Judith Thompson runs a florist shop on the Gulf Coast of Texas, specializing in floral arrangements for weddings and other special events. She advertises weekly in the local newspapers and is considering increasing her advertising budget. Before doing so, she decides to evaluate the past effectiveness of these ads. Five weeks are sampled, and the advertising in dollars and sales volume for each of these is shown in the following table.
| Sales (1,000) | Advertising (100) |
| 11 | 5 |
| 6 | 3 |
| 10 | 7 |
| 6 | 2 |
| 12 | 8 |
- Which is the independent variable?
Sales
Advertising
- Which is the dependent variable?
Sales
Advertising
- What is the mean of x?
- What is the mean of y?
- What is the value for b1?
- What is the regression model?
y = 4 + x
y = 1 + 4x
Using the model, how much will her sales be if she will allocate $1,200 in advertising? (in dollars)
- What is the value of SST?
- What is the value of SSE?
- What is the value of SSR?
- How reliable is the regression model? (in percent)
- How strong is the relationship between sales and advertising?
l. What type of relationship does sales and advertising have?
EXPERT ANSWER
a)
independent variable:
Advertising
…..
b) dependent variable:
Sales
……
c)
Sample size, n = 5
here, x̅ = Σx / n= 5.000
……
d)
ȳ = Σy/n = 9.000
.
e)
SSxx = Σ(x-x̅)² = 26.00
SSxy= Σ(x-x̅)(y-ȳ) = 26.00
estimated slope , ß1 = SSxy/SSxx = 26/26= 1.0
…….
f)
intercept,ß0 = y̅-ß1* x̄ = 9- (1 )*5= 4.0000
Regression line is, Ŷ= 4.0 + ( 1.0 )*x
Predicted Y at X= 12 is
Ŷ= 4.00000 + 1.00000 *12= 16.0
…………
g)
SSE= (SSxx * SSyy – SS²xy)/SSxx = 6.0000
h)
SSR= S²xy/Sxx = 26.0000
i)
SST=Ssyy=32
………..
j)
R² = (SSxy)²/(SSx.SSy) = 0.813
Approximately 81.25% of variation in observations of variable Y, is explained by variable x
.
k)
correlation coefficient , r = SSxy/√(SSx.SSy) = 0.90139
strong relationship
.
l)
linear and positive relationship
………….