Question

1. A medical statistician wanted to examine the relationship between the amount of sunshine (x) and incidence of skin cancer (y). As an experiment he found the number of skin cancers detected per 100,000 of population and the average daily sunshine in eight counties around the country. These data are shown below.

Average Daily Sunshine	5	7	6	7	8	6	4	3
Skin Cancer per 100,000	7	11	9	12	15	10	7	5

1. Find the least squares regression line.
2. Draw a scatter diagram of the data and plot the least squares regression line on it.
3. Calculate the coefficient of determination and interpret it.
4. Calculate the coefficient of correlation and what does the coefficient of correlation calculated tell you about the direction and strength of the relationship between the two variables?

Answer 1

Solution(a)
Least regression line can be written as
Y = a + bX
Number of country n = 8
Here Y is dependent variable i.e. Skin Cancer
X is independent variable i.e. Average Daily sunshine
a is intercept of regression line
b is slope of regression line
Slope of regression line can be calculated as
Slope = ((n*Xi*Yi)-(Xi * Yi))/((n*Xi^2)-(Xi)^2))

X	Y	X^2	Y^2	XY
5	7	25	49	35
7	11	49	121	77
6	9	36	81	54
7	12	49	144	84
8	15	64	225	120
6	10	36	100	60
4	7	16	49	28
3	5	9	25	15
46	76	284	794	473

Slope = ((8*473)-(46*76))/((8*284)-(46*46)) = 288/156 = 1.846
Intercept of regression line can be calculated as
Intercept = (Yi - Slope *Xi)/n = (76 - 1.846*46)/8 = -1.115
So regression equation can be calculated as
Y = -1.115 + 1.846*X
Solution(b)
Scatter diagram can be constructed as

Solution(c)
Coefficient of detemination can be calculated as
Coefficient of determination = (Correlation coefficient)^2
Correlation coefficient can be calculated as
Correlation coefficient = ((n*Xi*Yi)-(Xi * Yi))/sqrt(((n*Xi^2)-(Xi)^2))*((n*Yi^2)-(Yi)^2))) = ((8*473)-(46*76))/sqrt(((8*284)-(46*46))*((8*794)-(76*76)) = 288/sqrt(156*576) = 0.9607
Coefficient of determination = (0.9607)^2 = 0.9231
So Coefficient of determination can be interpreted as 92.31% variance is explained in skin cancer due to change in Average Daily Sunshine.
Solution(d)
Correlation coefficient can be calculated as
Correlation coefficient = ((n*Xi*Yi)-(Xi * Yi))/sqrt(((n*Xi^2)-(Xi)^2))*((n*Yi^2)-(Yi)^2))) = ((8*473)-(46*76))/sqrt(((8*284)-(46*46))*((8*794)-(76*76)) = 288/sqrt(156*576) = 0.9607
Correlation coefficient can be interpreted as both varibles skin cancer and Average daily sunshine are positively correlated with each other and both variables are strongly correlated with each other. If one variable increases than other variable also increase.