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 negative linear correlation between advertising cost and profit . Use a significance level of 0.01 and round all values to 4 decimal places. Advertising Cost Profit 3 18 4 22 5 16 6 29 7 24 8 31 9 22 10 29 11 25 Ho: ρ = 0 Ha: ρ < 0 Find the Linear Correlation Coefficient r = Find the p-value p-value = The p-value is Less than (or equal to) α Greater than α 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 a significant linear correlation between advertising expense and profit. There is a significant positive linear correlation between advertising expense and profit. There is insufficient evidence to make a conclusion about the linear correlation between advertising expense and profit.

Answer 1

The given data is:

Advertising Cost	Profit
3	18
4	22
5	16
6	29
7	24
8	31
9	22
10	29
11	25

We want to determine if there is a significant negative linear correlation between the advertising cost and profit.

Hence, the alternative hypothesis is that: r<0 and the null hypothesis is that r>=0

where r refers to the linear correlation coefficient. This coefficient ranges between -1 and 1. -1 signifies a strong negative correlation and +1 indicates a strong positive correlation.

In order to determine r, we first normalize both Advertising Cost (X) and Profit (Y). This is done by using the formula:

where X_bar and Y_bar refer to the mean of X and Y. Sigma(X) and Sigma(Y) refers to the standard deviation of X and Y respectively.

For the given data, we have:

Using these and the table above, we get the normalized X and Y to be:

Advertising Cost (X)	Profit (Y)	Z_X	Z_Y
3	18	-1.4606	-1.1767
4	22	-1.0955	-0.3922
5	16	-0.7303	-1.5689
6	29	-0.3652	0.9806
7	24	0	0
8	31	0.3652	1.3728
9	22	0.7303	-0.3922
10	29	1.0955	0.9806
11	25	1.4606	0.1961

Next, we take the products of Z_X and Z_Y. We get:

Advertising Cost (X)	Profit (Y)	Z_X	Z_Y	(Z_X*Z_Y)
3	18	-1.4606	-1.1767	1.7187
4	22	-1.0955	-0.3922	0.4297
5	16	-0.7303	-1.5689	1.1458
6	29	-0.3652	0.9806	-0.3581
7	24	0	0	0
8	31	0.3652	1.3728	0.5013
9	22	0.7303	-0.3922	-0.2864
10	29	1.0955	0.9806	1.0742
11	25	1.4606	0.1961	0.2864

Next, we solve for r. The formula for r is:

Summing the last column, we get:

The number of entries in the table are n= 9.

Hence, we have:

Now, there are n=9. Hence, df = 9-2 = 7.

Now, our alternate hypothesis is one-tailed. Hence, we have to multiply the significance level by 2 and our new significance level becomes 0.01*2 = 0.02.

Using a correlation coefficient, we get the critical value to be 0.7450.

To reject the null our r should be r<-0.7450 to have a negative correlation. And for the positive correlation, r>0.7450.

Clearly, that is not the case. Hence, we cannot reject the null hypothesis and conclude that the correlation is not significant.