Exam Preparation

Question One

A firm is interested in understanding how income influences ownership patterns for major appliances   It surveys 12 individuals and collects data on their annual income and the number of major appliances each owns.   The data are as follows:

Required :

1. Calculate a Pearson’s Correlation Coefficient for Annual Income and Number of Appliances owned.
2. Is this correlation coefficient statistically significant at the 5% level?
3. Estimate a linear regression using annual income to predict the number of appliances owned.  Describe your results.
4. If an individual had an annual income of $75,000 how many major appliances would you expect him or her to own?
5. Do you have any concerns about this regression?

Question Two

Part A

A Hospital Emergency Department has, on average, 30 new patients arrive each hour.  They plan human resources in ten minute intervals and want to ensure that they have sufficient doctors and nurses to cover most scenarios.

Required:

1. For a ten minute interval, what is the probability that exactly five patients arrive in the Emergency Department?
2. For a ten minute interval, what is the probability that three or fewer patients arrive in the Emergency Department?
3. For a ten minute interval, what is the probability that four or more patients arrive at the Emergency Department?

Part B

A restaurant gets 10% of its orders wrong.  During a lunch hour, the restaurant receives 30 orders.

Required:

1. What is the probability that none of the orders are wrong?
2. What is the probability that 2 or fewer orders are wrong?
3. What is the probability that 4 or more orders are wrong?

Part C

A computer manufacturer knows that 10% of its computers will have immediate faults and require replacement. One of the firm’s customers recently placed an order for 20 new computers.

Required:

1. What is the probability that none of the computers have a fault?
2. What is the probability that exactly two of the computers have a fault?
3. What is the probability that three or more of the computers have a fault?
4. What is the probability that four or fewer of the computers have a fault?

Part D

A restaurant is trying to plan for turnover of customers around its peak periods. On a typical day, the restaurant has 120 customers come in each hour during its peak period.

Required:

1. For a particular 2 minute interval, what is the probability that exactly 4 customers arrive?
2. For a particular 2 minute interval, what is the probability that 4 or fewer customers arrive?
3. For a particular 2 minute interval, what is the probability that 4 or more customers arrive?

Part E

A Claims officer in an insurance company knows that 25% of all claims processed are fraudulent. In a particular week he processes 20 new claims.

Required:

1. Use the Binomial distribution to determine the probability that none of the 20 new claims are fraudulent.
2. Use the Binomial distribution to determine the probability that 2 or fewer claims are fraudulent.
3. Use the Binomial distribution to determine the probability that 3 or more of the new claims are fraudulent.

Question Three

Part A

A bank is attempting to ‘bundle’ a number of financial products together to try to increase the proportion of each customer’s business they hold.   For a randomly selected number of new clients in a particular branch they offer a package which combines a current account, overdraft facility and a credit card.    The bank is interested in maximizing the total fees collected per customer.    The firm will only offer these bundles if it will generate significantly greater monthly fees per customer.  They collect the following data:

Required:

Based on the data collected would you recommend the bundle be offered to all customers (test at a 95% level of confidence)?

Part B

A precision instruments manufacturer is studying differences between two warehouses (A and B).  The company is particularly interested in the average time products spend in inventory at the two warehouses, particularly since Warehouse A has recently implemented an inventory management system that is designed to more efficiently tie production to customer’s demand.

Required

1. Estimate the population mean inventory time for Warehouse A and Warehouse B separately with 95% confidence (i.e. a 95% confidence interval)
2. If the company worries that the products risk becoming obsolete after 35 days, can we conclude that inventory time in Warehouse A is less than 35 days?  Test at a 5% level of significance.
3. If the company worries that the products risk becoming obsolete after 35 days, can we conclude that inventory time in Warehouse B is less than 35 days?  Test at a 5% level of significance.
4. Conduct a statistical test at the 5% level to determine if the inventory times between the two warehouses are significantly different from one another.  Based on this information, do you think the inventory management system should be expanded to other warehouses?

Question Four

Part A

You have been given data for a sample of 15 employees and the hours of overtime worked and the hours absence taken by each in the previous month.  The data are as follows:

Required:

1. Calculate the mean overtime
2. Calculate the median overtime
3. Calculate the variance and standard deviation for overtime
4. Calculate the 25th and 75th percentile for overtime.  What do each of these statistics tell us?
5. Calculate the coefficient of skewness and coefficient of variation for overtime? What do these statistics tell us?
6. Using a method you think appropriate display overtime hours graphically.
7. Calculate the correlation coefficient between overtime and hours of absence.
8. Calculate a t-statistic to show whether or not the correlation coefficient is statistically significant at the 5% level?
9. Estimate a linear regression using absenteeism to predict overtime and interpret your results.
10. If an individual had 10 hours of absence what would you predict her/his overtime to be?
11. Is there anything that concerns you about this regression?

Part B

A firm wants to forecast its sales for the next year so it can plan production. Sales for the previous three years are as follows:

Required:

1. Using a linear regression, calculate the trend in sales.
2. Incorporating seasonal effects, forecast sales for each of the four quarters of 2006.
3. Describe your results and the relative importance of trend and seasonal effects.
   Part C
   A firm is interested in forecasting production for the next year.  They provide you with data for the previous four years (all figures are in ‘000s of units):

   	1998	1999	2000	2001
   Quarter 1	37	36	34	34
   Quarter 2	59	58	57	57
   Quarter 3	25	23	25	25
   Quarter 4	33	32	31	20

   
   Required:
   1. Using an appropriate method, provide the production manager a forecast for each quarter of 2002.
   2. Discuss the relative importance of the trend and the cyclical factors in this forecast.
   3. What factors (not included here) might alter the prediction?

Part D

A firm is interested in compiling statistics on the length of time its customer service representatives spend on telephone calls with new clients.  It collects the following data (which you should treat as a population):

Required:

1. What is the mean call length for this population?
2. What is the median call length for this population?
3. What is the variance and standard deviation for the call length of this population?
4. What is the 25th and 75th percentile of call length for this population?
5. What is the coefficient of skewness of call length for this population?
6. What is the coefficient of variation of call length for this population?
   Question Five
   Part A
   A firm is interested in whether there is an association between rank in the hierarchy and Job Satisfaction.  They survey their employment records to count the number of individuals who are ‘Very Satisfied’ with their jobs, who are ‘Somewhat Satisfied’ with their jobs and who are ‘Dissatisfied’ with their jobs.  The data are as follows:

   	Very Satisfied	Somewhat Satisfied	Dissatisfied
   Managers	45	15	10
   Staff	100	50	25

   
   Required :
   1. Using an appropriate statistical test, determine if there is an association between rank and job satisfaction.

Part B

The instructor of this module is required to set three different exam papers each semester, which he calls A, B and C, respectively.   He is interested in seeing if there are differences in grades across the three papers.  After marking the exams he takes a random sample of three students from each of the different exam groupings.

Required :

1. Using Analysis of Variance, test at a 5% level of significance if the results on the three exam papers are significantly different from one another.

Question Six

Part A

A random sample of 20 individuals in the MBA program was drawn and their average GMAT score was 540 with a sample standard deviation of 120.

Required :

1.   Estimate the true population GMAT score with 95% confidence.

2.   Estimate the true population GMAT score with 99% confidence.

3.   Describe the difference between your results in 1 and 2.

4.   If a University Administrator believed the true average GMAT score for is 560, would you be able to reject this claim?

Part B

In a random sample of 300 managers in a large firm, 140 indicated that they offered a particular training program to their workers.

Required :

1.   If the firm wished to have a 50% participation rate in this program, does the evidence suggest that significantly less than 50% are participating (test at a 95% level of confidence)?

2.   If the firm wished to be confident of the population mean with a range of +/- 4%, 95 percent of the time, how many managers would it need to survey?

Question Seven

A production manager worries about having enough workers to staff the production line for each shift.  The organization hires ‘floating’ workers who are scheduled to come in for each shift to cover anticipated absences.  Recently absenteeism has been on the rise and on some days, the manager does not have sufficient workers to operate, which costs the firm a great deal in lost productivity.  The production manager collects data on the number of absences from the first three weeks of a month and seeks to forecast absences for Week 4.

Required :

1.   Using a linear regression, calculate the trend in absence over this three week period.

2.   Incorporating seasonal effects, forecast the number of days lost due to absence for each day in Week 4.

3.   Describe your results and the relative importance of trend and seasonal effects.

Question Nine

A production manager is interested in comparing the productivity of two different production lines.  He collects data on output from 15 shifts for each line.   The data is as follows:

Required:

1. For each production line calculate the sample mean and standard deviation.
2. What is the probability that a randomly drawn shift from production line A will produce more than 80 units?
3. What is the probability that a randomly drawn shift from production line B will produce more than 80 units?
4. What is the probability that the true mean production of production line A is more than 80?
5. What is the probability that the true mean production of production line B is more than 80?
6. Conduct a statistical test (at the 5% level) to show if there is a significant difference between the mean production of these two shifts.

Question Ten

Part A

A human resources manager knows that there is a 30% chance that a new individual hired into the organization will quit within a year.  The firm hires 15 new individuals during a recent recruiting campaign.

1. What is the probability that none of the individuals quit within a year?
2. What is the probability that exactly 4 quit within a year?
3. What is the probability that 3 or fewer quit within a year?
4. What is the probability that 4 or more quit within a year?

Part B

A telephone help line receives, on average, 180 calls per hour.   In a given 2 minute interval

1. What is the probability that there are 0 calls?
2. What is the probability that there are 3 or fewer calls?
3. What is the probability that there are 5 or more calls?

Part C

1. A health authority is concerned about the waiting times at emergency rooms in hospitals.  The authority intends to conduct a statistical test on this.  What sample size would be required to estimate the true (population) a bound of 10 minutes, at a 99% confidence level?  The population standard deviation is known to be 12 minutes.
2. A marketing firm is seeking to gauge the level of recognition of its product. It is currently seeking a 60% recognition rate. What sample should be surveyed to determine if the firm wanted to be 95% confident the actual recognition rate was within 3% of the sample recognition rate?

Question Eleven

Part A

A multi-site manufacturer is concerned that some facilities are not meeting production standards.  It examines three production runs from three different facilities and counts the number of ‘faults’ in each run.

Required :

1. Using an appropriate method determine whether there is a significant difference in the number of faults between the three sites

Part B

A firm is interested in the perception of training program directed towards sales staff.  In particular it wants to determine if there is a statistical association between satisfaction with the program and tenure in the organization.  It surveys 100 recent participants and identifies if they are ‘newer’ employees (having been with the firm less than five years) or ‘older’ employees (having been with the firm for at least five years).  It asks each of these 100 participants to indicate if they are ‘unsatisfied’, ‘moderately satisfied’ or ‘very satisfied’ with the new training program.  The results are presented in the table below:

Required :

1. Using an appropriate statistical test determine if there is a significant association between tenure in the organization and satisfaction with the training program.

Question Twelve

Part A

A firm is interested in examining the expenses paid to its male and female managers.  It collects the following data.

Required :

1. For each gender calculate the mean and median expenses.
2. For each gender calculate the variance and standard deviation of expenses.
3. Calculate the coefficient of skewness and coefficient of variation for males and females.  What do these statistics tell us?

Part B

A firm is interested in examining the salaries paid to its managers in two countries.  It collects the following data.

Required :

1. For each country calculate the mean and median income.
2. For each country calculate the variance and standard deviation of income.
3. One a single graph plot the Lorenz curves for income for each country. What does the Lorenz curve tell us?

Question Thirteen

A police officer monitors vehicle speeds at three different ‘checkpoints’.  On a particular day he draws a sample of 6 vehicle speeds from each of the three locations.  The data are as follows:

Required :

1. Using Analysis of Variance (ANOVA) test at the 5% level if there are significant differences in the means speeds across the three checkpoints.

Question Fourteen

A lecturer on a distance learning MBA programme is interested in whether there is a statistical difference in the grades of its students from two different countries. She draws a random sample of 15 students from Country A and Country B.  The data is below.

Required :

1. Construct a 95% confidence interval for the marks in each of the two countries.
2. What is the probability that a randomly selected sample of students has a mean of greater than 70 in each of the countries?
3. What is the probability that a random sample of students would have a mean of more than 60 but less than 80 in each of the countries?
4. Conduct a statistical test at the 5% level to show if there is a difference in marks between the two countries.

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