Forecasting with a Regression Line

If we want to use our regression line for forecasting, then we need to be aware where inaccuracies can come from.

Points do not lie exactly on a straight line – r gives a measure of how close the points are to the line. We can calculate confidence limits between which we can be reasonably certain the forecast value should lie.

Data might not fit a straight line – the straight line is often an approximation to a more complex curve.

Data might not continue to fit a straight line - in the future the situation may change.

The forecasting will be most accurate in the central values of x – you need to be very careful forecasting outside the range of values on which you have data. It may be that sales increase as advertising does, over a particular range, but at some point the sales will begin to level off and no longer increase at the same rate.

EXERCISE

The table below shows the data on gas consumption and mean daily temperature.

1. Draw a scatter chart and fit a regression line of gas consumption on temperature.

2. What is the intercept, and what does it mean in this case? What is the slope, and how could you interpret it?

3. Use your regression line to forecast the gas consumption in a month when the mean temperature is .

4. Suppose global warming were to take place and the mean monthly temperature in July was expected to be . What would your predicted gas consumption be? Would you have as much confidence in this prediction as in the earlier one?

Time in
Months
Mean Daily Temperature
Inland Energy Consumption
Natural gas million tonnes coal equivalent
January
1.3
9.8
February
2.8
9.3
March
5.0
10.4
April
8.7
6.6
May
11.0
5.7
June
12.9
4.9
July
16.4
3.3
August
15.0
3.4
September
14.9
4.8
October
11.6
4.9
November
4.8
8.3
December
6.7
10.0

5. Use the following set of data to calculate the equation of the least-squares regression line of y on x :

Y
x
12
10
16
12
17
14
17
16
16
18
19
20
19
22
22
24
18
26
21
28

6. Ten job applicants were ranked by an interviewer in order of preference from 1 (lowest) to 10 (highest). The ten applicants also sat an aptitude test. The interviewer’s rankings, the aptitude scores and the applicants’ ages are shown below:

Applicant:
A
B
C
D
E
F
G
H
I
J
Ranking:
5.5
7
2
10
9
1
3
8
5.5
4
Aptitude:
105
110
90
120
125
95
100
120
110
110
Age:
35
24
45
20
20
40
30
18
35
22

(a) Find the Pearson correlation coefficient between Aptitude and Age and comment on the result.
(b) Find the equation of the least-squares regression line, assuming that Aptitude is the dependent variable (y) and Age is the independent variable (x).
(d) Comment on the usefulness of the regression equation estimated in part (b).
(e) Rank the aptitude scores, giving 1 to the lowest score and 10 to the highest.

(f) Calculate Spearman’s rank correlation coefficient between the interviewer’s rankings and the aptitude rankings, and interpret the result.