Question

1. An Ad agency would like to know the results from ads placed based on the dollar amounts spent of a certain ad. The agency decides to run a pilot study in several small markets. The dollar amounts spent on the ad was 2 thousand, 6 thousand and 13 thousand and the sales were in tens of thousands. The data are in the table below. (Use the numbers in the table, don’t put all the zeros after the values or you will get large numbers). Use Regression for your answers

Ad expenditures	2	6	13
Sales	4	17	25
	5	19	27
	6	21	29

1. Test the null hypothesis that the means are all equal. Use an alpha of .05.
2. Find eta square and interpret.
3. Find the regression equation.
4. Test to see if the regression equation is significant. Again use an alpha of .05.
5. Find R square and interpret.

Answer 1

a)

Ho: µ1=µ2=µ3
H1: not all means are equal

treatment	2	6	13
count, ni =	3	3	3
mean , x̅ i =	5.000	19.00	27.00
std. dev., si =	1.0	2.0	2.0
sample variances, si^2 =	1.000	4.000	4.000
total sum	15	57	81				153	(grand sum)
grand mean , x̅̅ =	Σni*x̅i/Σni =	17.00
( x̅ - x̅̅ )²	144.000	4.000	100.000
							TOTAL
SS(between)= SSB = Σn( x̅ - x̅̅)² =	432.000	12.000	300.000				744
SS(within ) = SSW = Σ(n-1)s² =	2.000	8.000	8.000				18.0000

no. of treatment , k =   3  
df between = k-1 =    2  
N = Σn =   9  
df within = N-k =   6  
      
mean square between groups , MSB = SSB/k-1 =    744/2=   372.0000
mean square within groups , MSW = SSW/N-k =    18/6=   3.0000
      
F-stat = MSB/MSW =    372/3=   124.00

Decision:   p-value<α , reject null hypothesis    
      
there is enough evidence of significant mean difference among three treatments      

b)

eta square ,effect size = SSbet/SST=   0.9764 (large)

c)

x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
2	4	25.0000	169.0000	65.000
2	5	25.0000	144.0000	60.000
2	6	25.0000	121.0000	55.000
6	17	1.0000	0.0000	0.000
6	19	1.0000	4.0000	-2.000
6	21	1.0000	16.0000	-4.000
13	25	36.0000	64.0000	48.000
13	27	36.0000	100.0000	60.000
13	29	36.0000	144.0000	72.000

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	63.00	153.00	186.00	762.00	354.00
mean	7.00	17.00	SSxx	SSyy	SSxy

Sample size,   n =   9      
here, x̅ = Σx / n=   7.000          
ȳ = Σy/n =   17.000          
SSxx =    Σ(x-x̅)² =    186.0000      
SSxy=   Σ(x-x̅)(y-ȳ) =   354.0      
              
estimated slope , ß1 = SSxy/SSxx =   354/186=   1.9032      
intercept,ß0 = y̅-ß1* x̄ =   17- (1.9032 )*7=   3.6774      
              
Regression line is, Ŷ=   3.677   + (   1.903   )*x

d)

R² =    (SSxy)²/(SSx.SSy) =    0.8842  
Approximately    88.42%   of variation in observations of variable sales , is explained by variable Ad expenditure