In September of 1998, a local TV station contacted an econometrician to analyze some data for them. They were going to do a Halloween story on the legend of full moons affecting behavior in strange ways. They collected data from a local hospital on emergency room cases for the period from January 1, 1998 until mid-August. There were 229 observations. During this time, there were eight full moons and seven new moons (a related myth concerns new moons) and three holidays (New Year's day, Memorial Day, and Easter). If there is a full-moon effect, then hospital administrators will adjust numbers of emergency room doctors and nurses, and local police may change the number of officers on duty. Let \(T\) be a time trend \((T=1,2,3, \ldots, 229)\). Let the indicator variables \(H O L I D A Y=1\) if the day is a holiday, \(=0\) otherwise; \(\quad F R I D A Y=1\) if the day is a Friday, \(=0\) otherwise; \(\quad S A T U R D A Y=1\) if the day is a Saturday, \(=0\) otherwise; FULLMOON \(=1\) if there is a full moon, \(=0\) otherwise; \(N E W M O O N=1\) if there is a new moon, \(=0\) otherwise. Consider the model

a. What is the expected number of emergency room cases for day \(T=100\), which was a Friday with neither a full or new moon?

b. What is the expected number of emergency room cases for day \(T=185\), which was a holiday Saturday?

c. In terms of the model parameters, what are the null and alternative hypotheses for testing that neither a full moon nor a new moon have any effect on the number of emergency room cases? What is the test statistic? What is the distribution of the test statistic if the null hypothesis is true? What is the rejection region for a \(5 \%\) test?

d. The sum of squared residuals from the regression in (XR7.2.1) is 27109. If full moon and new moon are omitted from the model the sum of squared residuals is 27424 . Carry out the test in (c). What is your conclusion?

e. Using the model in equation (XR7.2.1), the estimated coefficient of SATURDAY is 10.59 with standard error 2.12, and the estimated coefficient for FRIDAY is 6.91, with standard error 2.11. The estimated covariance between the coefficient estimators is 0.75. Should the hospitals prepare for significantly more emergency room patients on Saturday than Friday? State the relevant null and alternative hypotheses in terms of the model parameters. What is the test statistic? What is the distribution of the test statistic if the null hypothesis is true? What is the rejection region for a test at the \(10 \%\) level? Carry out the test and state your conclusion?