Question

The following data represent a company's yearly sales volume and its advertising expenditure over a period of 5 years.

(Y) Sales in Advertising (X) Millions of Dollars in ($10,000)

15 32

16 33

18 35

17 34

16 36

(a) Compute the coefficient of determination for the estimated regression equation you got in the previous in-class problem.

(b) Interpret the meaning of the value of the coefficient of determination that you found in (a). Be very specific.

(c) Perform a t test and determine whether or not X and Y are related. Let = 0.05.

(d) Perform an F test and determine whether or not X and Y are related. Let = 0.05

Answer 1

R Code

y= c(15,16,18,17,16)

x=c(32,33,35,34,36)

lm=lm(y~x)
summary(lm)

R result

Call:
lm(formula = y ~ x)

Residuals:
1 2 3 4 5
-6.000e-01 -9.159e-16 1.200e+00 6.000e-01 -1.200e+00

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.8000 11.7881 0.238 0.828
x 0.4000 0.3464 1.155 0.332

Residual standard error: 1.095 on 3 degrees of freedom
Multiple R-squared: 0.3077,   Adjusted R-squared: 0.07692
F-statistic: 1.333 on 1 and 3 DF, p-value: 0.3318

Ans.

A) Y = 2.8+ (0.4*X)

B)

The sign of a regression coefficient tells you whether there is a positive or negative correlation between each independent variable the dependent variable. A positive coefficient indicates that as the value of the independent variable increases, the meanof the dependent variable also tends to increase. A negative coefficient suggests that as the independent variable increases, the dependent variable tends to decrease.

The coefficient value signifies how much the mean of the dependent variable changes given a one-unit shift in the independent variable while holding other variables in the model constant. This property of holding the other variables constant is crucial because it allows you to assess the effect of each variable in isolation from the others.

The coefficients in your statistical output are estimates of the actual population parameters. To obtain unbiased coefficient estimates that have the minimum variance, and to be able to trust the p-values, your model must satisfy the seven classical assumptions of OLS linear regression.

The Advertising (X) Millions of Dollars coefficient in the regression equation is 0.4. This coefficient represents the mean increase of Sales for every additional one Millions of Dollars. If Millions of Dollars increases by 1 , the average Sales increases by 0.4.

C)

cor.test(y,x)

   Pearson's product-moment correlation

data: y and x
t = 1.1547, df = 3, p-value = 0.3318
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.6415236 0.9647999
sample estimates cor 0.5547002

Here P value is 0.3318 which is greater than 0.05 so X and Y are related.

D) By the fitting between X and Y we got,

Call:
lm(formula = y ~ x)

Residuals:
1 2 3 4 5
-6.000e-01 -9.159e-16 1.200e+00 6.000e-01 -1.200e+00

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.8000 11.7881 0.238 0.828
x 0.4000 0.3464 1.155 0.332

Residual standard error: 1.095 on 3 degrees of freedom
Multiple R-squared: 0.3077,   Adjusted R-squared: 0.07692
F-statistic: 1.333 on 1 and 3 DF, p-value: 0.3318

So Here also P value is 0.3318 which is greater than 0.05 so X and Y are related.