Happy Lodge Ski Resorts tries to forecast monthly attendance. The management has
ID: 332585 • Letter: H
Question
Happy Lodge Ski Resorts tries to forecast monthly attendance. The management has noticed a direct re- lationship between the average monthly temperature and attendance.
(a) Given five months of average monthly temper- atures and corresponding monthly attendance, compute a linear regression equation of the relationship between the two. If next month’s average temperature is forecast to be 45 degrees, use your linear regression equation to develop a forecast.
(b) Compute a correlation coefficient for the data and determine the strength of the linear relationship between average temperature and attendance. How good a predictor is temperature for attendance?
Please do not use excel as my professor would like this done the old fashion way.
Month Average Temperature Resort Attendance ( in thousands) 1 24 43 2 41 31 3 32 39 4 30 38 5 38 35Explanation / Answer
Linear Regression Equation will be of the form
Y = a + bX where
Y: is the dependent variable – here it is the resort attendance
X: is the independent variable – here it is the temperature
a: is the y-intercept – to be calculated here
b: is the slope – to be calculated here
n: sample size = 5
We tabulate the data given using the above symbols
Month
Average Temperature (X)
Resort Attendance (Y)
1
24
43
2
41
31
3
32
39
4
30
38
5
38
35
The first step is to calculate the value of XY, X^2 andY2 for all the data points and find the sum of each column
Month
Average Temperature (X)
Resort Attendance (Y)
XY
X^2
Y^2
1
24
43
1032
576
1849
2
41
31
1271
1681
961
3
32
39
1248
1024
1521
4
30
38
1140
900
1444
5
38
35
1330
1444
1225
Total
165
186
6021
5625
7000
Thus ?X = 165; ?Y=186; ?XY=6021; ?X^2=5625; ?Y^2=7000
The next step is to calculate the value of a and b
a = ((?Y * ?X^2) – (?X*?XY))/(n*?X^2 – (?X)^2)
b = (n*?XY - ?X*?Y)/ (n*?X^2 – (?X)^2)
Thus,
a = 52785/900 = 58.65
b = -0.65
Thus, the regression equation is
Y( Resort Attendance) = 58.65 – 0.65*X (Average Temperature)
Thus, if next month’s average temperature is 45, then using the regression equation we get,
Resort Attendance would be: 58.65 – 0.65*45 = 29.4
b)
To calculate co-relation coefficient:
mean of X (X’) = ?X/n = 33
mean of Y (Y’) = ?Y/n = 186/5 = 37.2
Standard Deviation X (SDx) = Sqrt((?(x-x’)^2)/(n-1)) = 45
Standard Deviation Y (SDy) = Sqrt((?(y-y’)^2)/(n-1)) = 20.2
The formula for correlation co-efficient is
r = (?((x-x’)/SDx)* ((y-y’)/SDy))/(n-1)
Inserting the values and calculating, we get
r = 0.97
Since R>0.8, we can say that the strength of correlation is very good and that temperature is a very good predictor of attendance.
Month
Average Temperature (X)
Resort Attendance (Y)
1
24
43
2
41
31
3
32
39
4
30
38
5
38
35
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