Answer the following questions on the basis of what you learn from the data collected by the "Caligula's Castle" casino group. For questions (1)-(9), use only the five variables provided in the dataset.

- Estimate the mean age of Caligula's patrons, and give a 95%-confidence interval for your answer.
- Estimate the percentage of the casino group's patronage which is female, and give a 95%-confidence interval for your answer.
- (Session 3) Approximately how large a sample size would have been required in order for the margin of error in your estimate in (2) to have been only half as large?
- Predict casino revenues from a 45-year-old male guest who is given an incentive package consisting of $200 in direct incentives and $100 in indirect incentives.
- What is the margin of error (at the 95%-confidence level) in your prediction in (4)?
- Variations in age, sex, and the amounts of direct and indirect compensation provided to the patrons in the sample can explain what percentage of the overall observed variation in casino revenues across those patrons??
- How much per-patron casino revenue would you expect to be generated, on average, by patrons receiving a $400 incentive package divided evenly between direct and indirect compensation?
- What is the margin of error (at the 95%-confidence level) in your estimate in (7)?
- Give a 95%-confidence interval for your estimate of the average additional casino revenue generated by each dollar's-worth of direct compensation given away.

Now, we'll explore variations in our model.

- What two new variables would you put into your model in order to see if the effectiveness of direct or indirect incentives (on revenue generation) varies according to the sex of patrons?
- Add both of those variables to your model. Taking all the original explanatory variables into account as well, for which of the two newly-added variables is there not particularly strong evidence that it belongs in your model? Cite a significance level relevant to your answer.

From here onwards, We'll continue with our (new) full model, containing the original four explanatory variables and the variable added in (11) which seems to belong in our model.

- Caligula's decides to offer two different incentive packages of equal value. Package A consists primarily of direct incentives, and Package B consists primarily of indirect incentives. Which package would you prefer to award to male guests? Which to female guests? (Your answer could conceivably be to give the same package to guests of both sexes, or it could be better to give one type of package to men, and the other type to women. Your goal, of course, is to maximize casino revenues.)

Session 3:

- In order to see whether younger patrons (because they're poorer) and older patrons (preparing for retirement) generate lower revenues than patrons of intermediate age, what new variable would you add to your model?
- Add that variable to your model. Taking all the original explanatory variables into account, as well as this new variable and the variable added in (11), at what age does the age-related effect on casino revenues top out? (Please keep at least one decimal place in your answer.)
- Using this model, predict casino revenues generated by a 35-year-old woman given $100 in direct compensation and $300 in indirect compensation.