Stat170 - Introductory Statistics Semester 1 Assignment 2

Question 1   (12 marks)

The times taken to assemble a car in a manufacturing plant follow a normal distribution with a mean of 20 hours and a standard deviation of 2 hours. Use this information to answer the following questions.  Show your working, including an appropriate diagram.

a.    Find the probability that a randomly selected car takes between 21 and 24 hours to assemble.  

b.    Find the probability that the average assembly time for a randomly selected sample of 5 cars manufactured in the plant will be longer than 22 hours.

c.    69% of new cars purchased in Australia have automatic transmission. Find the probability that less than 40 new cars from a random sample of 50 have automatic transmission.

Question 2 (15 marks)

In 2013, a study of 50 cruise ships was undertaken to investigate the variables which determine the crew size of a cruise ship. For each of the 50 cruise ships, information was recorded on their most recent cruise. 

Some of the variables recorded on each cruise ship are described below.

Variable Name         Variable Description

Ship                       Name of cruise ship

Age                        Age of the cruise ship in years as of 2013

Cabins                    Number of cabins

Passengers              Number of passengers on most recent cruise

Crew                      Number of crew on most recent cruise

The following output was constructed from information recorded on the age and the crew size of the 50 cruise ships in the study. Consider this output for parts a. to f.

Regression Analysis: Crew versus Age

Model Summary

      S    R-sq  R-sq(adj)  R-sq(pred)

235.954  49.67%     48.62%      45.38%

Coefficients

Term        Coef  SE Coef  T-Value  P-Value   VIF

Constant  1301.3     87.2    14.92    0.000

Age       -35.74     5.19    -6.88    0.000  1.00

Regression Equation

Crew = 1301.3 - 35.74 Age

Question ­2 continued

a.  Comment on the relation between the age of a cruise ship and the number of crew employed on the ship:

b.    Did the oldest cruise ship have the largest crew size?

c.    What was the approximate age and crew size of the oldest cruise ship?

d.    6 of the cruise ships were 12 years old. What was the approximate crew size of the 12 year old ship with the smallest crew size?

e.    The 12 year old cruise ship with the smallest crew size had the largest residual. Calculate the approximate value of this residual, using your results from part d.

 6 Write down one reason (referring specifically to relevant part of the output on the previous page) why a linear model may not be a good model for predicting the number of crew required on a cruise ship from the age of the cruise ship:

 Question 2 continued

Now consider both the number of cabins and the number of passengers as possible predictors of the crew size. The following output was constructed from the 50 cruise ships in the sample. Use this information for the remainder of Question 4.

Crew vs Cabins:

Regression Analysis: Crew versus Cabins

Model Summary

      S    R-sq  R-sq(adj)  R-sq(pred)

81.3511  94.02%     93.89%      93.52%

Coefficients

Term        Coef  SE Coef  T-Value  P-Value  

Constant    96.6     26.3     3.67    0.001

Cabins    0.7791   0.0284     ****    ***** 

Crew vs Passengers:

Regression Analysis: Crew versus Passengers

Model Summary

      S    R-sq  R-sq(adj)  R-sq(pred)

94.2836  91.96%     91.80%      91.15%

Coefficients

Term          Coef  SE Coef  T-Value  P-Value  

Constant     118.3     29.9     3.95    0.000

Passengers  0.3726   0.0159    23.44    0.000 

Question ­2 (continued)

g.    Carry out an appropriate hypothesis test, using any relevant information from the previous page, to answer the following research question. (You should comment on any test assumption/s but you can assume that assumption/s are reasonably well satisfied.)

          Hypothesis Test:

h.    Now consider the following research question:

Calculate a 95% confidence interval to estimate the population slope of the line which relates the number of passengers and the crew size of a cruise ship. Interpret your interval in order to answer the research question. Note that the assumptions are satisfied for this linear model (see scatter plot and residual plots) – you do not need to comment on these here).

                      i.        Confidence Interval:

                     ii.        Conclusion:

.

Medical researchers carried out a study to compare various methods of pain relief for patients suffering different types of pain. 115 patients who took part in this study to test the effectiveness of slow release ‘pain patches’ applied directly to the skin. 52 of these subjects suffered from chronic back pain while the other 63 subjects were chronic migraine sufferers. Each subject had pain scores recorded before and again 2 hours after applying the ‘pain patch’ to the skin. Pain scores were a self-assessed score out of 10. A score of 0 indicated no pain and a score of 10 would indicate the most extreme possible pain. Questions 3 to 5 are based on this study.

Variable                             Description

ID                                ID number for patient

Condition                        Medical Condition: 1 = Back pain, 2 = Migraine

PainScoreBefore              Self-assessed pain score before pain patch treatment applied

PainScoreAfter                             Self-assessed pain score 2 hours after pain patch treatment applied

Question 3 (8 marks)

Using appropriate parts of the following output which was constructed using the results of this study to carry out a hypothesis test in the space provided on the following page to answer this research question:

Descriptive Statistics: PainScoreAfter_Back, PainScoreAfter_Migraine

Variable                  N   Mean  SE Mean  StDev  Minimum  Median  Maximum

PainScoreAfter_Back      52  5.933    0.155  1.116    3.500   6.000    8.500

PainScoreAfter_Migraine  63  4.999    0.118  0.938    3.000   5.000    7.000

Question 3

Hypothesis Test:

Question 4 (10 marks)

1. The file, PainScores.mtw, which is available on iLearn, contains some of the data from the study described on page 6. Use these data to carry out an appropriate hypothesis test to answer this research question above.  
2. Copy and paste appropriate Minitab output for this test, along with appropriate graphical summaries onto the following page. All Minitab output should be neatly presented and clearly labelled. Marks will be deducted for summaries which do not have appropriate titles and labels.
3. Write a report addressing the research question above on the page provided. Refer to the document ‘A Guide to Report Writing’ and, in particular, to the section headed ‘A Short Guide to Report Writing for STAT170 students’ which is provided on iLearn. Your report should be based on the results of an appropriate hypothesis tests as directed in this guide. Your report not exceed one A4 page and should be word-processed and presentedon the page provided. It should include the following headings:

Introduction

Methods

Results

Conclusion