# data decisions

Problem 1 Context

The employee manager of a large company would like to estimate the proportion of full-time employees who prefer adopting the basic plan of the available health care plans in the forthcoming annual enrollment period.

Given this context, please select the most appropriate answer to each of the following questions:

1) The alternative hypothesis usually represents:

the status quo

the perceptions of the sample population

the theory the researcher would like to prove

the preconceived ideas of the researcher

2) The employee manager believes that if the proportion of employees preferring the basic plan is greater than 19%, then that plan should be subsidized by the company. A 95% confidence interval for the proportion of employees preferring the basic plan yields (16.35%; 18.12%). Based on this interval, at the 5% level of significance what should the manager’s decision be regarding whether or not to subsidy the basic plan?

Subsidy the plan

Do not subsidy the plan

3) Explain your selection on the previous question.

4) Write out the null and the alternative hypotheses that the manager is testing in this example.

5) The manager is not satisfied with the accuracy of the confidence interval (16.35%; 18.12%). What are the options to increase the interval’s accuracy (at the same confidence level)? (Mark all that correspond)

Increase the sample size

Use a smaller multiple (critical value)

Use a larger multiple (critical value)

Reduce the sample size

6) What sample size should they collect to build a 95% confidence interval with a margin of error of 2 percentage points? Use 17% as an estimate for the proportion.

Problem 2 Context

A vendor of a software product claims that their new program/app called Algebra without Tears greatly enhances the understanding of key mathematical concepts among school students. They approached the Anne Arundel County School District with the suggestion that the program be adopted in county schools and integrated into the curriculum. While the program—which runs on PC/Mac/iPad—has a very nice user interface, it is expensive. There is a \$100,000 purchase fee, and an annual maintenance license cost of \$12,000. The county wants to know what the effect of adopting this program will be on student outcomes. They are aware that some students are already using the program (on personal licenses) and collect data from three schools. The data is described below:

Score: (Dependent Variable) Aggregate score on math exams over the last year (out of 400)

Income: Family income in dollars

Use: Whether software has been used for at least 12 months (1 if yes/ 0 if no)

School: School attended (identified as A, B, or C)

Age: Student age (in years)

The categorical variable School, is recoded using two dummy variables (School_A = 1 if School = A, and 0 otherwise; School_B = 1 if School = B, and 0 otherwise). The regression output is given below.

7) State the multiple regression equation.

8) What is the interpretation of the coefficient of Age?

9) The p-value for the School_A indicates (please choose the most appropriate answer)

That School A is not useful explaining Score, given the other variables in the model

That School A is a useful predictor

That School A provides good math education

That School A is statistically different from School B

10) Should the school district adopt Algebra without Tears and integrate it into the curriculum? Explain.

11) What would you expect the math score to be of a girl who is 8 years old, does not use the software in class, goes to school C, and has a family income of \$91 thousand?

Problem 3 Context: I am Ulysses

Uncle Ulysses operates a small business: he produces three types of houses to sell to people wanting to get back to nature. Based on your discussion with your uncle you construct the following table based on your understanding of his weekly availability of labor.

Item Profit Fabrication Assembly Painting Crating

Birdhouse \$15 1 2.5 1 0.25

Doghouse \$35 1.25 1.5 0.5 0.5

Outhouse \$58 1.75 4 2 0.75

Capacity30 hours 50 hours 20 hours 20 hours

Based on the information in the table, you formulate the problem as a linear program. The labor needed for fabrication, assembly, painting, and crating is all dedicated solely to each function, i.e., none of the workers have been cross-trained. The Excel sensitivity report is shown below.

12) The optimal production quantities for each product are (round to two decimals):

Birdhouses:

Doghouses:

Outhouses:

13) The maximum weekly profit that Uncle Ulysses can make is \$ (round to two decimals.)

14) Which additional resources would you recommend that Uncle Ulysses try to obtain? (Mark all that correspond)

paint

assembly

fabrication

crate

15) Suppose Uncle Ulysses is required by contract with the American Birding Association to make one birdhouse each week. How would this affect his profits?

Profits would decrease by \$17.15.

Profits would increase by \$15.

Profits would increase by \$0.00.

Profits would decrease by \$1E + 30.

16) Uncle Ulysses feels that his prices are too low, particularly for his birdhouses. How much would he have to charge for his birdhouses before it is profitable for him to make and sell them?

\$15.00

\$32.15

\$47.15

\$17.15

17) Some kiddos in the neighborhood have started to make and sell their own doghouses. To stay competitive Uncle Ulysses is considering lowering the price of his doghouse, which would reduce the profit per doghouse from \$35 to \$25. If he does, what would be the effect on this bottom line (total profit)?

His total profit will decrease by \$10.

It is not possible to say with the information provided.

His total profit will stay the same.

His total profit will decrease by \$153.80.

18) Uncle Ulysses’ available hours for crating have fallen from 20 hours to 15 hours because of a new “one-hour nap time” rule for this five-year-old worker. How will this affect his profits?

Profits will decrease by \$7.69.

Profits will not change.

Profits will increase by \$7.69.

Profits will decrease by \$12.31.

19) Uncle Ulysses can obtain an additional 10 hours of painting capacity free of charge by training his helper monkey to paint. If he did this, how would his profits be affected?

Profit would increase by \$6.92.

Profits would decrease by \$20.

Cannot tell from the information provided.

Profits would increase by \$1.58.

20) Uncle Ulysses is planning for next spring, and he is considering making only two products. Based on the results from the linear program, which two products would you recommend that he make?

birdhouses and outhouses

doghouses and outhouses

He should continue to make all three.

birdhouses and doghouses

21) Your sister is willing to help your uncle with fabrication after school for one hour, three times a week. She wants to get paid \$15 per hour of work. What is your recommendation for Uncle Ulysses?

Hire her.

Hire her for painting.

Don’t hire her.

Cannot make a recommendation with the information provided.

Problem 4 Context: Level 9

The management at Level 9 (the best ski clothing shop) is trying to decide the size of its order for women ski parkas. Depending on the amount of snow fall in the region, the demand will vary and the the payoff will be different.

The following table summarizes the payoffs the company expects to receive under each scenario.

Heavy Normal Light

Probability 0.3 0.5 0.2

Large 10 7 -1

Medium 7 8 3

Small 2 3 5

22) What order size should Level 9 select based on the maximax criterion?

Medium

Small

Large

23) What order size should Level 9 select based on the maximin criterion?

Medium

Large

Small

24) Complete the regret table.

Order size Heavy Normal Light

Large

Medium

Small

25) What order size should Level 9 select based on the minmax regret criterion?

Large

Medium

Small

26) What order size should Level 9 select based on the expected value criterion?