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. University survey
The table below summarizes the results of a survey that asked 5000 faculty and students at a Canadian university whether they support a proposed ban on the sale of junk food (chips, pop, candy, and chocolate bars) on campus.
Support Oppose Neutral Total
Faculty 250 40 10 300
Students 1000 3500 200 4700
Total 1250 3540 210 5000
Figure 1. Survey data.
If one of the 5000 individuals surveyed is selected at random:
a. Find the probability that the individual selected opposes the proposed ban. [2 marks]
b. Find the probability that the individual selected is a student. [2 marks]
c. Find the probability that the individual selected is a student and opposes the proposed ban. [2 marks]
d. Find the probability that the individual selected is faculty or supports the proposed ban. [3 marks]
e. Find the probability that the individual selected is a student given that the individual selected opposes the proposed ban. [2 marks]
f. Find the probability that the individual selected supports the proposed ban given that the individual selected is faculty. [2 marks]
g. Are the events “being a student” and “support the proposed ban” mutually exclusive events? Explain. [2 marks]
h. Are the events “being a student” and “oppose the proposed ban” independent events? Determine the answer by showing the appropriate math work. [4 marks]
Problem 2. Shirts sold
In analyzing the sales of the last 200 dress shirts sold at a large men’s wear retail outlet, the manager created the following frequency table.
Shirt Size Frequency
Small 15
Medium 70
Large 85
Extra Large 30
Figure 1. Frequency table.
When the next customer walks in to buy a dress shirt, find the probability that:
a. The customer will buy a large dress shirt. [2 marks]
b. The customer will not buy a small dress shirt. [2 marks]
c. What type (concept) of probability are you using in determining your answer to part a above? Briefly explain. [2 marks]
Problem 3. Jobs-for-All
The owner of Jobs-for-All, a privately run job readiness program, publicly claims that anyone who has successfully completed this program will have a 90% chance of finding full-time employment within a month after graduating.
To investigate the owner’s claim, a consumer protection agency contacted three people, at random, who had successfully completed Jobs-for-All. The agency was careful to ensure that these three people were contacted one month after graduating from the program. Assuming that the owner’s claim is accurate, compute the following probabilities.
a. What is the probability that all three people contacted obtained full-time jobs? [4 marks]
b. What is the probability that none of the three people contacted obtained full-time jobs. [4 marks]
c. What is the probability that at least one of the three people contacted obtained full-time work? [4 marks]
Problem 4. Curling competition
Twenty curling teams (each representing a different country) are competing for gold, silver, and bronze medals in a week-long world tournament. In how many different ways can the curling teams win gold, silver, and bronze? [4 marks]
Problem 5. Lottery
In a popular local lottery, you must select 3 numbers (in any order) correctly out of a possible 25 numbers to win the top prize. If you purchase one ticket (with your three numbers on it), what is the probability of your winning the top prize? Keep your final answer to six decimals. [4 marks]
Problem 6. Major Motors
Major Motors sells both new and used cars. Past sales records show the following customer purchasing behaviour: 60% of all sales tend to be of new cars, and 40% of used cars. When a new car is sold, there is a 50% chance that the customer will purchase an extended car warranty. When a used car is sold, there is a 75% chance that the customer will purchase an extended car warranty.
A new customer is about to purchase a car from Major Motors. As the salesman, you would like to look at all possibilities in terms of the person buying a used or new car with or without the extended warranty.
a. Draw a tree diagram to illustrate all possible customer purchase outcomes in terms of the customer buying a used or new car with or without the extended warranty. [3 marks]
b. Find the probability that the new customer will purchase a new car and the extended warranty. [4 marks]
c. Find the probability that the new customer will purchase an extended warranty. [4 marks]
d. Determine if the two events “buys a new car” and “purchases an extended warranty” are independent. [4 marks]
Assignment 2B. 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 Assignment2B. Use a word processing program that allows you to convert to a PDF file after you have completed all your solutions.
Make sure that you type the appropriate problem subheading (e.g., Problem 7a) before you copy and paste the related StatCrunch output or type solutions to the related interpretation questions in the Assignment2B word processing file.
Each time you make a change to a StatCrunch file or word processing file, be sure to save the file.
Problem 7. SumDice: At least 8/Less than 8
Open a new StatCrunch blank Data table.
a. Consider the following probability experiment. Two dice are tossed, and you are interested in observing the total of the two numbers that come up on the dice. For example, if a pair of ones (1,1) appears, the total is 2. If the two dice show a 5 on the first die and 3 on the second die (5,3), the total is 8.
Use StatCrunch to simulate this probability experiment of tossing a pair of dice simultaneously 1,000 times. In the process, you will use random numbers to generate two column variables: “Die1” and “Die2”. You will then create a third variable column called “Die1 + Die2” by using StatCrunch to compute the appropriate Expression.
After generating the variable column “Die1+Die2”, copy and paste the variable names and the first five rows of columns “Die1”, “Die2”, and “Die1+Die2” from StatCrunch to your word processing file Assignment2B, under the subheading Problem 7a.
Save the StatCrunch data file as “SumDice”, and the Word file as Assignment2B. [1 mark]
b. Refer to the probability experiment in Problem 7a, in which a pair of dice is tossed and the total of the two numbers that appear is observed. Consider the event “the dice total will be at least 8.” This means that the dice total could be 8, 9, 10, 11, or 12.
With the StatCrunch file “SumDice” open and the variable “Die1+Die2” displayed in the third column, create the recoded variable “Recode(Die1+Die2)” with two values: “At Least 8” and “Less Than 8”.
Copy and paste the four variable names: “Die1”, “Die2”, “Die1+Die2”, and “Recode(Die1+Die2)”, with the first five values for each of these variables, from StatCrunch to your Word file Assignment2B, under the subheading Problem 7b. [1 mark]
c. Using the StatCrunch data that you created in Problems 7a and 7b above, use StatCrunch to approximate the probability of the event “the dice total will be at least 8” by creating the appropriate frequency table for the variable “Recode(Die1 + Die2).”
Copy and paste the frequency table that you created from StatCrunch to the Word file Assignment2B under the subheading Problem 7c. Based on your frequency table created, the Probability (At Least 8) =? [1 mark] Remember to resave the StatCrunch data file “SumDice” and the Word file Assignment2B. d. Theoretically (based on the classical view of probability), you should expect to see the relative frequency of the “At Least 8” category to be close to . Explain. Type your answer under the subheading Problem 7d in your Word file Assignment 2B. [1 mark]
Problem 8. SumDice: At most 7/More than 7
Open the SumDice data table you created in problem 7 above.
a. Use StatCrunch to approximate the probability of the event “the dice total will be at most 7” based on the empirical concept of probability. That is, you want StatCrunch to compute the relative frequency related to the event “the dice total will be at most 7” based on repeating the probability experiment 1,000 times as you did in Problem7a above.
With the StatCrunch file “SumDice” open and the variable “Die1+Die2” displayed in the third column, create a second recoded variable “Recode(Die1+Die2)” with two values: “At Most 7” and “More Than 7”.
Use StatCrunch to create the appropriate frequency table for the second recoded variable “Recode(Die1+Die2).”
Copy and paste the frequency table you created to the Word file Assignment2B under the subheading Problem 8a.
Based on your frequency table created, the Probability (At Most 7) = ?
Remember to save both the StatCrunch data file “SumDice” and the Word file Assignment2B. [1 mark]
b. Theoretically (based on the classical view of probability), you should expect to see the relative frequency of the “At Most 7” category to be close to ? Type your answer under the subheading Problem 8b. Because random integers are used in the solution, solutions will differ. [1 mark]
Problem 9. Sun Exposure Survey
In this and subsequent problems, we will use the actual responses of 4,022 Canadians in Canada's first and only national Sun Exposure Survey, to determine and compute probabilities of interest.
Figure 1 presents two questions from the Sun Exposure Survey. Figure 2 provides an overview of the responses received from the 4,022 Canadians. These responses are stored in the file SunEx2, in the StatCrunch Groups folder AU Math216 Winter 2015.
Please circle the coded responses that apply to you.
Variable Code
- Have you ever been diagnosed to have skin cancer or a pre-cancerous skin condition? (Examples of pre-cancerous skin conditions: red spots or patches or suspicious moles.) [Q439, p. 18 in survey] PreCancer
Yes 1
No 2
Do not know 3 - What is your natural hair colour? [Q441, p. 19 in survey] HairColour
Blonde 1
Red 2
Light brown 3
Dark brown 4
Black 5
Do not know 6
Refuse to say 7
Figure 1. Selected Survey Questions from 1996 Canadian Sun Exposure Survey. Source: Section 12.0, Questionnaire. 1996 Sun Exposure Survey. The National Study on Sun Exposure and Protective Behaviours. Funded by: National Cancer Institute of Canada, The Canadian Dermatology Association, The Canadian Association of Optometrists, Environment Canada, Health Canada, BC Tel. (Special Surveys Division/ Division des enquêtes spéciales, Statistics Canada, Ottawa, Ontario, Canada, 1996). Archived by Statistics Canada at http://www23.statcan.gc.ca/imdb-bmdi/instrument/4419_Q1_V1-eng.pdf
Respondent
Number
PreCancer
HairColour
1 2 3
2 2 3
3 2 3
4 2 3
5 2 4
. . .
. . .
. . .
4019 2 2
4020 2 3
4021 2 3
4022 2 3
Figure 2. Data File: SunEx2
To check your understanding of the data table above, the first Canadian responding to the survey (Respondent Number 1) has not been diagnosed with pre-cancer and has light brown hair colour.
Open the StatCrunch dataset file SunEx2, in the StatCrunch Groups folder AU Math 216 Winter 2015. (See Navigation Hints if you need help finding this file.)
a. Recode each of the coded responses to each of the variables, PreCancer and HairColour, to more recognizable values, as shown in Figure 3. The recoded values will be displayed as new column variables Recode(PreCancer) and Recode(HairColour).
After recoding the variables, copy and paste the first five rows (along with the variable names) of the two recoded variable columns, PreCancer and HairColour, from the StatCrunch data table to the Word file Assignment2B, under the subheading Problem 9a. [1 mark]
Remember to re-save both the StatCrunch file and the updated word processing file.
Variable Original Code: New Code:
PreCancer 1
2
3 1_Yes
2_No
3_DoNotKnow
HairColour 1
2
3
4
5
6
7 1_Blonde
2_Red
3_LTBrown
4_DKBrown
5_Black
6_DoNotKnow
7_RefuseToSay
Figure 3. Recode Data Values in SunEx2 Data Table
b. Use StatCrunch to create a contingency table consisting of the two recoded variables Recode(PreCancer) and Recode(HairColour). Select Recode(PreCancer) as the row variable and Recode(HairColour) as the column variable. Display both the Counts and Row Percents. Copy and paste the contingency table to the word processing file Assignment 2B, under 9b. [2 marks]
c. Based on the contingency table you created, use your calculator (not StatCrunch) to compute the probability that a randomly selected adult Canadian responding to the survey will: (type your answers under the heading Problem 9c in your Word file.)
i. Have Red hair colour. [1 mark]
ii. Have been diagnosed with PreCancer. [1 mark]
iii. Have Dark Brown hair colour OR will be diagnosed with PreCancer. [1 mark]
iv. Have Black hair colour AND will be diagnosed with PreCancer. [1 mark]
v. Have Blonde hair colour GIVEN that have been diagnosed with PreCancer. [1 mark]
vi. Determine, by making the appropriate math calculations, if the events “Blonde hair colour” and “have been diagnosed with PreCancer” are independent events. [1 mark]
Remember to save your latest version of the StatCrunch file and your Word file.