusion:

The value of ${\left(\sum X\right)}^2$ is 121.

To determine

To Find: The value of $\sum (X+1)$.

Answer to Problem 19P

The value of $\sum (X+1)$ is 16.

Explanation of Solution

Given info:

X represents the scores.

The scores are 1, 2, 4, 1 and 3.

Calculation:

X  represents the scores.

Y represents the values when 1 is added from each score, that is:

$Y=X+1$

Then,

$\sum Y=\sum (X+1)$

Software procedure:

Step-by-step procedure to obtain the $\sum (X+1)$ 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 1.
• 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:

Descriptive Statistics
	N	Sum
Y	5	16.00
Valid N (listwise)	5

From the SPSS output, the value of $\sum Y=\sum (X+1)$ is 16.

Conclusion:

The value of $\sum (X+1)$ is 16.

To determine

To Find: The value of $\sum {(X+1)}^2$.

Answer to Problem 19P

The value of $\sum {(X+1)}^2$ is 58.

Explanation of Solution

Given info:

X represents the scores.

The scores are 1, 2, 4, 1 and 3.

Calculation:

X  represents the scores.

Y represents the values when 1 is added from each score and then squaring it that is:

$Y={(X+1)}^2$

Then,

$\sum Y=\sum {(X+1)}^2$

Software procedure:

Step-by-step procedure to obtain the $\sum {(X+1)}^2$ 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 (X+1)*(X+1) 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:

Descriptive Statistics
	N	Sum
Y	5	58.00
Valid N (listwise)	5

From the SPSS output, the value of $\sum Y=\sum {(X+1)}^2$ is 58.

Conclusion:

The value of $\sum {(X+1)}^2$ is 58.