Question: As a market analyst for a cereal company, you have been instructed to develop a regression equation that identifies the relationship between cereal sales and sugar content, with the goal of eventually increasing cereal sales. This must be done with minimal expenditure, thus you cannot undertake any sample surveys or other extensive data gathering techniques. The only available data that you have is the sugar content (measured in grams per serving) of the 28 different brands of cereal currently sold by the company and the sales figure (measured in thousands of boxes sold ) for each brand over the past year. The range of the sugar content for the 28 cereals is 1.2 grams per serving to 4.5 grams per serving. Using this data, you generate the following regression model and statistics.

Sales’ = -98.58 + 1459.2 SugarContent r 2 = 0.574, t = 3.65

___ 1. What could you conclude from these results? A significant regression model has been found, with t = 3.65, p = 0.00116 < 0.05. This means that the sugar content is a useful predictor of sales.

___ 2. How would you interpret the value of the slope coefficient? For an extra gram of sugar, the number of boxes sold increase by 1,459,200 on average.

___ 3. What factors lead you to this decision? The correlation between the variables is significant (p = 0.00116), and the amount of explained variation is reasonably high (57.4%).

___ 4. How would you interpret the value of the intercept term? The intercept is meaningless in this case, because it doesn’t make sense to think of a case of a cereal with 0 grams of sugar.

___ 5. How would you interpret the value of the r2? The amount of variation in Sales explained by Sugar Content is 57.4%.

___ 6. What would be the appropriate conclusion for you to provide to the head of the marketing department based on the regression results? The conclusion is that a significant regression model has been found, with t = 3.65, p = 0.00116 < 0.05. This means that the sugar content is a useful predictor of sales, and 57.4% of the variation in sales is explained by the sugar content.

___ 7. Suppose the marketing director has looked at your results and concluded that all cereals should now contain at least 9 grams of sugar per serving (which is the maximum amount feasible at current profit ratios). What would be the best response on your part in light of the statistical results? The regression results obtained wouldn’t be applicable to the case of 9 grams of sugar per serving, because that value would be outside of the range used for constructing the regression model, and hence, we would be extrapolating.

___ 8. What would be the projected sales of a new cereal with a sugar content of 3.4 grams of sugar per serving?

The predicted sales level is

Sales’ = -98.58 + 1459.2* 3.4 = 4,862.7

which corresponds to 4,862,700 boxes.

2)

You are a policy analyst who has been asked to determine the average level of automobile ownership for households in a metropolitan area. The city council has told you that it is prepared to spend $287 million on a new rapid rail transit system if your results show that this average is below a value of 1.24 automobiles per household. If the average is equal to or above this value, then the system will not generate enough revenue to be financially feasible. To obtain the information, you take a random sample of 700 households in the metropolitan area. This gives you a sample mean and a sample variance. From the federal government, you are able to determine the actual value of 2 for this variable in the population based on similar studies.

___ 1. What would be the most appropriate way to present the information you derive from the sample concerning the value of the mean?

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