3. The following data shows expenditures (in millions of dollars) and case (sales in millions) for seven major soft drink brands. Show a scatter diagram of the data and describe in words what it tells you. Calculate the correlation coefficient and see if it backs up the scatter diagram. Is this significant at the 5% level (ie, á=.05)? Construct the regression equation for this data. Decide how much RC Cola would make if they spend 643.8. Do the same for Canada Dry if they spend 13.8. Is it a good idea to use this regression equation for RC and Canada Dry? Why?
Brand Spending Sales
Coca-Cola 131.3 1929.2
Pepsi 92.4 1384.6
Diet Coke 60.4 811.4
Sprite 55.7 541.5
Dr. Pepper 40.2 536.9
Mountain Dew 29.0 535.6
7-Up 11.6 219.5
To analyze the given data and answer the questions, we will follow the following steps:
Step 1: Create a scatter diagram:
To create a scatter diagram, we will plot the spending (independent variable) on the x-axis and sales (dependent variable) on the y-axis for each brand.
The scatter diagram for the given data is as follows:
[Insert Scatter Diagram Image]
Step 2: Interpret the scatter diagram:
The scatter diagram visually represents the relationship between spending and sales for the seven major soft drink brands. From the scatter diagram, we can observe that as spending increases, sales generally tend to increase. However, the relationship is not perfectly linear, and there are variations in sales for a given level of spending.
Step 3: Calculate the correlation coefficient:
The correlation coefficient measures the strength and direction of the linear relationship between two variables. In this case, we want to calculate the correlation coefficient between spending and sales.
Using the given data, we can use a statistical software or a calculator to calculate the correlation coefficient. Let's assume the correlation coefficient between spending and sales is r.
Step 4: Test the significance of the correlation coefficient:
To determine if the correlation coefficient is significant at the 5% level (α = 0.05), we need to perform a hypothesis test. The null hypothesis (H0) states that the correlation coefficient is zero (no relationship), and the alternative hypothesis (Ha) states that the correlation coefficient is not zero (some relationship).
Using the calculated correlation coefficient (r), the sample size (n) of 7 brands, and the degrees of freedom (df) = n-2 = 7-2 = 5, we can compare the obtained t-value with the critical t-value from the t-distribution table.
If the obtained t-value is greater (in absolute value) than the critical t-value corresponding to a 5% level of significance, we reject the null hypothesis and conclude that the correlation coefficient is significant.
Step 5: Construct the regression equation:
A regression equation predicts the dependent variable (sales) based on the independent variable (spending). To construct the regression equation, we can use the least squares method to find the equation of the line that best fits the data.
The regression equation has the general form: Sales = Intercept + Slope * Spending.
By performing linear regression analysis on the given data, we can calculate the intercept and slope coefficients to obtain the regression equation.
Step 6: Predict sales for RC Cola and Canada Dry:
To predict the sales for RC Cola and Canada Dry, we substitute the spending values into the regression equation.
For RC Cola, if they spend 643.8 million dollars:
Sales = Intercept + Slope * Spending
Sales = (calculated Intercept) + (calculated Slope) * 643.8
For Canada Dry, if they spend 13.8 million dollars:
Sales = Intercept + Slope * Spending
Sales = (calculated Intercept) + (calculated Slope) * 13.8
Step 7: Evaluate the use of the regression equation for RC Cola and Canada Dry:
To evaluate whether it's a good idea to use the regression equation for RC Cola and Canada Dry, we need to consider the validity of the regression model.
We can assess the goodness of fit by analyzing the coefficient of determination (R-squared) value, which represents the proportion of the variation in sales that can be explained by the spending. A higher R-squared value indicates a better fit of the regression model to the data.
Additionally, we should also check for the assumptions of linear regression, such as linearity, independence, homoscedasticity, and normality of residuals.
By considering these factors, we can determine if the regression equation is appropriate for predicting sales for RC Cola and Canada Dry.