part A of assignment 1, math216 PDF

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this is the first assignment of part A of assignment 2, math216...

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(Revision 4)

Assignment 1 Assignment 1 consists of a theory component (1A, worth 80%) and a computer component (1B, worth 20%). The total marks allocated are summarized below.

Assignment

Grade weight

Assignment 1A. Theory component

96 marks

80%

Assignment 1B. Computer component

24 marks

20%

120 marks

100%

Total marks

You must submit Assignment 1A and Assignment 1B together as PDFs, through the appropriate drop box on the course home page. Submit: 1.

One PDF solution file (file) entitled Assignment1A containing all your answers to Assignment 1A, presented in the proper order. Your name and student ID number must be at the top of the first page of your solution file. and

2. One PDF solution file (file) entitled Assignment1B containing all your answers to Assignment 1B, presented in the proper order. We suggest that you print the Assignment 1 questions, so that you can conveniently review the questions with their solutions when you prepare for your exams.

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Assignment 1A. Theory Component Show your work for this component. Where relevant and unless otherwise instructed, keep your calculations and your final answer to at least four decimals.

Problem 1. Consumer survey Consider this consumer survey for a family shoe store: Survey

For Office Use Only

Please check off the boxes that apply to you.

Variabl e

1. Please indicate your marital status.

Marital

Code

married

1

single

2

2. Are you employed outside of the home?

Work

yes

1

no

2

3. How do you find out about our sales events?

Promo

newspaper promo

1

radio promo

2

TV promo

3

4. Please estimate how much you spend on shoes per year. $_________

Spend

5. Please indicate your age in years. _________

Age

Figure 1. Shoe store survey questions.

In the shoe store consumer survey, indicate: a.

which variables are qualitative and which are quantitative [5 marks]. A person marital status (label entry), employment status outside the home (attribute entry) and “how do you find out about our sales events” (attribute entry) are all qualitative dates.

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A person’s age and amount spend per year on shoes is considered a quantitative data entry. b. which variables are nominal variables [3 marks] The data set in the promo, work and marital options are all nominal variables since no mathematical computations can be made by them and since the data entry cannot be ranked. c.

which variables are ratio variables [2 marks] Ratio variables include data entry for age and spend since both are considered to have inherent zeroes.

d. Name two statistical methods (tools or measures) studied in Unit 1, that would be suitable for analyzing or summarizing the marital status variable. [4 marks] Either surveys or cluster samples would be suitable for analyzing or even summarizing the marital status variable

Problem 2. Oakford Speed Campaign The police service in the small city of Oakford mounted an intensive photo radar campaign to reduce drivers’ speeds in school zones. The speed limit in Oakford’s school zones is 30 kph. Past records (from before the radar campaign began) indicate that the average recorded speed in Oakford school zones was 34 kph, and that the majority of drivers exceeded this average recorded speed. Having included all the school zones in the city in their photo radar campaign, the Oakford police service recorded the speeds (in kph) of a random sample of 20 drivers from various school zones in Oakford, as follows. 21 44 39 24 24

28 22 44 28 44

34 28 24 23 22

39 34 28 29 26

Figure 1. Recorded speeds.

Based on the 20 data values above:

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4

a.

Compute the mean speed. [2 marks]

e.

Compute the median speed. [2 marks]

f.

Determine the modal speed. [2 marks]

g.

Compute the standard deviation of the 20 speed values. [6 marks]

Mathematics 216/ Assignment 1

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h. Of the drivers recorded, 25% showed speeds on or below what rate of speed (in kph)? [3 marks] 25% of driver were driving below the 24 kph speed limit. i.

Based on a comparison of the mean and media speeds computed in a and b above, comment on the skewness of the distribution of the 20 recorded speeds. [3 marks] Based of the comparison of the mean and median speed, we can conclude that we should have a right skew since the mean is greater than both the median. We conclude that majority of the drivers obey the speed limit and contribution to a positive skew.

j.

Construct a frequency distribution of the data, using equal class intervals of 5 (i.e., class width = 5). Make your first class 20 to 24. For each class in your distribution, show the midpoint, the frequency, the relative frequency (in percent), and the less than cumulative frequency. [6 marks]

k. Hand-sketch a frequency polygon based on the frequency distribution conducted in g above. [4 marks]

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l.

Based on your sketch of the frequency polygon, determine if the 20 speed values are positively or negatively skewed. How can you tell? [2 marks] From the graph, I conclude that the frequency polygon has a positive skew because it picks really early on at the 22 midpoint and generates a right extended tail.

m. Would the mean or the median be a better measure of central tendency for the 20 speed values? Explain. [2 marks] The median would be a better measure of central tendency since it helps show that half the people drove under the speed limit. n. Compare the mean and the skewness of the recorded speeds after the radar campaign with the mean and skewness of the recorded speeds before the radar campaign. Would you conclude that the photo radar campaign was a success or a failure? Briefly explain. [6 marks] Before the recording of speed with the radar, the average speed was 34 kph, meaning the mean was less than the median at that point. We can conclude that since the median is larger then the mean, more people drove above the speed limited resulting in a left skew. After the recording of speed with radar, the average speed reduces, meaning the mean is greater than the median. We will have a right skew since the major of drivers drove slower.

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Problem 3. Monthly Salaries A sample of 100 graduates who completed a 6-month security guard training program reported the following monthly starting salaries (in security guard related employment). Monthly Salary $1,001-$1,400

Number of Graduates 1

$1,401-$1,800

11

$1,801-$2,200

14

$2,201-$2,600

38

$2,601-$3,000

36

Total

100

Figure 1. Graduate Salaries.

a.

Compute the (approximate) mean monthly starting salary. [4 marks]

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o. Compute the (approximate) standard deviation of the monthly starting salaries. [6 marks]

(note: a lot of the work for part b was done in part a) p. Hand-sketch a relative frequency histogram, using a percentage scale on the Y-axis. Do not be concerned with drawing the graph exactly to scale. Based on the sketch, determine if the distribution of the salaries is positively or negatively skewed. What does the skewness reveal, in terms of the distribution of starting salaries? [6 marks]

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From the graph we have a negative skew, this means mean is less than the median. We can conclude that since the median is larger than the mean, a small number of graduates are receiving less compared to the vast majority.

Problem 4. InfoMart Report InfoMart is a research firm that collects and analyzes information for business firms, government agencies, non-profit organizations, and consumer clients. InfoMart has just completed a report comparing the demographic features of two cities that have similar size populations. In this question, you will examine some of the summary statistics that InfoMart computed with respect to two demographic variables: monthly household income, and age of the homeowner. These summary statistics are presented in the table below. (Homeowner age for Mountainview is not shown in the table, as it is not required to complete this problem.)

Computed Statistic

Mountainview: Household inc.

Computer-oriented Approach to Statistics

Rivercity: Household inc.

Rivercity: Homeowner age

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Mean

4,300

4,300

45

Minimum

0

0

20

Maximum

10,200

10,200

70

First Quartile (Q1)

1,500

4,000

25

Second Quartile (Q2)

2,000

4,500

50

Third Quartile (Q3)

8,000

4,800

55

Standard Deviation

3,750

100

20

Figure 1. InfoMart data.

a.

Determine the median monthly household income for each city as shown in the table above. [2 marks] The median monthly income as shown for the Mountainview household is 2,000 while the monthly income for the Rivercity household is 4,500.

q. In Rivercity, which variable, income or age, exhibits the greater degree of relative variability? Compute the appropriate statistics. [4 marks]

Rivercity income has a Coefficient of Variability of 2.32% and Rivercity age has a CV of 44.44%, therefore Rivercity age has more variation, relative to its mean. r.

In Mountainview, within what range of monthly household income does the middle 50% of households fall? [2 marks] The interquartile range is a measure of variation that gives the range of the middle portion of any data. The IQR is the difference between the third and first quartiles IQR = Q3 - Q1 IQR = 8000 -1500 IQR = 6500

s.

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Compute the variance for the age variable for Rivercity. [2 marks]

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t.

In Rivercity, 75% of the households surveyed exceed what age? [2 marks] Since ¾ of the data fall below the third quartile, 75% of the age surveyed exceed beyond the third Quartile, this is 55.

u. In Mountainview, 75% of the households surveyed earned less than what monthly income? [2 marks] Since ¾ of the data fall below the third quartile, 75% of the household earns surveyed less then the third Quartile, which is less than 8000.

Problem 5. Utility Bills The mean monthly utility bill for all households in a mid-sized city is $100, with a standard deviation of $20 per month. The monthly utility bills have a bell-shaped distribution. The Thompson family received a monthly bill of $150. Compute the Thompson family’s z-score, and determine if their monthly utility bill would be considered statistically unusual. Briefly explain how your answer relates to the empirical rule. [6 marks]

Because the z-score of the Thompson’s utility bill is 2.5 their monthly is higher than the mean. As a result, their monthly utility bill is not statistically exceptional. The empirical rule states 95% of data lie with -2 and 2,and since 2.5 lies outside the 95% deviations.

Problem 6. Student Transcript

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Joe Smith, a student at Metro University, completed the courses Biology 101, Psychology 210, Chemistry 250, and English 200 in the fall semester of 2015. He received the grades transcript below.

Fall Semester 2015, Transcript for Joe Smith

Course

Course Credit (weight)

Course Grade

Biology 101

45

86

Psychology 210

30

74

Chemistry 250

45

88

English 200

60

77

Figure 1. Transcript

Compute Joe Smith’s overall average grade (often called grade point average) for the fall semester. [4 marks]

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Mathematics 216/ Assignment 1

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Problem 7. Travel Times An employee in a university’s parking office wants to consider typical one-way travel times from home to the university for those applying for on-campus parking. The employee randomly selects 25 student parking applications and sees the following one-way travel times (in minutes) from home to the university: 22 65 64 65 54

45 35 57 47 46

32 48 68 45 55

45 54 52 66 52

55 36 61 69 63

Figure 1. Travel times.

Construct a finished stem-and-leaf plot, with the stems ranging from 2 to 6. [4 marks]

Assignment 1B. Computer Component As you work through each computer problem, use StatCrunch to generate all computer-related solutions. Do not round off the results you get from StatCrunch. Make sure that, for each computer problem, you copy and paste the output generated by StatCrunch (as is) as requested, into a single word processing file called Assignment1B. Use a word processing program that allows you to convert to a PDF file after you have completed all your solutions. Be sure to type the appropriate problem subheading (e.g., Problem 8a) before you copy and paste the related StatCrunch output or type solutions to the related interpretation questions in the Assignment1B word processing file. Each time you make a change to a StatCrunch file or word processing file, be sure to save the file.

Problem 8. WR Rodeo: Create and Recode Data File WR Rodeo is a world-renowned rodeo held annually in a western Canadian city. It is a week-long event that provides a variety of rodeo and social events, games and rides, gift shops, and fast food outlets to its attendees. The rodeo sells an advance discount package that gives event discounts on rodeo events

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and related entertainment activities. The municipal government hosting the rodeo presented the survey below to people attending the rodeo.

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Please circle the survey code or fill in the numeric values that apply to you below.

Variable

1. Please indicate where you live.

Home

Text Code

Rural Area

1

Urban Area

2

2. Do other members of your family attend rodeo-related events, functions, and entertainment activities?

Family

Code

Yes

1

No

2

3. Did you purchase the rodeo’s advance discount package?

Discount

Code

Yes

1

No

2

4. Which one of the following do you feel is most effective in attracting attention to WR Rodeo's various events and functions?

Promo

Code

Newspaper promo

1

Radio promo

2

TV promo

3

5. How much do you (and your family) spend each day on rodeo-related events?

Spend

$ _____

6. During rodeo week, how many rodeo-related events, functions, and/or activities do you or other family members attend?

Attend

_______

7. Please indicate your monthly family income before taxes.

Income

$ ______

8. Please indicate your age.

Age

______

Figure 1. Survey questions.

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The responses of 25 regular attendees who responded to this survey are given in the table below.

16

Resp#

Home

Family

Discount

Promo

Spend

Attend

Income

Age

1

1

1

1

2

350

13

7400

45

2

1

1

1

1

400

14

8600

31

3

2

2

2

1

150

2

3600

62

4

1

2

1

1

300

12

8700

32

5

2

2

1

1

100

3

8800

33

6

2

1

2

2

75

2

5300

53

7

2

2

2

2

100

3

4100

64

8

1

1

1

1

500

13

8300

34

9

1

1

1

1

500

15

8800

35

10

2

2

2

1

100

3

4400

65

11

1

1

1

1

200

10

6500

51

12

1

1

1

2

350

10

6400

52

13

2

2

2

1

50

2

5200

54

14

1

1

1

1

300

10

7200

44

15

1

1

1

1

450

12

7400

43

16

1

2

1

2

450

14

7900

37

17

2

2

2

1

50

2

3400

72

18

1

1

1

1

300

10

7300

42

Mathematics 216/ Assignment 1
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