Question

In: Statistics and Probability

Listed below are paired data consisting of amounts spent on advertising (in millions of dollars) and...

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.

Solutions

Expert Solution

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.


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