Question

In: Statistics and Probability

Sir Francis Galton, in the late 1800s, was the first to introduce the statistical concepts of...

Sir Francis Galton, in the late 1800s, was the first to introduce the statistical concepts of regression and correlation. He studied the relationships between pairs of variables such as the size of parents and the size of their offspring.

Data similar to that which he studied are given below, with the variable x denoting the height (in centimeters) of a human father and the variable y denoting the height at maturity (in centimeters) of the father's oldest (adult) son. The data are given in tabular form and also displayed in the Figure 1 scatter plot, which gives the least-squares regression line as well. The equation for this line is =y+88.570.52x

.

Height of father, x
(in centimeters)		 Height of son, y
(in centimeters)
160.6 170.6
181.9 179.2
190.0 195.4
175.1 179.5
162.4 167.9
201.7 190.5
184.5 189.0
157.2 174.9
173.8 170.7
182.0 188.3
172.4 182.3
200.4 190.7
193.6 188.9
176.7 174.5
190.1 188.7
187.9 176.2
      

x

150

160

170

180

190

200

210

y

150

160

170

180

190

200

210

0
Figure 1

Answer the following:

1. Fill in the blank: For these data, heights of sons that are greater than the mean of the heights of sons tend to be paired with heights of fathers that are _____ the mean of the heights of fathers. Choose onegreater thanless than
2. According to the regression equation, for an increase of one centimeter in father's height, there is a corresponding increase of how many centimeters in son's height?
3. From the regression equation, what is the predicted son's height (in centimeters) when the height of the father is 187.9 centimeters? (Round your answer to at least one decimal place.)
4. What was the observed son's height (in centimeters) when the height of the father was 187.9 centimeters?

Solutions

Expert Solution

The dependent variable 'y' is the son's height and the independent variable 'x' is the fther's height. We use regression to predict 'y' based on 'x'.

The regression equation is given by

Where the slope = 0.52

Slope =

intercept = 88.57

intercept =

1. Fill in the blank: For these data, heights of sons that are greater than the mean of the heights of sons tend to be paired with heights of fathers that are _____ the mean of the heights of fathers.

Exp: Since the slope is positive, we can say that 'y' would increase with an increase in 'x'. There is positive correlation, therefore both will be greater if one is greater than the mean and vice versa.

 Greater than

2. According to the regression equation, for an increase of one centimeter in father's height, there is a corresponding increase of how many centimeters in son's height?

Exp: Slope determines the magnitude of change in 'y' due to a change in 'x'.

 0.52

3. From the regression equation, what is the predicted son's height (in centimeters) when the height of the father is 187.9 centimeters? (Round your answer to at least one decimal place.)

Exp: Simply sub x = 187.9 in the regression equation since it says prediction.

 185.448

4. What was the observed son's height (in centimeters) when the height of the father was 187.9 centimeters?

Exp: It says 'pbserved' means the height that is recorded and present in the data.

 176.2

orchestra answered 2 years ago
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