Effectiveness of a Diet Regimen - Standard Normal Distribution - Statistics Assessment Answer

Statistics Assessment Task

Answer the following questions by yourself in as much detail as possible in the time permitted. In the exam format, partial credit will be awarded in the event that the reasoning is sufficiently clear to uncover the parts of a question correctly answered. Answer the questions to the best of your ability and be systematic in solving the problems. This will always take you far. A Workbook called CPHire2019.xlsx contains the necessary data. The data have been imported from Excel into R and saved as CPHire2019.RData also. You are free to consult materials in your possession and on the WISE site but you may not consult with classmates or receive any other form of material assistance. Use the 90% confidence level if one is not given. Two reminders are in order. First, missing data are ubiquitous and adding na.rm=T will remove them should NA answers emerge. Second, R can be made to limit scientific notation by issuing the command options(scipen=8) for example. This will force R to print 8 digits. Remember that Excel is periodically of use in reformatting data and importing it and each dataset is in that format as well.

R Code without interpretation is always an incomplete answer.

Preliminaries

1. Find the ???? such that the area in red is exactly equal to 0.9 as shown in the Figure below.

 

2. What is the probability less than 1.5 in a normal distribution with mean zero and standard deviation 2?

3. Z defines the standard normal distribution (Normal(????=0,????=1). ????2 has a chi-square distribution ????2 with parameter df=1. Chi-square (????2) random variables can only be positive. The sum of k independent random Z-squared is a chi-square ????2 with df=k. In R, the chi-squared ????2 distribution is chisq with p, d, r, and q just like the others. Plot 1000 random draws from a chi-square distribution with 7 degrees of freedom.

4. What is the probability in a chi-squared (1) distribution less than or equal to 3.84.

5. What is the 90% upper bound value of chi-square(df=4)?

6. In each of the following scenarios, determine if the data are paired or not paired.

a. We would like to know if Intel’s stock and Southwest Airlines’ stock have similar rates of return. To find out, we take a random sample of 50 days, and record Intel’s and Southwest’s stock on those same days.

b. We randomly sample 50 items from Target stores and note the price for each. Then we visit Walmart and collect the price for the same 50 items.

c. Compare pre- (beginning of semester) and post-test (end of semester) scores of students.

d. Compare artery thicknesses at the beginning of a study and after 2 years of taking Vitamin E for the same group of patients.

e. Assess effectiveness of a diet regimen by comparing the before and after weights of subjects.

f. A school board would like to determine whether there is a difference in average SAT scores for students at one high school versus another high school in the district. To check, they take a simple random sample of 100 students from each high school.

g. Assess gender-related salary gap by comparing salaries of randomly sampled men and women.

h. A marketing team has called upon focus groups to compare two products. These customer ratings Ratings [from 0 to 100 with 0 as worst and 100 as best] are provided for 50 consumers that evaluated both products.

i. A sample of data on net income for firms in the tech sector.

The Case

You are employed by a human resources consulting firm. A client – a call center staffing firm – is preparing to promote an existing manager to replace the retiring Director of Operations. There are two viable candidates for the position: Chris and Pat, both longtime employees of the firm that have both moved up to their current position of Senior VP for Operations from front line customer service. We have three pieces of data to inform a choice between Chris and Pat.

A Revenue Comparison [20 pts]

The call center is organized into a large number of small identically sized units and units are specialized; units are quite different from one another but are easily comparable to themselves over time for revenues. To assess their performance, a business analyst has carefully compiled performance data on the revenue of units (measured in dollars per week) that have been managed by both Chris and Pat within a small time window. From left to right in Revenue, we have the unit’s three digit HR number [ID], revenue when Chris was manager [ChrisRev], revenue when Pat was manager [PatRev]. Use Revenue data to answer the following questions.

1. Summarise revenue for Chris and for Pat. Provide an appropriate numerical summary with the answers in the appropriate metric.

2. What is a 90% confidence interval for the average revenue while Pat managed these units given this precise sample? Provide both boundaries.

3. Assume that the underlying population is normal with the population mean and population standard deviation exactly equal to the sample mean and standard deviation that you have estimated from the sample of units managed by Pat. What is the probability of a weekly observed revenue of $160,000.00 or higher when Pat is the manager?

4. [TRUE/FALSE] With 90% confidence, units managed by Pat have average revenue of $160,000.00 or higher?

5. What is a 90% confidence interval of the average difference in revenue?

6. Is either manager better at generating higher average revenues? Yes or No. If so, Who is it?

7. Revenue volatility may be a bad characteristic of managers. Provide a graphic and a sentence comment on which manager has more volatility/greater variation in revenues!

Expenses for Operating Divisions [25 pts]

We have random samples of Expenses in dollars per week for units currently managed by Chris and for units currently managed by Pat. It is worth noting that divisions only have one manager at a time and identical sizes for units mean that expenses are very much under the manager’s control but do not depend on the unit’s function. The data appear in worksheet Expenses of the workbook on WISE. There are two recorded variables: ChrisExp is the sample of weekly expenditures in the units currently managed by Chris, PatExp is the sample of weekly expenditures in the units currently managed by Pat.

1. Show and interpret an informative plot that compares expenses for Pat and Chris.

2. Provide an appropriate 95% confidence interval for average expenses per week for units managed by Chris in the appropriate metric.

3. Average expenses per week for units managed by Chris could be $130,000 per week at the 90% level of confidence? [True/False]

4. Provide an appropriate 95% confidence interval for average expenses per week for units managed by Pat.

5. Average expenses per week for units managed by Pat are $130,000 [or not] at the 90% level of confidence? [True/False]

6. Construct a 90% confidence interval for the difference in average expenses among managers. Provide both boundaries of the interval and the metric.

7. Does either of the managers have higher expenses with at least 90% confidence? Which one? What is the probability of no difference?

Integrating the Financials

8. You now have estimated differences among the managers in both revenues and expenses over common time periods. Is there a difference in net income? If time permits, provide a 90% confidence interval [higher and lower bounds] for the difference between their weekly net incomes.

Is there a difference? Yes or No. If so, who is better? What is the associated interval for net income:

Employee Survey [20 pts]

Both Chris and Pat are well known to employees of the firm. Management conducted a survey on two independent random samples of employees. The question posed was, Would you approve of Chris/Pat being promoted to the position of Director of Operations? Employees could Approve or Disapprove of promoting the candidate in question. The survey responses are recorded in the Worksheet Survey on WISE. PatLike records Approve

or Disapprove of promoting Pat, ChrisLike records Approve or Disapprove of promoting Chris. With reference to these data, you are to answer the following questions.

1. Provide an appropriate 95% confidence interval for the true proportion believing that Chris should be promoted.

2. The true proportion of employees believing that Pat should be promoted is 0.5 or less with 90% confidence. True or False, when compared to the alternative that it is greater than 0.5.

3. Could we say, with 95% confidence, that there is a difference (as opposed to no difference) between the proportion of employees approving of Chris and Pat being promoted? Provide the boundaries of the two-sided 95% confidence interval and which alternative is preferred if one is preferred?

4. Assuming the true probability is 0.5, how many respondents would have been required to estimate the probability of responding to within plus-or-minus 0.07 with 90% confidence?

5. The survey was sent to independent random samples of size 110. Not everyone responded. Suppose that responding to the survey or not is a binomial random variable. Management is concerned that employee nonresponse could invalidate the usefulness of the data. Assess this.

a. First, how many people responded to the question concerning Pat? Chris?

b. A 90% lower bound for the probability of responding for Pat is (at least)?

c. A 90% lower bound for the probability of responding for Chris is (at least)?

d. Is there a difference in the probability of responding with 90% confidence, when compared to no difference? True or False __________________. Provide a two-sided 90% confidence interval for the difference.

Turnover Concerns

Because a promotion involves managing larger numbers of people, management is concerned about turnover under the two managers. Employees turnover at a rate of 5 per week. Excessively high turnover could be a disaster. If the manager’s observed turnover level [or higher] occurs with probability less than 0.05, this should disqualify that candidate because management takes this as evidence that the manager in question has a true turnover rate that is higher than 5 per week. Does it matter if the true turnover rate is actually 5.5?

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