1. Consider a multinomial experiment

1. Consider a multinomial experiment involving n = 300 and k = 5 cells. The observed frequencies resulting from the experiment are shown in the companying table, and the null hypothesis to be tested is as follows:

H_o P₁= .1, P₂= .2, P₃ = .3, P₄ = .2 , P₅ = .2

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Test the hypothesis at the 1% significance level.

Cell 1 2 3 4 5

Frequency 24 64 84 72 56

2. A random sample of 50 observations yielded the following frequencies for the standardized intervals:

Interval Frequency

Z ? -1 6

-1 < Z ?0 27

0 < Z ?1 14

Z > 1 3

Can we infer that the data are not normal? Use ? =.10

3. Conduct a test to determine whether the two classifications r and c are independent, using the data in the following contingency table. Use ? =.10.

C₁ C₂ C₃

R₁ 40 32 48

R₂ 30 48 52

4. University students often complain that universities reward professors for research, but not for teaching, and argue that professors react to this situation by devoting more time and energy to the publication of their findings and less time and energy to classroom activities. Professors counter that research and teaching go hand in hand: More research makes better teachers. A student organization at one university decided to investigate the issue. They randomly selected 50 economics professors who are employed by a multi-campus university. The students recorded the salaries (in $ thousands) of the professors, their average teaching evaluations (on a 10-point scale), and the total number of journal articles published in their careers. Perform a complete analysis (produce the regression equation, access it, and report your findings). Data are listed below. Use a 5% significance level.

5. A study performed by a Columbia University professor (descried in Report on Business, august 1991) counted the number of times per minute professors from three different departments said “uh” or “ah” during lectures to fill gaps between words. The data derived from observing 100 minutes from each of the three departments were recorded. If we assume that the more frequent use of “uh” or “ah” results in more boring lectures, can we conclude that some departments’ professors were more boring than others? Data recorded below:

*NOTE* Frequent use of “uh” and “ah” could signify nervousness, a lack of rehearsal, or speaking without a memorized script, it should not automatically mean the professor is boring. However, it serves as a distractor. It could be annoying, and unappealing. The listener will be so attuned to each “ah” and “uh” that he or she will focus on that, and away from the lecture itself. That could lead to the belief that the lecture is boring.

6. A newspaper publisher, trying to pinpoint his market’s characteristics, wondered whether the way people read a newspaper is related to the reader’s educational level. A survey asked adult readers which section of the paper they read first, and asked to report their highest educational level. These data were recorded (column 1 = first section read where 1 = front page, 2 = sports, 3 = editorial, and 4 = other; and column 2 = educational level where 1 = did not complete high school, 2 = high school graduate, 3 = university or college graduate, and 4 = post graduate degree). Is there sufficient evidence to conclude that the four methods differ in their success? Data below:

*NOTE* There could be various factors that affect how a reader reads the newspaper. They may not start with the same section each time. There could be a reason for beginning with the sports section on one day, and the editorial on another. The survey did not include these variables. If they were not factored in, the data gathered by the survey would be inconclusive.

7. A statistics practitioner conducted a two-factor analysis of variance experiment with a = 4, b = 3,

and r = 8. The sum of the squares are listed here:

SS(total) = 9420 SS(A) = 203 SS(B) = 859 SS(AB) = 513

a. Test at the 5% significance level to determine whether factors a andb interact.

b. Test at the 5% significance level to determine whether differences exist between the levels of factor a.

c. Test at the 5% significance level to determine whether differences exist between the levels of factor b.

8. To determine how the number of housing starts is affected by mortgage rates, an economist recorded the average mortgage rate and number of housing starts in a large country for the past 10 years. These data are listed here: (*NOTE: The CD reflects “8” as the last “Rate” entry, instead of “9.0” as it shows in the Unit Assessment .”)

Rate: 8.5 78 7.6 7.5 8.0 8.4 8.8 8.9 8.5 9.0

Starts: 115 111 185 201 206 167 155 117 133 150

a.) Determine the regression line.

b.) What do the coefficients of the regression line tell you about the relationship between mortgage rates and housing starts?

9. The administrator of a school board in a large country was analyzing the average mathematics test scores in the schools under her control. She noticed that there were dramatic differences in the scores among the schools. In an attempt to improve the scores of all the schools, she decided to determine the factors that account for the differences. Accordingly, she took a random sample of 40 schools across the country and, for each, determined the mean test scores last year, the percentage of teaches in the percentage of teachers in each school who hold at least one university degree in mathematics, the mean age, and the mean annual income (in $ thousands) of mathematic teachers. The data is below. Use a 5% significance level.

a.) Conduct a regression analysis to develop the equation.

b.) Is the model valid?

c.) Interpret and test coefficients.

d.) Predict with 95% confidence the test scores at a school where 50% of the mathematic teachers have mathematic degrees, the mean age is 43, and the mean annual income is 48,300.

10. A developer who specializes in summer cottage properties is considering purchasing a large tract of land adjoining a lake. The current owner of the tract has already subdivided the land n/..to separate building logs and has prepared the lots by removing some of the trees, The developer wants to forecast the value of each lot From previous experience she knows that the most important factors affecting the price of the lot are size, number of mature trees, and distance to the lake. From a nearby area, she gathers the relevant data for 60 recently sold lots. The data are below:

Use a 5% significance level.

a. ) Find the regression equation

b.) What is the standard error of estimate? Interpret its value.

c.) What is the coefficient of determination? What does this statistic tell you?

QUESTION 11: Suppose you are on a game show called “Let’s Make a Deal”. The host just has given you a very unpopular prize that is not worth very much money. You would like to trade your prize for a much better one. Instead of offering a straight trade for your prize, he offers you the chance to gamble for a better prize. He shows you three curtains, A, B, and C. Behind one of them is a brand new car worth $20,000; and behind the other two curtains is nothing.

What is the probability that the car is behind curtain A?

What is the probability that the care is behind curtain B? What’s the probability it is behind curtain C?

Before the host asks you to actually make a choice of the curtain for which you want to trade your prize, he opens curtain C to reveal there is nothing behind it.

Now that you know nothing is behind curtain C; what is the probability that the car is behind curtain A?

What is the probability that the car is behind curtain B?

At this time, the host asks you to make a choice between curtains A or B. You can trade your current prize for one of the curtains, knowing that you could choose the curtain with nothing. That means you will lose your original prize and end up with nothing…you can keep your original prize, and forego the gamble altogether. Which curtain do you choose?

QUESTION 12: CASE: In the last few years, colleges and universities have signed exclusivity with a variety to private companies. These agreements bind the university to sell that company’s products exclusively on the campus. Many of the agreements involve food and beverage firms. A large university with a total enrollment of about 50,000 students has offered Pepsi-Cola an exclusivity agreement, which would give Pepsi exclusive rights to sell its products at all university facilities for the next year and an option for future years. In return the university would receive 3Soio of the on-campus revenues and an additional lump sum of $200,000 per year. Pepsi has been given 2 weeks to respond.

The management at Pepsi quickly reviews what it knows. The market forsoft drinks is measured in terms of the equivalent of 1 2-ounce cans. Pepsi currently sells an average of 22,000 cans or their equivalents per week [over the 40 weeks of the year that the university operates). The cans sell for an average of $.75 each. The costs including labor amount to $.20 per can. Pepsi is unsure of its market share but suspects it is considerably less than Exclusivity Agreement University 50%. A quick analysis reveals that if their current market share were 25%, then with an exclusivity agreement Pepsi would sell 88,000 cans per week. Thus, annual sales would be or 3,520,000 cans per year (calculated as 88,000 cans per week for 40 weeks). The gross revenue would be computed as follows:

Gross revenue: 3,520,000 cans X $.75 revenue/can : $2,640,000 This figure must be multiplied by 65% since the university would rake in 35% of the gross. Thus, 65% x $2,640,000: $1,716,000 The total cost of 20 cents per can (or $704,000) and the annual payment to the university of $200,000 is subtracted to obtain the net profit.

Net profit = $ 1 ,716,000 – $704.000 – $200,000 = $812,000

Their current annual profit is: 40 weeks X 22,000 cans/week X $.55/can : $484,000.

lf the current market share is 25%, the potential gain from the agreement is $812,000- $484,000 = $328,000

The only problem with this analysis is that Pepsi does not know how many soft drinks are sold weekly at the university. ln addition, Coke is not likely to supply Pepsi with information about its sales, which together with Pepsi’s line of products constitutes virtually the entire market.

A recent graduate of a business program believes that a survey of the university’s students can supply the needed information. Accordingly, she organizes a survey that asks 500 students to keep track of the number of soft drinks they purchase on campus over the next 7 days.

Perform a statistical analysis to extract the needed information from the data. Estimate with 95% confidence the parameter that is at the core of the decision problem. Use the estimate to compute estimates of the annual profit. Assume that Coke and Pepsi drinkers would be willing to buy either product in the absence of their first choice.

QUESTIONS:

a.) Describe the analysis

b.) On the basis of maximizing profits from sales of soft drinks at the university, should Pepsi agree to the exclusivity agreement?

QUESTION 13:Nutrition education programs which teach their clients how to lose weight or
reduce cholesterol levels through better eating patterns have been growing in popularity. The nurse in charge of one suchprogram at a local hospital wanted to know whether the programs actually work. A random sample of 33 clients who attended anutrition education program for those with elevated cholesterol levels was drawn. The study recorded the weight, cholesterollevels, total dietary fat intake per average day, total dietary cholesterol intake per average day, and percent of dailycalories from fat. These data were gathered both before and 3 months after the program. The researchers also determined thegender, age, and height of the clients. The data are stored in the following way:

Column 1: Gender (1= female, 2 = male)

Column 2: Age

Column 3:Height (in meters)

Columns 4 and 5: Weight, beforeand after (in kilograms)

Columns 6 and 7: Cholesterol level, beforeand after

Columns 8 and 9:Total dietary fat intake per average day, before and after (in grams)

Columns 10 and 11:Dietary cholesterol intake per average day, before and after (in milligrams)

Columns 12 and 13: Percent daily caloriesfrom fat, before and after

QUESTIONS:

The nurse would like the following information:

a). In terms of each of weight, cholesterol level, fat intake, cholesterol intake, and calories from fat, is the program a success?

b.) Does gender affect the amount of reduction in each of weight, cholesterol level, fat intake, cholesterol intake, and calories from fat?

c.) Does age affect the amount of reduction in weight, cholesterol level, fat intake, cholesterol intake, and calories from fat cholesterol?