7. Regression

Question 1

Least Squares Regression Equation

Answer 1

Y’ = bX + a

Y’ = the predicted value
X = the known value
a and b = numbers calculated from the original correlation analysis.

b = r √(SSy / SSx)

a = ̅Y – b ̅X

Question 2

Give the 5 steps for determining the least squares regression equation

Answer 2

1. Determine values of SSx, SSy, and r by referring to the original correlation analysis.
2. Substitute numbers into the formula and solve for b.
3. Assign values to ̅X and ̅Y by referring to the original correlation analysis.
4. Substitute numbers into the formula and solve for a.
5. Substitute numbers for b and a in the least squares regression equation.

Question 3

Least Squares Regression Equation

Answer 3

The equation that minimizes the total of all squared prediction errors for known Y scores in the original correlation analysis.

Question 4

Assume that an r of .30 describes the relationship between educational level (highest grade completed) and estimated number of hours spent reading each week. More specifically:

Educational Level (X):
 ̅X = 13
SSx = 25

Weekly Reading Time (Y)
̅Y = 8
SSy = 50

r = .30

Determine the least squares equation for predicting weekly reading time from educational level.

Answer 4

b = r √(SSy/SSx)
b = 0.30 √(50/25) = .42

a =  ̅Y – b ̅X
a = 8 – (.42)(13) = 2.54

Y' = bX + a
Y' = .42X + 2.54

Question 5

Assume that an r of .30 describes the relationship between educational level (highest grade completed) and estimated number of hours spent reading each week. More specifically:

Educational Level (X):
 ̅X = 13
SSx = 25

Weekly Reading Time (Y)
̅Y = 8
SSy = 50

r = .30

Faith’s education level is 15. What is her predicted reading time?

Answer 5

Y’ = .42X + 2.54

Y’ = (.42)(15) + 2.54 = 8.84

Question 6

Assume that an r of .30 describes the relationship between educational level (highest grade completed) and estimated number of hours spent reading each week. More specifically:

Educational Level (X):
 ̅X = 13
SSx = 25

Weekly Reading Time (Y)
̅Y = 8
SSy = 50

r = .30

Keegan’s educational level is 11. What is his predicted reading time?

Answer 6

Y’ = .42X + 2.54

Y’ = (.42)(11) + 2.54 = 7.16

Question 7

Standard Error of Estimate (Definition Formula)

Answer 7

Sy|x = √[Sy|x / (n – 2)]

= √[∑(Y – Y’)² / (n – 2)]

Question 8

Standard Error of Estimate (Computation Formula)

Answer 8

Sy|x = √[SSy (1 – r²) / (n – 2)]

SSy = ∑Y² – (∑Y)²/n

Question 9

Standard Error of Estimate (Sy|x)

Answer 9

A rough measure of the average amount of predictive error.

Question 10

Give the 2 steps for the calculation of the standard error of estimate, Sy|x

Answer 10

1. Assign values to SSy and r by referring to previous work with the least squares regression equation.
2. Substitute numbers into the formula and solve for Sy|x

Question 11

Calculate the standard error of estimate assuming that the correlation of .30 is based on n = 35 pairs of observations and supply a rough interpretation of the standard error of estimate.

Educational Level (X):
 ̅X = 13
SSx = 25

Weekly Reading Time (Y)
̅Y = 8
SSy = 50

r = .30

Answer 11

Sy|x = √[SSy (1 – r²) / (n – 2)]

Sy|x = √[50 (1 – 0.30²) / (35 – 2)] 
        = √[50 (0.91) / 33]
        = √45.5/33 
        =√1,38
        = 1.17

Roughly indicates the average amount by which the prediction is in error.

Question 12

Squared Correlation Coefficient (r²)

Answer 12

The proportion of the total variability in one variable that is predictable from its relationship with the other variable.

Question 13

r² interpretation

r² (computation formula)

Answer 13

r² = SSy’ / SSy = (SSy – Sy|x) / SSy

r² = [SPxy / √(SSxSSy)]²

Question 14

Assume that an r of .30 describes the relationship between educational level and estimated hours spent reading each week.

According to r², what percent of the variability in weekly reading time is predictable from its relationship with educational level?

Answer 14

9 % predicted

Question 15

Assume that an r of .30 describes the relationship between educational level and estimated hours spent reading each week.

What percent of variability in weekly reading time is not predictable from this relationship?

Answer 15

91 % not predicted

Question 16

Assume that an r of .30 describes the relationship between educational level and estimated hours spent reading each week.

Someone claims that 9 percent of “each” person’s estimated reading time is predictable from the relationship. What is wrong with this claim?

Answer 16

9 % refers to the variability of “all” estimated reading times.

Question 17

The correlation between the IQ scores of parents and children is .50, and that between the IQ scores of foster parents and foster children is .27.

Does this signify, therefore, that the relationship between foster parents and foster children is about one-half as strong as the relationship between parents and children?

Use r² to compare the strengths of these two correlations.

Answer 17

No.

The r² of .25 for parents and children is about four times greater than the r of 0.07 for foster parents and foster children.

Question 18

Multiple Regression Equation

Answer 18

A least squares equation that contains more than one prediction or X variable.

Question 19

Regression Toward the Mean

Answer 19

A tendency for scores, particularly extreme scores, to shrink toward the mean.

Question 20

Regression Fallacy

Answer 20

Occurs whenever regression toward the mean is interpreted as a real, rather than a chance, effect.

Question 21

After a group of college students attended a stress-reduction clinic, declines were observed in the anxiety scores of those who, prior to attending the clinic, had scored high on a test for anxiety.

Can this decline be attributed to the stress-reduction clinic? Explain your answer.

Answer 21

No, because the observed decline could be due to regression toward the mean, given that the students scored high on the anxiety test prior to attending the clinic.

Question 22

After a group of college students attended a stress-reduction clinic, declines were observed in the anxiety scores of those who, prior to attending the clinic, had scored high on a test for anxiety.

What type of study, if any, would permit valid conclusions about the effect of the stress-reduction clinic?

Answer 22

An experiment where students who score high on the anxiety test are randomly assigned either to attend the stress-reduction clinic or to be in a control group.