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4th Edition

HEALEY + 1 other

Publisher: Cengage Learning,

ISBN: 9781305093836

Chapter 13, Problem 13.3P

Textbook Problem

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Twelve families have been referred to a counselor, and she has rated each of them on a scale that measures family cohesiveness. Also, she has information on family income and number of children currently living at home. Take family cohesion as the dependent variable.

Family |
Cohesion Score |
Income |
Number of Children |

A | 10 | 30,000 | 5 |

B | 10 | 70,000 | 4 |

C | 9 | 35,000 | 4 |

D | 5 | 25,000 | 0 |

E | 1 | 55,000 | 3 |

F | 7 | 40,000 | 0 |

G | 2 | 60,000 | 2 |

H | 5 | 30,000 | 3 |

I | 8 | 50,000 | 5 |

J | 3 | 25,000 | 4 |

K | 2 | 45,000 | 3 |

L | 4 | 50,000 | 0 |

**a.** Find the multiple regression equations (unstandardized).

**b.** What level of cohesion would be expected in a family with an income of $20,000 and six children?

**c.** Compute beta-weights for each independent variable and compare their relative effect on cohesion. Which was the more important factor?

**d.** Compute

**e.** Write a paragraph summarizing your findings.

To determine

**(a)**

**To find:**

The unstandardized multiple regression equation with family cohesion as the dependent variable.

**Given:**

The table for a sample of twelve families is given below.

Family | Cohesion Score | Income | Number of Children |

A | 10 | 30, 000 | 5 |

B | 10 | 70, 000 | 4 |

C | 9 | 35, 000 | 4 |

D | 5 | 25, 000 | 0 |

E | 1 | 55, 000 | 3 |

F | 7 | 40, 000 | 0 |

G | 2 | 60, 000 | 2 |

H | 5 | 30, 000 | 3 |

I | 8 | 50, 000 | 5 |

J | 3 | 25, 000 | 4 |

K | 2 | 45, 000 | 3 |

L | 4 | 50, 000 | 0 |

**Formula used:**

Let the data values be

The formula to calculate mean is given by,

The formula to calculate standard deviation is given by,

Where,

The formula for calculating the correlation coefficient *r* is given as,

Where, *X* and *Y* are the two variables.

The formula to calculate the partial slopes for the independent variables is given by,

And,

Where,

The least square multiple regression line is given by,

The formula to calculate the

**Calculation:**

From the given information,

The cohesion scores is the dependent variable. Thus, it is represented as *Y*.

The rest of the variables, i.e. income and number of children are the independent variables and are represented as

The mean is given by,

For income, substitute 12 for

The mean of income is 42917.

For children, substitute 12 for

The mean of children is 2.75.

For cohesion score, substitute 12 for

The mean of cohesion score is 5.5.

Consider the following table of sum of squares of income and Cohesion Score.

Family | Income |
Cohesion Score |
|||||

A | 30, 000 | 166848889 | 10 | 4.5 | 20.25 | ||

B | 70, 000 | 27, 083 | 733488889 | 10 | 4.5 | 20.25 | 121873.5 |

C | 35, 000 | 62678889 | 9 | 3.5 | 12.25 | ||

D | 25, 000 | 321018889 | 5 | 0.25 | 8958.5 | ||

E | 55, 000 | 12, 083 | 145998889 | 1 | 20.25 | ||

F | 40, 000 | 8508889 | 7 | 1.5 | 2.25 | ||

G | 60, 000 | 17, 083 | 291828889 | 2 | 12.25 | ||

H | 30, 000 | 166848889 | 5 | 0.25 | 6458.5 | ||

I | 50, 000 | 7, 083 | 50168889 | 8 | 2.5 | 6.25 | 17707.5 |

J | 25, 000 | 321018889 | 3 | 6.25 | 44792.5 | ||

K | 45, 000 | 2, 083 | 4338889 | 2 | 12.25 | ||

L | 50, 000 | 7, 083 | 50168889 | 4 | 2.25 | ||

Totals | 515000 | 66 |

Substitute

Square the both sides of the equation.

Proceed in the same manner to calculate

The formula for calculating the correlation coefficient *r* is given as,

Substitute

The correlation coefficient between income and cohesion score is

The formula to calculate standard deviation is given by,

For income, substitute

The standard deviation of income is 13913.17.

For cohesion score, substitute

The standard deviation of cohesion score is 3.09.

Consider the following table of sum of squares of number of children and cohesion score.

Family | Number of Children |
Cohesion Score |
|||||

A | 5 | 2 |

To determine

**(b)**

**To find:**

The expected cohesion in a family with an income of $20, 000 and six children.

To determine

**(c)**

**To find:**

The beta weights for each given independent variable and the effect of the variable on turnout.

To determine

**(d)**

**To find:**

The coefficient of multiple determinations.

To determine

**(e)**

**To explain:**

The relationship among the given three variables.

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