Final Exam a Qms 102 Fall 2010

Business Statistics ICQMS 102

Final ExaminationFall 2010

1. This is a closed book examination except students may use:- Calculators- Three sheets (8.5 x 11 inch) of notes.

2. There are 10 questions with a total of 100 marks

3. Where indicated the final answer must be placed on the drawn line provided. Only the final answer(s) for questions/parts of questions will be marked.

4. Time allotted: Two and one half hours.

5. Write neatly and legibly.

6. There are 17 pages (including this cover sheet)

7. Do not remove any pages from this examination package

8. Fill in the information requested below:

Name:________________ Section No:_______________ Student No:__________________

THE G. RAYMOND CHANG SCHOOL OF CONTINUING EDUCATION

350 Victoria Street, Toronto, ON, Canada M5B 2K3 Tel: 416.979.5035 Fax: 416.979.5277 E-mail:

[email protected] www.ryerson.ca/ce

Name_____________ Student Number: ______________ Page 2 of 18

Question 1 (4 marks)

1. The table below lists the Farm product price index for Canada 1997=100. Perform the following analysis:

October 2008 124

November 2008 121

December 2008 120.2

January 2009 117.8

February 2009 118.4

March 2009 120.5

April 2009 120.4

May 2009 120.7

June 2009 116.5

July 2009 115.7

August 2009 113

September 2009 105.1

October 2009 103.7

November 2009 103.2

December 2009 103.8

January 2010 105.4

February 2010 106

March 2010 107

April 2010 109.4

May 2010 110.9

June 2010 107.5

July 2010 106.2

August 2010 105.6

Construct the stem and leaf diagram for the above data. (4 marks)


Farm product Index

Tens Ones10 33310 5556677911 0311 567812 000014

2. Listed below is a modified Stats Canada Table “Individuals by total income level” for 2008, listing the number of individuals in various income brackets. Note that the class widths are not equal. (11 marks)

1. Estimated upper limit.

a. Complete the blank columns in the table (3 marks) __________

Class LimitsClass

MidpointsNumber of Individuals

Relative Frequency

Cumul Rel. Freqency

Under $5,000 2500 2,037,550 0.082386930.082387

$5,000 to under $10,000 7500 2,063,570 0.083439040.165826

$10,000 and under $15,000 12500 2,445,550 0.098884130.26471

$15,000 and under $20,000 17500 2,482,960 0.100396780.365107

$20,000 and under $25,000 22500 1,983,710 0.080209950.445317

$25,000 and under $35,000 30000 3,348,590 0.135397940.580715

$35,000 and under $50,000 43500 3,974,750 0.160716290.741431

$50,000 and under $75,000 62500 3,523,180 0.142457360.883888

$75,000 and under $100,000 87500 1,528,680 0.061811130.9457

$100,000 and under $150,000 125000 862,480 0.034873790.980573

$150,000 and under $200,000 175000 223,700 0.009045160.989618

$200,000 and under $250,000 225000 90,840 0.003673050.993292

$250,000 and under $350,0001 300000 165,910 0.006708461

Total, all income groups 24,731,470 1

Median total income 28,920


b. Find the Grouped Mean (2 marks) ___39775.73______

c. Estimate the first quartile (Q1) and the third quartile (Q3) (4 marks) ___________Q1: 14256.2Q3:50503.8

d. Comment on the shape of the distribution. Is it left or right skewed or symmetrical? (2 marks)

____Right Skewed – notice when u chart relative frequency to the midpoint.



It is known from past experience that only 20% of all new customers subscribing to the QMS Business Statistics magazine renew their subscription after the first year. A random sample of 30 new customers is selected for statistical analysis.

a. Find the probability that fewer than 20% of the new customers in the sample of 30 renew their subscription after the first year by answering the following:

1. First define the variable X: (1 mark)

The variable X is suppose to represent the number of individuals that renew their subscription to QMS.

2. Then state what type of probability distribution is applicable for this problem: (1 mark)

______________Binomial________________________

3. Then compute and state the required probability that fewer than 20% in the sample of 30 will renew subscription after the first year: (2 marks)

___________________________________0.6070__________________

b. What is the probability that between 4 and 8 (inclusive) in the sample of 30 will renew their subscription after the first year? (2 marks)

__________________________________0.7487____________________________________

______________________________________________________________________________

c. What is the most likely number of renewed subscriptions after the first year in the sample of 30? (2 marks)

_6 ------When exploring various X values we find that the probability is the highest at X=6


Question 4 (10 marks)An engineering project has fallen behind schedule and therefore required a lot of overtime work by a number of engineers over the last month. The project manager generated the following table of the overtime hours and associated costs for ten engineers that charged overtime.

Engineer Overtime hours Pay ($/hr)1 32 632 21 563 16 484 23 595 10 776 45 547 78 428 76 669 56 5610 37 47

a. What was the average number of hours of overtime worked per engineer? (2 marks)

39.4 hours

b. What was the average hourly overtime pay rate per engineer? (2 marks)

56.8 hours

c. What was the average pay per hour of overtime? (3 marks)

55.04

d. What was the median pay per hour of overtime? (3 marks)

Median: 5 and 65: 17396: 2016Median =1877.5


Question 5 (12 marks)Student grades on a statistics test follow an approximately normal distribution with a mean of 63% and a standard deviation of 7%.

a. What is the probability that a randomly selected student will score between 55% and 65% on the test? (5 marks)

1. First define the variable X: (1 mark)

____________X is the the probability that a student will be chosen

2. Then state what type of probability distribution is applicable for this problem: (1 mark)

_____________Normal Distribution___________________

3. Then compute and state the required probability value: (2 marks)

The required probability value is 0.4859.

b. What percentage of all the students who wrote the test will score more than 70%? (2 marks)

_________________________0.1587_________

c. What mark will be exceeded by only 7% of the students? (2 marks)

______________73.33%____________________


d. Calculate the first quartile (Q1) and the third quartile (Q3) for the distribution of student marks (2 marks)

Q1 = InvN(Left tail, 0.25, 7, 63) = 58

Q3 = InvN(Left tail, 0.75, 7, 63) = 67.7 = 68

_________________________________________________________________

e. If thirty six (36) students write this statistics test, what is the probability that the average mark will be between 62% and 64%? (2 marks)

Sampling distribution question, sigma xbar = sigma / sqrt(36) = 7/6 =

P(62 < xbar < 64) = Ncd(62,64,1.1666, 63) = 0.6086

_________________________________________________________________



Defects in the insulation of an electrical cable occur randomly at an average rate of 1.5 per 20-meter length of the cable.

a. Find the probability that in a randomly chosen 20-meter length of cable, there are exactly two defects by answering the following:

First define the variable X below: (1 mark)

The variable X stands for the number of defects founds

Then state what probability distribution is applicable? (2 marks)

_________Poisson distribution_______________________________________

Then compute and state the required probability value. (2 marks)

________ Ppd(2,1.5) = 0.2510_________________________

b. Find the probability that in a randomly chosen 20-meter length of cable there are at most 2 defects. (2 marks)

Pcd (2,1.5) = 0.8088

c. What is the probability that in a 40-m length of this cable there will be at least 3 defects? (3 marks)

________1 – Pcd(2,3) = 1 – 0.4232 = 0.5768


Question 7 (12 marks distributed as follows: 2, 6, 4)

The Unforgettable Desserts pastry café sells boxes of assorted pastries. The pastry boxes are delivered each Monday morning from the bakery to the pastry shop. At the start of each week, the shop owner must decide how many pastry boxes to order so as to maximize profit. The shop owner pays $3.50 to the bakery for each pastry box. The café sells each pastry box for $6.50. However, all pastry boxes not sold by the end of the week must be discarded and fresh pastry boxes must be ordered for the following week. Experience has shown that the weekly demand for the pastry boxes by café customers may be modeled by a random variable X with a probability distribution as shown below:

x P(x)0 0.091 0.172 0.253 0.224 0.195 0.08

a. Calculate the expected demand per week for the pastry boxes. ____2.49 (2.5)__________

b. Complete the table below assuming the demand probability distribution remains unchanged, but the pastry shop owner orders 3 pastry boxes per week.

x P(X=x)

No. Sold Sales Revenue ($)

Costs ($) Profits ($)

0 0.09 0 0 10.50 -10.501 0.17 1 6.50 10.50 -4.002 0.25 2 13.00 10.50 2.503 0.22 3 19.50 10.50 9.004 0.19 3 19.50 10.50 9.005 0.08 3 19.50 10.50 9.00

c. What is the expected profit when the pastry shop owner orders 3 pastry boxes per week?

________________$3.41_______________________


Question 8 (10 marks, 2 per part)

The observed frequency ogive below provides an approximation of the annual individual gross income in Canada for 2008 (Statistics Canada).

Use the frequency ogive to answer the following:

a. The median gross income is: approximately 30000 is the income._______

b. Complete the frequency distribution below:

Gross Income ($1000s) Frequency (millions)0 to under 25 1125 to under 50 7.350 to under 75 3.675 to under 100 1.5100 to under 125 0.6125 to under 150 0.3150 to under 175 0.1175 to under 200 0.1

Total 24.5

c. What percentage (correct to two decimal places) of individuals had an income of at least $50,000?

____________=(6.2/24.5) *100 = 25.31%________


d. Complete the table below:

Gross Income (X in $1000s) Frequency (millions)X < 25 1125 ≤ X < 75 10.9X ≥ 75 2.6

e. Assume income tax paid on gross income was in accordance with the following table:

Gross Income (X in $1000s) Income Tax Rate (percentage)X < 25 1525 ≤ X < 75 25X ≥ 100 45

What is the mean income tax rate (as a percent correct to two decimal places) paid by a tax payer in Canada?22.63


Question 9 (13 marks, 1 per part except where otherwise indicated)

The following represent the weights in lbs of garden soil delivered by trucks from the Green Backyard warehouse to customers on Tuesday:

600 625 550 495 510 3000 1080 800 589200 615 475 90 1200 2000 200 2450 4700

Help the manager of Green Backyard construct a box-whisker plot for the data by stating the value of each of the following:

a. The mean weight_________1121.056_________________

b. The first quartile____495______________________

c. The third quartile __________1200_______________

d. The median _________607.5____________________

e. The interquartile range ___705__________________

f. The right inner fence _______2257.5_____________LIF:_0___

g. The right outer fence ________3315_____________LOF:__

h. The left whisker starts at ________90_________________

i. The right whisker ends at_______2000_________________

j. State the value of each outlier: (if no outliers exist state “None”):

________________________3000, 2450________________________________

k. State the value of each extreme: (if no extremes exist state “None”):

_________________________________4700_________________________


l. Draw the horizontal box-whisker plot for the data. Draw the diagram within the given rectangle with the box-whisker plot centered on the given horizontal center line. Indicate the mean with “+” and the right fences with broken vertical lines. Do not draw the left fences. (2 marks)


Question 10 (10 marks, 1 mark per part unless otherwise indicated)

Circle the letter of the most appropriate response:

1. Which of the following involves inferential statistics as opposed to descriptive statistics?a. The average age of all members in a club is 32.1 years.b. Based on a sample of 500 voters, 60% of all voters prefer a certain political party.c. A basketball team scored an average of 67.2 points this season.d. A town reported 17 burglaries this month.e. 21% of the students in a class received an A grade.

2. Which of the following is most likely a statistic as opposed to a parameter?a. The average salary for all employed workers.b. The proportion of all drivers who have insurancec. The average income of tax payers sampled in a polld. The range of ages of all students attending universitiese. The most frequently occurring number of telephones in a Canadian household.

3. A questionnaire asks the respondent to report his or her gender. The scale of measurement is:a. Nominalb. Ordinalc. Intervald. Ratioe. None of the above

4. A questionaire asks the respondent to indicate his or her height. The scale of measurement is:a. Nominalb. Ordinalc. Intervald. Ratioe. None of the above

5. In a survey the respondent is asked to rate a grocery chain: 1= strongly dislike, 2= dislike, 3=neutral, 4=favor, 5=strongly favor. The scale of measurement used is:a. Nominalb. Ordinalc. Intervald. Ratioe. None of the above


6. A visual display that plots points on the horizontal axis based on the upper endpoints of the class and on the vertical axis according to the cumulative number of data values up to and including that class, and then connects the plotted points is called a(n):a. Histogramb. Frequency polygonc. Relative frequency polygond. Ogivee. None of the above

7. Suppose that a population has a mean of 100 and a standard deviation of 20. The standardized value (z score) corresponding to a data value of 64 would be:a. -1.80b. -0.44c. 0.44d. 1.80e. 3.20

Use the control chart factors table (last page of exam) and the following data to answer the next two questions (#8 and #9):

Five samples, each consisting of four measurements, were taken to evaluate whether a process was in control. The results were:

Sample Number Mean Range1 25 72 22 53 19 94 23 55 24 6

8. The centerline, upper control limit, and lower control limit for a 3-sigma range chart should be set at:a. Rbar = 6.0, UCL = 13.7, LCL = 0.0b. Rbar = 6.0, UCL = 13.7, LCL = 6.0c. Rbar = 6.0, UCL = 12.7, LCL = 6.0d. Rbar = 6.4, UCL = 14.6, LCL = 0.0e. Rbar = 6.4, UCL = 14.6, LCL = 6.4


9. The centerline, upper control limit, and lower control limit for a 3-sigma mean chart should be set at:

a. xbar = 22.0, UCL = 24.9, LCL = 20.3b. xbar = 22.0, UCL = 27.3, LCL = 17.9c. xbar = 22.6, UCL = 24.9, LCL = 20.3d. xbar = 22.6, UCL = 27.3, LCL = 17.9e. xbar = 22.6, UCL = 29.0, LCL = 16.2

10. The central limit theorem states that:a. The mean of the sample (xbar) approaches the mean of the population (μ) as the sample size increases.b. The distribution of the sample means (xbar) approaches the normal distribution as the sample size increases, regardless of the shape of the distribution of x.c. The distribution of the sample means remains constant as the sample size increases, regardless of the shape of the distribution of x. d. The binomial distribution approaches the normal distribution as the sample size increases.e. Both b and d are true


Control Chart FactorsNumber of observations in sample

d2 d3 D3 D4 A2

2 1.128 0.853 0 3.267 1.8803 1.693 0.888 0 2.575 1.0234 2.059 0.880 0 2.282 0.7295 2.326 0.864 0 2.114 0.5776 2.534 0.848 0 2.004 0.483

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Final Exam a Qms 102 Fall 2010