Case 12.3 Estimating Total Medical Costs. (Page 436-437)
a. We are looking for the interval (LCL, UCL) that describes the population of medical costs per person, categorized by age groups. The sample size large enough that we approximate the critical value (t*) as
using a table of commonly used critical values.
b. To get the total medical costs for all people above 45 for the years listed, we multiply the number of people (in thousands) expected to be alive for each age group in each year, by the medical costs per person for that age group, then add the numbers across the age groups for a particular year. Symbolically:
For example, the 95% confidence interval for the total cost, using the LCL and UCL costs calculated in part a is:
LCL=1784*9718+4381*2644+7877*1600+15572*639 = $51.5 billion Canadian
UCL=1877*9718+4606*2644+8272*1600+16342*639 = $54.1 billion Canadian
Repeating this process for all the years, we get:
Cost (Billion $ CA) 95% confidence interval
51.5 – 54.1
57.5 – 60.5
64.2 – 67.4
72.4 – 76.1
81.5 – 85.7
Case 13.2 Ambulance and Fire Department Response Interval Study (Page 504-505)
In this problem, we want to know what we can conclude from the data for each fire department. Two questions that we might ask for each fire department / ambulance pair are:
a) Is the mean difference between the arrival time of the ambulance and the arrival time of the fire department greater than 1 minute? Is our answer different at the boundaries of the 95% confidence interval?
There are at least 150 samples of each department, so we use the approximation
For this part, we calculate the mean difference in arrival times for each fire, ambulance department pair. To simplify presentation, we define .
At the 95% confidence interval, the fire departments in districts C and K respond consistently at least 1 minute faster (<= -1.0) than the ambulance service. In district W, only near the 5% extrema of cases do we see that the fire department arrived more than one minute before the ambulance service.
Next, we consider whether the fire department arrives in under 8 minutes more frequently than the ambulance service. To test significance, we choose our variable to be:
where service is either ambulance or fire. Thus:
The null hypothesis is that the probability that the ambulance arrives in less than 8 minutes is not significantly different from the probability that the fire department arrives in under 8 minutes:
Significance level: 3% (0.03)
We use a Z-test to test the validity of this hypothesis because we are comparing the frequency of an event in two populations:
Because the number of samples is large, we use the binomial approximation for
P(Tservice < 8)
Thus, the difference in the frequency of the fire department arriving in less than 8 minutes compared to the ambulance is significant at Z < 0.03 for all departments. Based on the previous assessment of the difference in means between the departments, we conclude that district C and K should train their fire-fighters as paramedics (rejecting the null hypothesis), while district W should hold off for now because the mean difference of time was < -1 only near the lower end of the confidence interval.
As more data becomes available, the question of whether to train the firefighters in district W should be revisited, because the lower limit of the confidence interval was below -1. If we had set our confidence interval at 80%, the lower limit would be above -1, so this is a borderline case. Since the mean difference in arrival time is about -0.7, for now fire department W does not need training.
Case 16.1 Insurance Compensation for Lost Revenues (Page 663)
a) Calculate the regression coefficients. To clarify why museum attendance should be a function of amusement park attendance, suppose museum attendance and amusement park attendance depends on the number of tourists in town model:
From this model, the attendance at the museum and amusement park should be correlated at times that both are open because we could solve for T(w) in APA(w), then MA would be solely a function of the attendance at the amusement park (APA). This also elucidates why the correlation between attendance at the two attractions might change over time: tourists might become more likely to visit the amusement park than the museum, such as when the museum closed.
a and b) Calculation of the regression coefficients. Here xi is the attendance at the amusement park for week i, yi is the attendance at the museum for week i. The summation goes over the time specified from the point of view (POV) of the insurance company and the museum. Formulas:
During the duration of the fire and rebuilding (147 weeks)
. From the insurance company’s point of view, this indicates that the museum should be compensated for lost revenue of:
(0.6935*7682.5+16.23)*(147 weeks)=785574 tickets.
From the museum’s point of view, it should be compensated for a lost attendance of:
To: Insurance Company Claims Representative
Re: Lost revenue claim for museum
I have completed my analysis of the lost revenue claim made by the rock and roll museum. Using the data from the 32 weeks preceding the fire to determine the number of tickets lost by the museum (as per your suggestion), the museum would need to be compensated for 785574 tickets worth of lost revenue. Using the last two quarters of data, as the museum had suggested, indicates that the museum may have lost as many as 1162994 tickets worth of lost revenue.
The museum’s argument is the more problematic. The regression line before the fire was
# of museum patrons = #of amusement park patrons * 0.69 + 16.23
after the reopening, the regression line is
# of museum patrons = #of amusement park patrons * 0.97 + 459.5
Assuming that the amusement park did not attract a smaller percentage of tourists after the fire, these lines suggest that the museum attracted a new loyal crowd of around 440 people that just came to see it (more precisely, that the museum had 460 more loyal patrons than the museum after the fire). Additionally, the increase in the slope suggests that almost everyone who went to the amusement park also went to the museum, whereas only 70% did so before.
In defense of the museum, the regression coefficients may have had little meaning in the first 32 weeks that the amusement park was open. The average number of tickets sold by the amusement park following its opening was lower during its first quarter of business than any quarter since, perhaps because the amusement park had yet to fully penetrate the local market. Therefore, the number of tickets sold by the amusement park may have been governed by dissimilar factors than those that affected ticket sales at the museum. For example, in the early days of the amusement park, it may have featured a grand opening that attracted a local crowd that would not have decided to see the museum, while it took a longer period of time to penetrate the regional market, whom would have wanted to see several of the attractions that the city had to offer.
The most fair option would be to obtain data from recreational businesses that had been open longer or the city’s department of commerce to quantify the increase in tourism during the past few years, then perform the regression with the museum’s data compared to the city’s tourism data, and extrapolate using the city’s data on tourism as the independent variable.