Discussion Question 1: Look back at the company you analyzed for revenue trends in Week 3. Now, assume there is a 50% chance that the revenue is what you predicted in Week 3, a 30% chance that it is the same as last year, and a 20% chance that it lower than last year by a small amount (you choose how much). Using this information, how can you create an estimate of revenue that considers these different probabilities? Why would that estimate be better than any of the original three?

Since I don’t have the original data, I’ll explain how to work this problem. The problem with the three estimates (week 3, last year, and slightly lower) is that they each only happen with a certain probability. In other words, there is only a 50% chance that the revenue will be what you predicted in week 3.

What we need to do is find the expected revenue. The way that we do this is by finding the product of the estimate with the probability of getting that estimate and add all those up for each possible outcome. This gives us an expected value. For example, consider the following table. I’m making up these values, since I don’t have what you’re working with.

Revenue / Probability / Rev.*prob.
Week 3 / 100000 / 0.5 / 50000
Last Year / 120000 / 0.3 / 36000
Slight decrease / 90000 / 0.2 / 18000
add these up----> / 104000

So our expected revenue is 104,000. Now, we can also consider the variability of this value as a measure of the certainty of this estimate.

Discussion Question 2: Go to . This is an index based on how people feel about the economy. All indexes are created in a similar fashion. A base is selected, in this case the base was 1985 and set at 100. So if the index is at 57.2 in May 2008, and October 1992 was at 54.6, are people feeling better or worse than they did in 1992?

They are feeling better.

If the index was at 62.8 in April 2008, what does that mean for May?

It means that people were feeling worse in May than they were in the previous month. That is, the higher the CCI is, the more confident people are feeling in the stability of their incomes and the economy. Consequently, the higher the CCI, the more likely consumers are to make purchases. For this reason, high CCI generally bode well for the economy. A lower CCI generally means that consumers are less confident. That is, they are spending less and saving more. This generally does not bode well for the economy.

More specifically, the index is set arbitrarily to 100 at the base year. For any consecutive year, the value of the index is the ratio of the current year’s measure to the base year times 100.

Why do we use indexes?

Index numbers are used to compare and understand complex data such as business activity, the cost of living, and employment. The CCI, for example, considers Current business conditions, Business conditions for the next six months, Current employment conditions, Employment conditions for the next six months, Total family income for the next six months. These complex concepts are bundled neatly into the CCI.

Discussion question 3: There are four people being considered for the position of chief executive officer of Dalton Enterprises. Three of the applicants are over 60 years of age. Two are female, of which only one is over 60.

a. What is the probability that a candidate is over 60 and female?

Let’s make a table.

under 60 / over 60 / TOTAL
woman / 1 / 1 / 2
man / 0 / 2 / 2
TOTAL / 1 / 3 / 4

The cells in yellow are the ones given in the problem. We get the rest by subtracting and adding.

So, we can easily see that P(over 60 & female) = ¼ = 0.25

b. Given that the candidate is male, what is the probability he is less than 60?

Since there are no men under 60 under consideration, the probability is 0.

c. What is the probability the person is female and over 60?

This is also read from the table. P(female & over 60) = ¼ = 0.25

Note, this is the same question as part a. Perhaps this question is to show that P(A&B) = P(B&A) or perhaps it’s a typo. If it’s the latter, just let me know.

Discussion question 4: Suppose your statistics instructor gave six examinations during the semester. You received the following grades (percent correct): 79, 64, 84, 82,92, and 77. Instead of averaging the six scores, the instructor indicated he would randomly select two grades and report that grade to the student records office.

a. How many different samples of two test grades are possible?

b. List all possible samples of size two and compute the mean of each.

Score 1 / Score 2 / average
79 / 64 / 71.5
79 / 84 / 81.5
79 / 82 / 80.5
79 / 92 / 85.5
79 / 77 / 78
64 / 84 / 74
64 / 82 / 73
64 / 92 / 78
64 / 77 / 70.5
84 / 82 / 83
84 / 92 / 88
84 / 77 / 80.5
82 / 92 / 87
82 / 77 / 79.5
92 / 77 / 84.5

c. Compute the mean of the sample means and compare it to the population mean.

The mean of these sample means is 79.667, and the average of the 6 test scores is 79.667

If you were a student, would you like this arrangement? Would the result be different from dropping the lowest score? Write a brief report

Since the population score and the average score obtained using this “sample of two” approach are the same, the student is faced with something of a quandary. The population average (the average of the first 6 tests) is a fixed quantity. That is, it’s a parameter. It doesn’t change. The student is assured of getting a 79.667. On the other hand, if the student opts to use the “sample of two” method, there is a particular probability that he will do better that 79.667 and a particular probability that he will do worse that 79.667. In fact, since 79.667 is not a possible score in the “sample of two” method, the student will either do better or worse in this method.

To help us decide, let’s look at the sorted list of scores present in part b:

88

87

85.5

84.5

83

81.5

80.5

80.5

79.5

78

78

74

73

71.5

70.5

In this case, there are 8 scores that are above 79.667 and there are 7 scores that are below 79.667. That is, if the student opts to use the “sample of two” method, there is a 53.3% chance that he will improve his score and a 46.7% chance that he will decrease his score.

This variability in the “sample of two” method can be considered a measure of risk. In this case, I believe that it is too risky to use the “sample of two” method. I would ask to keep the simple average of my 6 test scores.

Dropping the lowest score gives a third result. It produces an average of 82.8. Only 1/3 of scores obtained in the “sample of two” method are better than this score. If this were an option, it is clearly preferable to the other two methods.