QBM117 Business Statistics
Assignment 3
Due date: 22 January 2017 Value: 12%
Rationale
Assignment 3 is designed to assess the following learning outcomes:
• be able to explain the concepts of statistical inference, and apply these to confidence intervals and tests of hypotheses;
• be able to use a statistical package to analyse data appropriately, and then interpret the output;
• be able to evaluate if the assumptions underlying statistical techniques are valid in a given scenario;
• be able to apply basic principles of survey design, such as determination of appropriate sample sizes and sampling techniques.
• be able to explain the standard uses of Statistics in the media and in business environments, and judge whether the statistical methodology and conclusions drawn are appropriate;
Presentation
The assignment must be neatly handwritten with any Excel output inserted where required at the appropriate place in the assignment not in an Appendix at the back of the assignment.
Marks will be deducted for assignments which do not follow these guidelines.
The assignment is to be uploaded to EASTS as a single Word or PDF file. Assignments submitted in non-printable formats such as a ZIP file or as a collection of images will not be marked. If your scanner produces separate graphics files, please paste them into a Word document before submitting to EASTS. Pages must be numbered, and your name and student number must be included on every page.
Once you have submitted the assignment, please view your uploaded assignment to ensure that all the images, graphs etc are visible and formatted correctly
Question 1 (9 marks) 8 marks plus 1 mark for a sentence which answers the question for each part.
The amount of time a bank teller spends with each customer is normally distributed with a mean of 4.2 minutes and a standard deviation of 0.40 minutes.
a. What is the probability a teller spends less than 4 minutes with a randomly selected customer?
(4 marks)
b. A random sample of 20 customers is selected. What is the probability that the average time spent per customer will be at least 4 minutes?
(4 marks) Marking Criteria
0 marks 1 mark 2 marks 3 marks 4 marks
a. 0 of 4 criteria listed under 4 marks provided or correct 1 of 4 criteria listed under 4 marks provided or correct 2 of 4 criteria listed under 4 marks provided or correct 3 of 4 criteria listed under 4 marks provided or correct Correct expression of the required probability as a mathematical inequality; along with relevant fully labelled diagram; correct calculations and answer.
b. 0 of 4 criteria listed under 4 marks provided or correct 1 of 4 criteria listed under 4 marks provided or correct 2 of 4 criteria listed under 4 marks provided or correct 3 of 4 criteria listed under 4 marks provided or correct Correct expression of the required probability as a mathematical inequality; along with relevant fully labelled diagram; correct calculations and answer.
Sentence answering the question for both parts not included. Sentence answering the question for both parts.
Question 2 (10 marks)
A random sample of 210 of the applicants for a degree course in Policing Practice was selected and 43 of them were women.
a. Construct a 95% confidence interval estimate of the true proportion of women who applied to do this course and write a sentence which interprets the confidence limits.
(6 marks)
b. Use the appropriate Excel workbook from Estimators.xls to construct a 99% confidence interval estimate of the true proportion of women who applied to do this course. Include the resulting Excel output in your answer. Highlight the upper and lower limits of the confidence interval and write down whether this interval is narrower or wider than the interval in part a.
(4 marks) Marking Criteria
0 marks 1 mark 2 marks 3 marks 4 marks
a.
Construction of Interval None of 4 aspects listed under 4 marks correct or included 1 of the 4 aspects listed under 4 marks correct or included 2 of the 4 aspects listed under 4 marks correct or included 3 of the 4 aspects listed under 4 marks correct or included Correct choice of formula for CI; any necessary conditions stated and checked; correct substitution into formula; correct calculations.
Interpretation of interval. Limits not stated or incorrect; no interpretation of interval in the context of this question. Limits correct but interpretation incorrect or missing; Limits correct; interpretation correct in context of this question.
b. None of 4 aspects listed under 4 marks correct or included. 1 of the 4 aspects listed under 4 marks correct or included. 2 of the 4 aspects listed under 4 marks correct or included. 3 of the 4 aspects listed under 4 marks correct or included. Correct choice of workbook presented; correct substitution into workbook; upper and lower limits highlighted; correct assessment
of width of interval.
Question 3 (6 marks)
A random sample of 36 packets was selected from a day’s production of 200 gm packets of cooking chocolate. The weight of each packet was measured and recorded. The mean weight of the sample was 203.4 gm with a standard deviation of 11 gm.
Construct a 99% confidence interval estimate of the true mean weight of the 200 gm packets of cooking chocolate produced that day. Interpret the confidence limits.
Assume the weights of the packets to be normally distributed.
(6 marks) Marking Criteria
0 marks 1 mark 2 marks 3 marks 4 marks
Construction of Interval None of 4 aspects listed under 4 marks correct or included 1 of the 4 aspects listed under 4 marks correct or included 2 of the 4 aspects listed under 4 marks correct or included 3 of the 4 aspects listed under 4 marks correct or included Correct choice of formula for CI; correct value from tables, correct substitution into formula; correct calculations.
Interpretation of interval/limits. Limits not stated or incorrect; no interpretation of interval in the context of this question. Limits correct but interpretation incorrect or missing; Limits correct; interpretation correct in context of this question.
Question 4 (15 marks)
Long stays in the Intensive Care Units (ICU) of hospitals are costly and burdensome to patients, their families and society. Much research is being done into the factors which influence the length of stay in ICU in an effort to find ways of reducing the number of days spent by patients in ICU. In one hospital, the average length of stay in ICU was 9.5 days with a standard deviation of 1 day. It was proposed that better communication between the healthcare teams in ICU and the families could reduce the average length of time spent in ICU. A specialised communications team that included the medical registrar and the ICU manager was established to increase communication between the healthcare personnel and the patients’ families. After 12 months of this team operating, a study found that the average length of stay was now 6.1 days.
a. Use a 5% level of significance, to test whether there is evidence to support the use of a specialized communications team to reduce the average length of stay in ICU.
(9 marks)
b. Use the appropriate Excel macro in Test Statistics.xls to obtain the output you would use to carry out the hypothesis test in part a. Include the Excel output as your answer to part b.
(2 marks)
c. Use the output produced in part b. to test the same hypotheses using a 1% level of significance. (Hint: This means you will have to use a decision rule in terms of a p-value.) You must re-state your hypotheses, include the decision rule in terms of a p-value and make the appropriate comparison in order to come to a decision whether to reject the null hypothesis or not. You must also add a written conclusion to answer the question asked.
(4 marks)
Marking Criteria
0 mark 1 mark 2 marks 3 marks 4 marks
a. Hypotheses No correct hypotheses. One correct hypothesis including correct parameter, and equality or inequality sign. Both hypotheses correct with correct parameter and equality or inequality sign.
Test statistic Incorrect choice of test statistic. Correct choice of test statistic.
Decision rule 0 of 2 aspects listed under 2 marks correct. 1 of 2 aspects listed under 2 marks correct. Correct critical value and decision rule.
Calculation of Test statistic 0 of 2 aspects listed under 2 marks correct. 1 of 2 aspects listed under 2 marks correct. Correct substitution into formula and correct numerical calculation
Decision and conclusion in the context of the questions 0 of 2 aspects listed under 2 marks correct. 1 of 2 aspects listed under 2 marks correct. Correct decision and written conclusion in the context of the question
b. Neither of 2 aspects listed under 2 marks correct or included. 1 of the 2 aspects listed under 2 marks correct or included. Correct choice of workbook presented; correct substitution into workbook; .
c. None of 4 aspects listed under 4 marks correct or included. 1 of the 4 aspects listed under 4 marks correct or included. 2 of the 4 aspects listed under 4 marks correct or included. 3 of the 4 aspects listed under 4 marks correct or included. Hypotheses restated; correct decision rule in terms of p-value; correct decision and conclusion in terms of the question asked.
Question 5 (16 marks)
Many consumers use credit cards for everyday purchases. There are over 16 million credit cards in Australia. A major bank is interested in the proportion of their credit card customers who have additional credit cards from other financial institutions. Approximately 50% of all credit card holders have only one credit card. A survey of the bank’s credit card holders in the age group 18 to 35 olds found that 61% of this age group had only one credit card.
a. Use a 5% level of significance to test the hypothesis that for 18 to 35 year olds, the proportion of credit card holders who have only one credit card is more than the proportion for all credit card holders.
(10 marks)
b. Use the appropriate Excel workbook from Test Statistics.xls to produce the relevant output to perform the hypothesis test in part a. Include the Excel output as your answer for this part.
(2 marks)
c. Suppose the same hypothesis test were to be carried out using a level of significance of 0.01 (instead of 0.05). Use the Excel output you obtained as your answer to part b. to carry out this hypothesis test. You must re-state your hypotheses from part a., include the decision rule in terms of a p-value and make the appropriate comparison in order to come to a decision whether to reject the null hypothesis or not. You must also add a written conclusion to answer the question asked. It is not necessary to recheck any assumptions.
(4 marks)
Marking Criteria
a. 0 mark 1 mark 2 marks 3 marks 4 marks
Hypotheses No correct hypotheses. One correct hypothesis including correct parameter, and equality or inequality sign. Both hypotheses correct with correct parameter and equality or inequality sign.
Test statistic and required conditions checked 0 of 2 aspects listed under 2 marks correct. 1 of 2 aspects listed under 2 marks correct. Required conditions checked and correct choice of test statistic.
Decision rule 0 of 2 aspects listed under 2 marks correct. 1 of 2 aspects listed under 2 marks correct. Correct critical value and decision rule.
Calculation of Test statistic 0 of 2 aspects listed under 2 marks correct. 1 of 2 aspects listed under 2 marks correct. Correct substitution into formula and correct numerical calculation
Decision and conclusion in context of question asked 0 of 2 aspects listed under 2 marks correct. 1 of 2 aspects listed under 2 marks correct. Correct decision and written conclusion in the context of the question
b. Neither of 2 aspects listed under 2 marks correct or included 1 of the 2 aspects listed under 2 marks correct or included ed. Correct choice of workbook presented; correct substitution into workbook
c. None of 4 aspects listed under 4 marks correct or included. 1 of the 4 aspects listed under 4 marks correct or included. 2 of the 4 aspects listed under 4 marks correct or included 3 of the 4 aspects listed under 4 marks correct or included. Hypotheses restated; correct decision rule in terms of p-value; correct decision with reference to p-value, and conclusion in terms of the question asked.
Question 6 (5 marks)
An office administrator in charge of a company’s call centre, wishes to estimate the length of time that callers who abandon their calls to its call centre wait before abandoning their calls. The administrator would like to estimate this to within 45 seconds of the true length of time with 95% confidence. A survey found that customers who abandoned their calls waited on average 6.5 minutes with a standard deviation of 16 minutes. Use this information as appropriate, to calculate the required sample size for the estimate.
Marking Criteria
0 marks 1 mark 2 marks 3 marks 4 marks 5 marks
None of 5 aspects listed under 5 marks correct or included. 1 of the 5 aspects listed under 5 marks correct or included. 2 of the 5 aspects listed under 5 marks correct or included. 3 of the 5 aspects listed under 5 marks correct or included. 4 of the 5 aspects listed under 5 marks correct or included. Correct sample size supported by correct formula, substitution, numerical calculations and written answer in the context of the question.