Question

Listed below are paired data consisting of amounts spent on advertising (in millions of dollars) and the profits (in millions of dollars). Determine if there is a significant positive linear correlation between advertising cost and profit . Use a significance level of 0.05 and round all values to 4 decimal places.

Advertising Cost	Profit
3	14
4	17
5	20
6	27
7	17
8	31
9	27

Ho: ρ = 0
Ha: ρ > 0

Find the Linear Correlation Coefficient
r =

Find the p-value
p-value =   

The p-value is

• Greater than αα
• Less than (or equal to) αα

The p-value leads to a decision to

• Do Not Reject Ho
• Reject Ho
• Accept Ho

The conclusion is

• There is a significant negative linear correlation between advertising expense and profit.
• There is insufficient evidence to make a conclusion about the linear correlation between advertising expense and profit.
• There is a significant positive linear correlation between advertising expense and profit.
• There is a significant linear correlation between advertising expense and profit.

Answer 1

X	Y	XY	X²	Y²
3	14	42	9	196
4	17	68	16	289
5	20	100	25	400
6	27	162	36	729
7	17	119	49	289
8	31	248	64	961
9	27	243	81	729

Ʃx =	42
Ʃy =	153
Ʃxy =	982
Ʃx² =	280
Ʃy² =	3593
Sample size, n =	7
x̅ = Ʃx/n = 42/7 =	6
y̅ = Ʃy/n = 153/7 =	21.85714286
SSxx = Ʃx² - (Ʃx)²/n = 280 - (42)²/7 =	28
SSyy = Ʃy² - (Ʃy)²/n = 3593 - (153)²/7 =	248.8571429
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 982 - (42)(153)/7 =	64

Null and alternative hypothesis:

Ho: ρ = 0

Ha: ρ > 0

Correlation coefficient, r = SSxy/√(SSxx*SSyy) = 64/√(28*248.85714) = 0.7667

Test statistic :  

t = r*√(n-2)/√(1-r²) = 0.7667 *√(7 - 2)/√(1 - 0.7667²) = 2.6704

df = n-2 = 5

p-value = T.DIST.RT(2.6704, 5) = 0.0222

Conclusion:

p-value < α Reject the null hypothesis.

Conclusion :

• There is a significant positive linear correlation between advertising expense and profit.