An airline, believing that 5% of passengers fail to show up for flights, overbooks (sells more tickets than there are seats).

Question 1
An airline, believing that 5% of passengers fail to show up for flights, overbooks (sells more tickets than there are seats). Suppose a plane will hold 265 passengers, and the airline sells 275 tickets. Let X denote the number of passengers that do show up.
a) (2) Solve by hand for the probability that 266 passengers show up.
b) (2) Solve by hand for the expected number of passengers that will show up.
c) (2) (Use R) Solve for the probability that the airline will not have enough seats for the passengers that show up.
d) (4) (Use R) The airline faces a penalty when a passenger shows up and is not seated. In particular, for every passenger that shows up and is not seated the airline has to pay them $500. Let Y denote the total amount in penalties paid. Create a table capturing the possible values of Y as well as the probability function of Y. For example, if 266 passengers show up then Y=$500 and this occurs with a probability of P(X=266) as found in part a).
e) (2) (Do by hand) Use your findings in part d) to solve for the expected amount that the airline will pay in penalties.

Question 2
A call center receives an average of 4.5 calls every 5 minutes. Each agent can handle one of these calls over the 5minute period. If a call is received, but no agent is avalible to take it, then that caller will be placed on hold. Assuming that the calls, X, follow a Poisson distribution,
a) (2) What is the probability that less than 3 calls are received in 5 minutes?
b) (2) What is the probability that at least 10 calls are received in a quarter of an hour?
c) (4) What is the minimum number of agents needed on duty so that calls are placed on hold at most 10% of the time? i.e find the value k = number of agents, such that
P(X > k) 0.1. (Hint: create a table of the cumulative distribution function of X).

Question 3
A few months ago, Dina was thinking about getting a dog. To help make a decision she looked in to the annual cost of medical care for dogs. Let X denote the annual medical cost of a dog. The Canadian Veterinary Medical Association claims that the annual cost is Normally distributed with an average of $100 and standard deviation of $30.
a) (2) What is the probability that the annual cost of a dog is at least $60?
b) (2) What is the probability that the annual cost of a dog ranges between $80 and $110?
c) (3) Solve for the inter-quartile range of the annual cost. The Canadian Veterinary Medical Association also claims that the annual cost of medical care for cats, Y, is Normally distributed with an average of $120 and standard deviation of $35.
d) (3) Solve for the expected difference in the cost of medical care for dogs and cats? What is the standard deviation of that difference assuming that the costs are independent.
e) (3) It can be shown that the difference of two Normal random variables is also a Normal random variable. Using this solve for the probability that the absolute difference in the cost of medical care for dogs and cats will exceed $15? Dina is now thinking about getting 2 dogs and a cat. Define a new variable capturing the total annual costs that Dina might incur expressed as a function of X and Y.
f) (3) Solve for the expected total cost of medical care for 2 dogs and a cat. What is the standard deviation of that total assuming that the costs are independent.
g) (2) It can be shown that the sum of independent Normal random variables is also a Normal random variable. Using this solve for the probability that total expenses will be less than $400?
h) (2) For Dina to better budget solve for the minimum amount of money she should put aside in a year to ensure that she is covered for the total costs with probability 0.85.

Question 4
Referring back to Question 3 describing veterinary medical costs for cats and dogs. Dina decided to randomly sample 20 dogs and 15 cats and ask their owners for an estimate of their annual medical expenses.
a) (3) Solve for the probability that the average medical cost of the dogs is found to be more than $85?
Dina summarised the data collected, and the values are presented in the table below.
               Mean            Standard
                                    Deviation
Dogs          $95               $25
Cats          $110             $30
b) (4) Compute separate 99% confidence intervals for the average annual medical cost for dogs and cats. Include a conclusion statement for each interval.
c) (4) Perform a hypothesis test at a 1% level to significance to test if the average annual medical cost for cats is greater than that for dogs. Clearly state your hypothesis and conclusion. (Note: from Question 4 we note that the population variances are unequal).

Question 5
In a usability study, two versions (A, B) of a company website were compared with respect to the time it takes to retrieve certain information from the site. One hundred subjects were randomly selected from the population, and 50 subjects were randomly assigned to each version. Sample means and standard deviations of retrieval time for the two versions are provided below.
Version       Mean         Standard Deviation
A                   209                      37
B                   225                      41
a) (5) Is there any difference in mean retrieval time between the two sites? Answer this
question by carrying out an appropriate 10% hypothesis test, assuming that the population variances are equal. Be sure to include the null and alternative hypotheses, value of the test statistic, p-value, and conclusion in the context of the study.
b) (3) Create a confidence interval for the mean retrieval time between the two sites and use it to test if there is a statistically significant difference between the two sites. Create this confidence interval using an appropriate confidence level to match your significance level in part a). Clearly state your conclusion.

Question 6
A retail chain is considering installing devices that resemble cameras, saving the expense of wiring and recording video. To test the benefit of this decoy system, it picked 40 stores, half to get the decoy and the other half to serve as a comparison group (control group). Stores were matched based on typical levels of sales, local market size, and demographics. The comparison lasted for 3 months during the summer. At the end of the period, the retailer used its inventory system to compute the amounts lost to theft in the stores. The data is stored in the file “AS2-Q6- Decoy.xls”.
a) (2) Why is it important that all stores measure theft during the same time period?
b) (5) Compute separate 95% confidence intervals for the amount lost to theft with and without the decoy cameras. In comparing the intervals, is there evidence of a statistically significant difference?
c) (4) Perform a hypothesis test to compare the decoy system versus the stores without them. In particular test at a 5% level of significance whether the decoy has reduced the average, clearly state your hypothesis and conclusion.