Assignment:
Probability
1. According to Investment Digest ("Diversification and the Risk/Reward Relationship", Winter 1994, 1-3), the mean of the annual return for common stocks from 1926 to 1992 was 15.4%, and the standard deviation of the annual return was 24.5%. During the same 67-year time span, the mean of the annual return for long-term government bonds was 5.5%, and the standard deviation was 6.0%. The article claims that the distributions of annual returns for both common stocks and long-term government bonds are bell-shaped and approximately symmetric. Assume that these distributions are distributed as normal random variables with the means and standard deviations given previously.
- Find the probability that the return for common stocks will be greater than 7%.
- Find the probability that the return for common stocks will be greater than 20%.
Hint: There are many ways to attack this problem in the HW. If you would like the normal distribution table so you can draw the pictures (my preferred way of learning) then I suggest you bookmark this site:
Confidence Interval Estimation
2. Compute a 95% confidence interval for the population mean, based on the sample 15, 17, 13, 14, 15, 14, and 59. Change the number from 59 to 14 and recalculate the confidence interval. Using the results, describe the effect of an outlier or extreme value on the confidence interval.
Hypothesis Testing
3. The director of admissions at the University of Maryland , University College is concerned about the high cost of textbooks for the students each semester. A sample of 25 students enrolled in the university indicates that X (bar) = $330.4 and s = $45.20.
a. Using the 0.10 level of significance, is there evidence that the population mean is above $300?
b. What is your answer in (a) if s = $90 and the 0.05 level of significance is used?
c. What is your answer in (a) if X (bar) = $305.10 and s = $45.20?
d. Based on the information in part (a), what decision should the director make about the books used for the courses if the goal is to keep the cost below $300?
4. A large hat manufacturer, MALLORY HATS, is concerned that the mean weight of their signature Kentucky Derby hat is not greater than 2.5 pounds. It can be assumed that the population standard deviation is .5 pounds based on past experience. A sample of 196 hats is selected and the sample mean is 2.55 pounds. Using a level of significance of .10, is there evidence that the population mean weight of the hats is greater than 2.5 pounds? Fully explain your answer.
Hypothesis Testing
1. The MBA department is concerned that dual degree students may be receiving lower grades than the regular MBA students. Two independent random samples have been selected 150 observations from population 1 (dual degree students) and 200 from population 2 (MBA students). The sample means obtained are X1(bar)=86 and X2(bar)=88. It is known from previous studies that the population variances are 4.8 and 5.2 respectively. Using a level of significance of .10, is there evidence that the dual degree students are receiving lower grades? Fully explain your answer.
Simple Regression
2. A CEO of a large pharmaceutical company would like to determine if he should be placing more money allotted in the budget next year for television advertising of a new drug marketed for controlling diabetes. He wonders whether there is a strong relationship between the amount of money spent on television advertising for this new drug called DIB and the number of orders received. The manufacturing process of this drug is very difficult and requires stability so the CEO would prefer to generate a stable number of orders. The cost of advertising is always an important consideration in the phase I roll-out of a new drug. Data that have been collected over the past 20 months indicate the amount of money spent of television advertising and the number of orders received.
The use of linear regression is a critical tool for a manager's decision-making ability. Please carefully read the example below and try to answer the questions in terms of the problem context. The results are as follows:
|
Month
|
Advertising Cost (thousands of dollars)
|
Number of Orders
|
|
1
|
$68.93
|
3,902,000
|
|
2
|
72.62
|
3,893,000
|
|
3
|
79.58
|
5,299,000
|
|
4
|
58.67
|
4,130,000
|
|
5
|
69.18
|
4,367,000
|
|
6
|
70.14
|
3,111,000
|
|
7
|
93.37
|
3,923,000
|
|
8
|
68.88
|
4,935,000
|
|
9
|
82.99
|
5,276,000
|
|
10
|
75.23
|
4,654,000
|
|
11
|
91.38
|
4,598,000
|
|
12
|
52.90
|
2,967,000
|
|
13
|
61.27
|
3,999,000
|
|
14
|
79.19
|
4,345,000
|
|
15
|
90.03
|
3,934,000
|
|
16
|
78.21
|
4,653,000
|
|
17
|
83.77
|
5,625,000
|
|
18
|
62.53
|
3,978,000
|
|
19
|
98.76
|
4,999,000
|
|
20
|
72.64
|
3,834,000
|
a. Set up a scatter diagram and calculate the associated correlation coefficient. Discuss how strong you think the relationship is between the amount of money spent on television advertising and the number of orders received. Please use the Correlation procedures within Excel under Tools > Data Analysis. The Scatterplot can more easily be generated using the Chart procedure.
NOTE: If you do not have the Data Analysis option under Tools you must install it. You need to go to Tools select Add-ins and then choose the 2 data toolpak options. The original Excel CD will be required for this installation. It should take about a minute.
b. Assuming there is a statistically significant relationship, use the least squares method to find the regression equation to predict the advertising costs based on the number of orders received. Please use the regression procedure within Excel under Tools > Data Analysis to construct this equation.
c. Interpret the meaning of the slope, b1, in the regression equation.
d. Predict the monthly advertising cost when the number of orders is 5,100,000. (Hint: Be very careful with assigning the dependent variable for this problem)
e. Compute the coefficient of determination, r2, and interpret its meaning.
f. Compute the standard error of estimate, and interpret its meaning.
Hypothesis Testing on Multiple Populations
3. The Course Manager for AMBA 610 wants to use a new tutorial to teach the students about business ethics. As an experiment she randomly selected 15 students and randomly assigned them to one of three groups which include either a PowerPoint presentation created by the faculty, AuthorGen Presentation created by the faculty, or a well known tutorial by the ABC company. After completing their assigned tutorial, the students are given a Business Ethics test. At the .01 significance level, can she conclude that there is a difference between how well the different tutorials work for the students?
Students Grades on the Business Ethics Test following the Tutorial
|
PowerPoint Tutorial
|
AuthorGen Tutorial
|
ABC Tutorial
|
|
88
|
79
|
65
|
|
85
|
86
|
83
|
|
91
|
72
|
78
|
|
87
|
92
|
86
|
|
88
|
91
|
81
|