Question

The following data represent a company's yearly sales volume and its advertising expenditure over a period of 5 years. (Y) Sales in Millions of Dollars 15 16 18 17 16 (X) Advertising in ($10,000) 32 33 35 34 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

a) Coefficient of determination =

Table of totals=

	y	x	xy	x^2	y^2
	15	32	480	1024	225
	16	33	528	1089	256
	18	35	630	1225	324
	17	34	578	1156	289
	16	36	576	1296	256
total	82	170	2792	5790	1350

The formula for coefficient of determination is as follows,

By substituting the values we get,

b) Interpretation =

The coefficient of determination is used to explain how much variability of one factor can be caused by its relationship to another factor. It is relied on heavily in trend analysis and is represented as a value between 0 and 1.

An R-squared of 0.554700, i.e it means that 55% of the dependent variable is predicted by the independent variable.

i.e 55% sales is predicted by the advertising.

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

let,   ρ = coefficient of determination

Performing the Hypothesis Test

• Null Hypothesis: H₀: ρ = 0 i.e There is not a significant linear relationship between x and y in the population.
• Alternate Hypothesis: H_a: ρ ≠ 0 i.e  There is a significant ralationship between x and y in the population.

There are two methods of making the decision. The two methods are equivalent and give the same result.

• Method 1: Using the p-value
• Method 2: Using a table of critical values

but we use the method 1.

by using the minitab software , we get the p value which is

Pearson correlation of y and x = 0.555
P-Value = 0.332

If the p-value is less than the significance level (α = 0.05)

• Decision: Reject the null hypothesis.
• Conclusion: “There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero.”

If the p-value is NOT less than the significance level (α = 0.05)

• Decision: DO NOT REJECT the null hypothesis.
• Conclusion: “There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is NOT significantly different from zero”

Here our p value > 0.05 ,Hence we accept H0 at 5% l.o.s.

Conclusion= There is a significant linear relationship between x and y .

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

• Null Hypothesis: H₀: There is not a significant linear relationship between x and y in the population.
• Alternate Hypothesis: H_a: There is a significant ralationship between x and y in the population.

By using the minitab software,we analyse the given data.

Regression Analysis: y versus x

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value
Regression 1 1.600 1.600 1.33 0.332
x 1 1.600 1.600 1.33 0.332
Error 3 3.600 1.200
Total 4 5.200

Model Summary

S R-sq R-sq(adj) R-sq(pred)
1.09545 30.77% 7.69% 0.00%

Coefficients

Term Coef SE Coef T-Value P-Value VIF
Constant 2.8 11.8 0.24 0.828
x 0.400 0.346 1.15 0.332 1.00

Regression Equation

y = 2.8 + 0.400 x

Here our p value > 0.05 ,Hence we accept H0 at 5% l.o.s.

Conclusion= There is a significant linear relationship between x and y .