Mathematics 216 Computer-oriented Approach to Statistics

Self-Test B (Computer Component) Solutions

Self-Test 1B Solutions

Each Guided Solution below summarizes key Quick Reviews (QR) steps—but not all steps—needed to complete the problem in Self‑Test1B. These QRs will be useful when you are preparing for the computer components of the assignments, midterm exam, and final exam. For more detailed steps, see the relevant Computer Lab and Activity.

QRs are in italics; steps are separated by arrows →.

Problem 1. Create a data file and a solutions file.

1a. Create a StatCrunch data file called TastyExpress. Input the 25 customer responses to the TastyExpress survey (based on Figure 2).

Guided Solution 1a

Computer Lab 1A, Activity 2.Open the StatCrunch software from within the StatCrunch website.
www.StatCrunch.com → Sign in → Open StatCrunch

Computer Lab 1A, Activity 3.Collect data with a survey; enter and save the data in StatCrunch; and sign out of StatCrunch.
Open StatCrunch → Enter variable names in data file → Enter data values in data file → Data → Save → File Name: TastyExpress → Delimiter: space → Save → StatCrunch → Sign out

Computer Lab 1A, Activity 4.Open a data file that you have saved in your My Data folder on the StatCrunch website.
www.StatCrunch.com → Sign in → My StatCrunch → My Data → TastyExpress

Computer Lab 1A, Activity 6.Export data from a StatCrunch data file to a spreadsheet file.
With data displayed in a StatCrunch data file → Data → Export → Select all variables → File name → Delimiter: comma

Computer Lab 1A, Activity 7.Copy and paste data from a spreadsheet file to a Word file.
With data displayed in the spreadsheet → Select all variables and data in the spreadsheet (Ctrl+A) → Copy the selected variables and data (Ctrl+C) → Open the Word file → Paste the selected variables and data into the Word file (Ctrl+V)

Data file:

1b. Export all the quantitative variables along with the first five data values for each quantitative variable from the StatCrunch file to a csv comma delimited file (spreadsheet file).

Guided Solution 1b

Computer Lab 1A, Activity 4.Open a data file that you have saved in your My Data folder on the StatCrunch website.
www.StatCrunch.com → Sign in → My StatCrunch → My Data → TastyExpress

Computer Lab 1A, Activity 6.Export data from a StatCrunch data file to a spreadsheet file.
With data displayed in the StatCrunch data file: Data → Export → Select quantitative variables → File name → Delimiter: comma

Computer Lab 1A, Activity 7.Copy and paste data from a spreadsheet file to a Word file.
With data displayed in the spreadsheet → Select the quantitative variables and related five data values in the spreadsheet → Copy the selected variables and data (Ctrl C) → Open the Word file → Paste the selected variables and data into the Word file (Ctrl V)

First five data values for the quantitative STAT101 survey variables pasted from a spreadsheet to a Word file:

Problem 2. Recode variables to text descriptions and export them to a spreadsheet file.

Guided Solution 2

Computer Lab 1A, Activity 4.Open the TasyExpress data file that you have saved in your My Data folder on the StatCrunch website.
www.StatCrunch.com → Sign in → My StatCrunch → My Data → TastyExpress

Computer Lab 1B, first part of Activity 5.Recode qualitative variable.s

Computer Lab 1A, Activity 6.Export data from a StatCrunch data file to a spreadsheet file.
With data displayed in a StatCrunch data file → Data → Export → Select recoded variables → File Name → Delimiter: comma

Computer Lab 1A, Activity 7.Copy and paste data from a spreadsheet file to a word file.
With data displayed in the spreadsheet → Select the recoded variables and related five data values in the spreadsheet → Copy the selected variables and data (Ctrl C) → Open the Word file → Paste the selected variables and data into the Word file (Ctrl V) → Save the updated Word file.

Save the recoded StatCrunch data file under the file name TastyExpress Recoded.
Data → Save → File Name: TastyExpress Recoded → Delimiter: space → Save

First 5 recoded variables pasted to Word file:

Problem 3. Create and interpret a pie chart.

3a. Construct a pie chart for the Gender variable: Relative Frequency.

Guided Solution 3a

Computer Lab 1B, Activity 5. Construct a pie chart to analyze qualitative survey data.
Graph → Pie Chart → With data → Recode(Gender) → Percent of Total

Computer Lab 1B, second part of Activity 5.Copy the pie chart into your Word file.
Click Options on the graph window → Copy → Right-click on the graph window → Copy image → Click in the Word file → Paste Special → Device Independent Bitmap

3b. Females are the most frequent gender category surveyed (56% vs. 44%).

Problem 4. Determine and interpret modes of qualitative variables.

4a. Display the mode for the variables Satisfy, Child, and Coupon.

Guided Solution 4a

Computer Lab 1A, Activity 8.Work with measures of central tendency: mean, median, mode.
Stat → Summary Stats → Columns → Select variables → Select Mode

Computer Lab 1A, second part of Activity 8.
With the StatCrunch summary statistics table displayed → Click in Summary Statistics table →
Ctrl A → Ctrl C → With the Word file displayed and your cursor under Problem 4a subheading → Ctrl V

Pasted summary statistics table:

4b. Type the mode for variables Satisfy, Child, and Coupon, and interpret the results in terms of the appropriate recoded values.

Problem 5. Create and interpret a Pareto chart.

5a. Construct a Pareto chart for the Recode(Satisfy) variable.

Guided Solution 5a

Computer Lab 1B, Activity 6. Construct a Pareto chart for qualitative data.
Graph → Bar Plot → With Data → Select Column variable → Select Frequency as Type → Count Descending Order → Display Value above bar

Computer Lab 1B, second part of Activity 6.Paste the Pareto chart into a Word file.
Click Options on the graph window → Copy → Right-click on the graph window → Copy Image → Click in the Word file → Paste Special → Device Independent Bitmap

5b. The Pareto chart indicates a relatively high level of consumer satisfaction, with the categories very satisfied and satisfied making up 17 of the 25 responses.

Problem 6. Create and interpret a frequency table.

6a. Construct a frequency table for Age, using 6 bins with a starting value of 20 and a fixed bin width of 5.

The frequency table should have 6 bins(classes) with a starting value of 20 and a fixed bin width of 5 where each bin includes the left endpoint. The frequency table should display: Frequencies, Relative Frequencies, and Cumulative Relative Frequencies. Copy and paste the frequency table to your Word file.

Guided Solution 6a

Computer Lab 1B, Activity 1.Construct a frequency distribution with classes.
Create a bins column: Data → Bin → Select Column → Define Bins: Use Fixed Width bins, Start-at Value, Bin-width Value → Bin Edges: Include Left Endpoint

Create a frequency table: Stat → Tables → Frequency → Select Bin Column →
Use Ctrl key to select multiple statistics: Frequency, Relative Frequency, Cumulative Relative Frequency etc... → Order by Value Ascending

Computer Lab 1B, second part of Activity 1.Copy and paste the frequency table into a Word file.
With the StatCrunch frequency table displayed → Click in the table → Ctrl A → Ctrl C → With the Word file open and your cursor under Problem 6a subheading → Ctrl V

Frequency table results for Bin(Age). Count = 25

6b. The 40–45 age category is the most frequent class.

6c. 28% of those surveyed were between 25 to 30 years of age.

6d. 56% of those surveyed were under 35 years of age.

Problem 7. Create and interpret a histogram.

7a. Construct a histogram for Visits using 4 bins with a starting value of 0 and a fixed bin width of 5.

The frequency table should display: Relative Frequencies. Copy and paste the histogram to your Word file.

Guided Solution 7a

Computer Lab 1B, Activity 2.Construct a frequency histogram.
Graph → Histogram → Select Column: Visits → Select Type: Relative Frequency → Define Bins →
Copy and paste the graph into a word file:
With the frequency histogram window displayed in StatCrunch, click Options on the graph window → Copy → Right-click on the graph window → Copy Image → Click in the Word file → Paste Special → Device Independent Bitmap

7b. The 10–15 visits per month category is the most frequent class.

7c. 40% of those surveyed visit TastyExpress between 10 and 15 times per month.

Problem 8. Analyze the variable Visits.

8a. Use StatCrunch to determine the mean, median, standard deviation, variance, first quartile, median, third quartile, and interquartile range for the variable Visits in the TastyExpress survey.

Copy and paste the Summary Statistics Table displaying all these statistics to your Word file.

Guided Solution 8a

Computer Lab 1B, Activities 8, 9, 10.Measures of Central Tendency, Variation, and Position.
Stat → Summary Stats → Columns → Select Columns: Visits → Select Statistics: Mean, Median, Std. dev., Q1, Q3, IQR

Computer Lab 1B, second part of Activity 1.Copy and paste the table into a Word file.
With the StatCrunch Summary Statistics table displayed → Click in the table → Ctrl A → Ctrl C → With the Word file open and your cursor under Problem 8a subheading → Ctrl V

Pasted summary statistics table:
Summary statistics

8b. 75% of the customers visited TastyExpress more than 6 times per month (Q1).

Problem 9. Analyze the monthly visits.

9a. Use StatCrunch to compute the mean and standard deviation monthly visits for each of the Gender subsets: male and female customers.

Copy and paste the Summary Statistics table displaying these statistics to your Word file.

Guided Solution 9a

Computer Lab 1B, Activity 12. Apply tools of descriptive statistics to subsets of data.
Stat → Summary Stats → Columns → Select Columns: Visits → Select Statistics: Mean, Std. dev. → Select variable to group by: Recode(Gender)

Computer Lab 1B, second part of Activity 1.Copy and paste the table into a Word file.
With the StatCrunch Summary Statistics table displayed → Click in the table → Ctrl A → Ctrl C → With the Word file displayed and your cursor under Problem 9a subheading → Click Ctrl V

Pasted Summary Statistics Table: Mean and standard deviation, Visits by Gender:
Summary statistics for Visits. Group by: Recode(Gender)

9b. Under the subheading 9b, type:

The number of monthly visits made by female customers (13.85714) significantly exceeds the number of monthly visits made by male customers (6.4545).

Problem 10. Analyze the monthly amount spent.

10a. Use StatCrunch to compute the mean and standard deviation monthly amount spent for each of the Gender subsets: male and female customers.

Copy and paste the Summary Statistics table displaying these statistics to your Word file.

Guided Solution 10a

Computer Lab 1B, Activity 12. Apply tools of descriptive statistics to subsets of data.
Stat → Summary Stats → Columns → Select Columns: Spend → Select Statistics: Mean, Std. dev. → Select variable to group by: Recode(Gender)

Computer Lab 1B, second part of Activity 1.Copy and paste the table to your Word file.
With the StatCrunch Summary Statistics table displayed → Click in the table → Ctrl A → Ctrl C
With the Word file displayed and your cursor under Problem 9a subheading → Click Ctrl V

Pasted Summary Statistics Table: Mean and Standard Deviation Spent By Gender:
Summary statistics for Spend. Group by: Recode(Gender).

10b. Under the subheading 10b, type:

The monthly amount spent by female customers ($196.71429) significantly exceeds the number of monthly visits made by male customers ($33.545455).

Problem 11. Create and interpret box-and-whisker plots.

11a. Use StatCrunch to create two box-and-whisker plots.

The plots must compare the monthly number of visits made by customers who frequently bring children to TastyExpress with the monthly number of visits made by customers who do NOT frequently bring children to TastyExpress. Copy and paste the box-and-whisker plots to your Word file.

Guided Solution 11a

Computer Lab 1B, Activity 11.Construct a box-and-whisker plot for quantitative data.
Graph → boxplot → Select Columns: Visits → Group By: Recode(Child) → Draw boxes horizontally

Computer Lab 1B, second part of Activity 2.Copy and paste the graph into a Word file.
With the graph window displayed in StatCrunch → Click Options on the graph window → Copy → Right-click on the graph window → Copy Image → Click in the Word file → Paste Special → Device Independent Bitmap

11b. If you move your cursor over each box plot in StatCrunch, you will see that the median monthly visits for customers who frequently bring children is 13, while the median monthly visits for customers who do NOT frequently bring children is 6.

Problem 12. Create and interpret a contingency table.

12a. Use StatCrunch to create a contingency table.

The contingency table must examine the relationship between the two variables Recode(Gender) and Recode(Satisfy). Select Recode(Gender) as the row variable and Recode(Satisfy) as the column variable. Display the Row percentage in the table.

Guided Solution 12a

Computer Lab 1B, Activity 13.Contingency table analysis: Relationship between two survey variables.
Stat → Tables → Contingency → With data → Select Row variable, Column variable → Select Row percent → Copy and paste the table into the Word file SelfTest1B, under the subheading Problem 12a.

With the StatCrunch Contingency table displayed → Click in the table → Ctrl A → Ctrl C → With the Word file displayed and your cursor under Problem 12a subheading → Ctrl V

Pasted Contingency Table:

Contingency table results:
Rows: Recode(Gender)
Columns: Recode(Satisfy)

12b. While 71.43% of the female customers are very satisfied with TastyExpress, 0% of the males are very satisfied with TastyExpress. Put differently, 72.73% of the males are less than satisfied with Tasty Express.

Problem 13. Simulate coin tossing.

13a. Generate random integers so as to simulate tossing a coin 200 times.

Let 1 represent heads, and 2 represent tails. Allow repeats. Copy and paste all the numbers generated in the StatCrunch Random Number window to the first variable column in the blank StatCrunch data file, with the first pasted value located in Row 1. Type Coin as the name of this first variable.

Recode the 1 to show as Heads and the 2 to show as Tails and store the results in the second column, with the variable name displayed as Recode(Coin).

Create a frequency table for the variable Recode(Coin), which displays the frequency and relative frequency.

Copy and paste the frequency table from StatCrunch to the Word file SelfTest1B, under the subheading Problem 13a.

Guided Solution 13a

Computer Lab 1A, Activity 8.Technology exercises for random numbers.
Open StatCrunch → Applets → Random numbers → Once the 200 random numbers display in the Random number window, hold down the left mouse button and scroll down to select all 200 numbers → Press Ctrl C to copy the numbers → Click in row 1 of the first variable column in the data file → Press Ctrl V to paste all 200 numbers into the first 200 rows of the first column in the data file → Type Coin as the column name.

Computer Lab 1B, first part of Activity 5.
Data → Recode → Coin → Compute → Heads, Tail

Computer Lab 1B, second part of Activity 1.Construct a frequency distribution (no bins required here).
Stat → Tables → Frequency → Select Recode(Coin) → Use Ctrl key to select multiple statistics: Frequency, Relative frequency,... → Order by Value Ascending

Computer Lab 1B, third part of Activity 1.Copy and paste a table into a Word file.
With the Frequency Table window displayed in StatCrunch, copy and paste the table into the Word file SelfTest1B, under the subheading Problem 13a.

Click Options on the frequency table window → Copy → Click on the frequency table window → Ctrl A → Ctrl C → Click in the Word file under Problem 12a → Ctrl V

Pasted frequency table for simulated coin toss:

Frequency table results for Recode(Coin).
Count = 200

13b. Theoretically, you would expect to see the relative frequency of the Heads category to be close to 0.50 or 50%. Because random integers are used, solutions will vary.

Self-Test 2B Solutions

Each Guided Solution below summarizes key Quick Reviews (QR) steps—but not all steps—needed to complete the problem in Self‑Test2B. These QRs will be useful when you are preparing for the computer components of the assignments, midterm exam, and final exam. For more detailed steps, see the relevant Computer Lab and Activity.

QRs are in italics; steps are separated by arrows →.

Problem 1. Work with random numbers and probability.

1a. Use StatCrunch to generate random numbers between 1 and 6, to simulate the tossing of one die 500 times.

Create the StatCrunch data file OneDie, showing the results of the simulation of 500 repetitions of a toss of one die. Copy and paste the variable name and the first 5 die numbers that were observed from StatCrunch to a Word file called SelfTest2B.

The results will differ for each student. The results observed by the course author are shown below:

Guided Solution 1a

Computer Lab 1, Activity 2.Open the StatCrunch Software from within the StatCrunch website.
www.StatCrunch.com → Sign in → Open StatCrunch

Computer Lab 2, Activity 3.Simulate the outcomes of the game of chance using random numbers:
Applets → Random Numbers → Type Minimum Value → Type Maximum Value → Type Sample Size → Allow Repeats → Compute

Computer Lab 2, Activity 3.Copy and paste random numbers to column in data file:
Select all random numbers generated in options window → Ctrl C → Click in first row of data file columns → Ctrl V

Computer Lab 1, Activity 7.Copy and paste data from a StatCrunch data file to Word file:
With data displayed in a StatCrunch data file:
Select the variable name and related five data values in the data file → Copy the selected variable and data (Ctrl C) → Open a Word file → Paste the selected variable and data into the Word file (Ctrl V) → Save the updated file.

1b. Referring to Problem 1a above, consider this event: The die number that comes up will be at least 3. This number could be 3, 4, 5, or 6.

In the StatCrunch data file OneDie, create the recoded variable Recode(Die) with two values: “At Least 3” and “Less Than 3”. Use StatCrunch to compute the approximate probability of the event “At Least 3” by creating a frequency table for the variable Recode(Die).

Copy and paste the two variable names, Die and Recode(Die), and the first five values for each of these variables from StatCrunch to the Word file SelfTest2B under the subheading Problem 1b.

Copy and paste the frequency table you created from StatCrunch to the Word file SelfTest2B under the subheading Problem 1b.

The results that the course author obtained are below:

Based on the frequency table created by the course author (below), the Probability (At Least 3) = 0.706

Frequency table results for Recode(Die):
Count = 500

Guided Solution 1b

Computer Lab 2, Activity 3.Recode the random numbers pasted to the data file:
Data → Recode → Select column variable → Compute → Recode each numerical value to text-based description

Computer Lab 2, Activity 3.Create a relative frequency table:
Stat → Tables → Frequency → Select Recode variable → Select Relative Frequency → Compute

Computer Lab 1B, second part of Activity 1.Paste a table into a Word file:
With the StatCrunch frequency table displayed → Click in table → Ctrl A → Ctrl C → With the Word file displayed and your mouse pointer under Problem 1b subheading → Ctrl V

1c. Theoretically (based on the classical view of probability), you should expect to see the relative frequency of the “At Least 3” category to be close to ____?

Solution 1c

Theoretically, you should expect to see the relative frequency of the “At Least 3” category to be close to (4 outcomes/6 possible outcomes) = 0.67.

Problem 2. Probability and random numbers

2a. Consider the following probability experiment:

A pair of dice is tossed and you are interested in observing the totalof 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.

Create a StatCrunch data file called TwoDice, which contains the results from simulating 1,000 repetitions of a toss of two dice and observing the total of each set of two numbers.

Copy and paste the 3 variable names and the first 5 rows of the three Columns Die1, Die2, and Die1+Die2 from StatCrunch to the Word file SelfTest2B under the subheading Problem 2a. the results will differ for each student.

The results observed by the course author are below.

Guided Solution 2a

Computer Lab 1, Activity 2.Open StatCrunch through the StatCrunch website:
www.StatCrunch.com → Sign on → Open StatCrunch

Computer Lab 2, Activity 3.Simulate the outcomes of the game of chance using random numbers:
Applets → Random numbers → Type minimum value → Type maximum value → Type sample size → Allow repeats → Click compute

Computer Lab 2, Activity 3.Copy and paste random numbers to column in data file:
Select all random numbers generated in options window → Ctrl C → Click In first row of data file columns → Ctrl V

New Activity: Create a new column variable by calculating row totals of existing variables:
With the 1000 random values displayed in each of the two variable Columns Die1 and Die2: Data → Compute → Expression → In the Compute Expression box, click Build to display the Expression box → In the Columns section, click Die1 → click Add Column → Click the + sign on the virtual calculator keyboard → In the Columns section, click Die2 → click the Add Column button (see figure below) → Click Okay, then click Compute, to create the third variable column Die1+Die2

Computer Lab 2, second part of Activity 1:Copy and paste data from a StatCrunch data file to Word file:
With data displayed in a StatCrunch data file → Select the three variable names and related five data values in the data file → Copy the selected variables and data (Ctrl C) → Open a Word file → Paste the selected variables and data into the Word file (Ctrl V) → Save the updated document.

2b. Refer to the probability experiment in Problem 2a, 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 10.” This means that the dice total could be 10, 11, or 12.

In the StatCrunch data file TwoDice, create the recoded variable Recode(Die1+Die2) with two value: “At Least 10” and “Less Than 10”. Use StatCrunch to compute the approximate probability of the event “At Least 10” by creating a frequency table for the variable Recode(Die1+Die2).

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 the Word file SelfTest2B under the subheading Problem 2a.

Copy and paste the frequency table that you created from StatCrunch to the Word file SelfTest2B under the subheading Problem 2b.

The results the course author obtained are shown below. Based on the frequency table, the Probability (At Least 10) = 0.191.

Frequency table results for Recode(Die1+Die2):
Count = 1000

Guided Solution 2b

Computer Lab 2, Activity 3.Recode the random numbers pasted to the data file:
Data → Recode → Select column variable → Compute → Recode each numerical value to text-based description →

Computer Lab 2, Activity 3.Create a relative frequency table:
Stat → Tables → Frequency → Select Recode variable → Select Relative Frequency → Compute

2c. Theoretically (based on the classical view of probability), you should expect to see the relative frequency of the “At Least 10” category to be close to ____?

Solution 2c

Theoretically (based on the classical view of probability), you should expect to see the relative frequency of the “At Least 10” category to be close to 6/36 = 0.17. This is based on the observation that there are 6 (out of 36) different possible outcomes that can occur for the event “At Least 10” to occur: ((4,6), (6,4), (5,5), (5,6), (6,5), (6,6))

Problem 3. Relative frequency of an event

3a. In the StatCrunch data file TwoDice, create a second recoded variable Recode(Die1+Die2) with two values.

In the StatCrunch data file TwoDice, create a second recoded variable Recode(Die1+Die2) with two values: “At Most 5” and “More than 5”. Use StatCrunch to compute the approximate probability of the event “At Most 5” by creating a frequency table for the second recoded variable Recode(Die1+Die2). Copy and paste the frequency table you created from StatCrunch to the Word file SelfTest2B under the subheading Problem 3a.

Based on the frequency table created by the course author (below), the Probability (At Most 5) = 0.282.

Frequency table results for Recode(Die1+Die2):
Count = 1000

Use StatCrunch to create a frequency table for the second recoded variable Recode(Die1+Die2) that displays the relative frequency for the two values “At Most 5” and “More Than 5”.

Guided Solution 3a

Computer Lab 2, Activity 3.Recode the dice totals as “At Most 5” and “More than 5”:
Data → Recode → Select Column variable → Compute → Recode each numerical value to text-based description

Computer Lab 2, Activity 3.Create a relative frequency table for recoded variable:
Stat → Tables → Frequency → Select Recode variable → Select Relative Frequency → Compute

3b. Theoretically (based on the classical view of probability) you should expect to see the relative frequency of the “At Most 5” category to be close to ____?

Solution 3b

Theoretically (based on the classical view of probability), you should expect to see the relative frequency of the “At Most 5” category to be close to 10/36 = 0.2777. This is based on the observation that there are ten (out of 36) different possible outcomes that can occur for the event “At Most 5” to occur: ((1,1), (2,1), (1,2), (1,3), (3,1), (2,2), (2,3), (3,2) (4,1), (1,4))

Problem 4. Sun Exposure survey

4a. Recode the responses.

Open the StatCrunch data-set file SunEx1 which is in the StatCrunch Groups folder AU Math216 2020.

With the StatCrunch SunEx1 data file displayed, create two recoded variables: Recode(Melanoma) and Recode(SkinColour).

Copy and paste the two recoded variables Recode(Melanoma) and Recode(SkinColour) and the first five values for each of these variables from StatCrunch to the Word file SelfTest2B under the subheading Problem 4a.

The results are shown below:

Guided Solution 4a

Computer Lab 2, Activity 3.Recode variables Melanoma and SkinColour.
Data → Recode → Select column variable → Compute → Recode each numerical value to text-based description

4b. Use StatCrunch to construct a contingency table consisting of the two recoded variables Recode(Melanoma) and Recode(SkinColour).

Use StatCrunch to construct a contingency table consisting of the two recoded variables Recode(Melanoma) and Recode(SkinColour). Select Recode(Melanoma) as the row variable and Recode(SkinColour) as the column variable. Display both the Counts and Row Percents.

Copy and paste the contingency table to the Word file SelfTest2B under the subheading Problem 4b.

The results are shown below:

Contingency table results:
Rows: Recode(Melanoma)
Columns: Recode(SkinColour)

Chi-Square test:

Guided Solution 4b

Computer Lab 2, Activity 2, Empirical Probability.Computing probabilities involving multiple variables:
Create the appropriate contingency table.
Stat → Tables → Contingency → With Data → Select row variable → Select column variable → Display Row Percent → Compute

4c. 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:

Solution 4c

Self-Test 3B Solutions

Each Guided Solution below summarizes key Quick Reviews (QR) steps—but not all steps—needed to complete the problem in Self‑Test3B. These QRs will be useful when you are preparing for the computer components of the assignments, midterm exam, and final exam. For more detailed steps, see the relevant Computer Lab and Activity.

QRs are in italics; steps are separated by arrows →.

Problem 1. Dozens Bet in roulette

1a. Create the data file DozensBet, and the Word file Self Test3B.

Create a StatCrunch data file called DozensBet, which contains the probability distribution of the Dozens Bet in a roulette game. Copy and paste the probability to a Word file SelfTest3B.

The pasted table is displayed below.

Guided Solution 1a

Computer Lab 1A, Activity 2.Open StatCrunch through Pearson MyLab.
www.StatCrunch.com → Sign in → Open StatCrunch

Create the variables and enter the values that describe the population distribution:
Type NetPayoff as the first column variable → Enter the two values of NetPayoff → Type P(X) as the second column variable → Enter the two values of P(X)

Copy and paste the probability distribution table to the Word file:
Select the probability distribution table → Ctrl C → Ctrl V to paste to the Word file.

1b. Use StatCrunch to compute the mean and standard deviation of the probability distribution related to the Dozens Bet.

Use StatCrunch to compute the mean and standard deviation for the DozensBet probability distribution. Copy and paste the graph, mean and standard deviation of the probability distribution from StatCrunch to the Word file SelfTest3B under subheading Problem 1b. Your results should look the figure below.

Guided Solution 1b

Computer Lab 3A, Activity 1.Find the mean and standard deviation of a discrete random variable.
With the NetPayoff and P(X) column variables displayed in the data file:
Stat → Calculators → Custom → Select column variable X → Select column variable P(X) → Compute

To copy and paste the probability distribution graph to a Word file:
Click Options Copy → Click right mouse button in StatCrunch graph window → Copy Image → Paste Special → Device Independent Bitmap

1c. Interpret the meaning.

Guided Solution 1c

The long run average net payoff that you can expect to achieve when playing the Dozens Bet roulette game many, many times is $-.0526 per game. If you played this game 1,000 times, and you bet $1 each time, you would lose approximately  1000 × $.0526 = $52.60.

Problem 2. Multiple-choice test

Compute the following binomial probabilities, with n = 100 and p = 0.25.

2a. Find the probability of getting exactly 50 correct answers.

The probability of getting exactly 50 correct answers = 0.00000005

2b. Find the probability of getting at most 40 correct answers.

The probability of getting at most 40 correct answers = 0.99967603

2c. Find the probability of getting less than 30 correct answers.

The probability of getting less than 30 correct answers = 0.85045895

2d. Find the probability of getting more than 80 correct answers.

The probability of getting more than 80 correct answers = 0.0

2e. Find the probability of getting between 20 and 30 correct answers.

The probability of getting between 20 and 30 correct answers.

2f. Find the probability of passing the exam (50 or more correct answers).

The probability of passing = 0.0000007

Guided Solution 2

Computer Lab 3A. Activity 2.Find binomial probabilities.
Stat → Calculators → Binomial → Click Standard or Between → Type N in the N box → Type P in the P box → Type X values in the P(X) box → Compute

To copy and paste the binomial probability distribution graph to a Word file:
Options → Copy → Click right mouse button → Copy Image → Paste Special → Device Independent Bitmap

Problem 3. Knee transplant

Given that the waiting times for a knee transplant are normally distributed with a mean of 122 days and a standard deviation of 34 days, use StatCrunch to find the following

3a. Find the probability that the waiting time will be at least 120 days.

The probability of waiting at least 120 days = 0.52345367

3b. Find the probability that the waiting time will be at most 90 days.

The probability of waiting at most 90 days = 0.17330722

3c. Find the probability that the waiting time will be less than 60 days.

The probability of waiting less than 60 days = 0.03411162

3d. There is a 90% probability that the waiting time will be at least how many days?

There is a 90% probability that the waiting time will be at least 78.427247 days.

3e. There is an 85% probability that the waiting time will be at most how many days?

There is an 85% probability that the waiting time will be at most 157.23874 days.

Guided Solution 3a, b, c

Computer Lab 3B, Activity 1.Find normal probabilities.
Stat → Calculators → Normal → Click Standard or Between → Type the mean → Type the standard deviation → Type the appropriate X values in the P(X) box → Compute

To copy and paste the binomial probability distribution graph to a Word file:
Options → Copy → Click right mouse button → Copy Image → Paste Special → Device Independent Bitmap

Guided Solution 3d, e

Computer Lab 3B, Activity 2.Normal distributions: Find X values.
Stat → Calculators → Normal → Click Standard or Between → Type the mean → Type the standard deviation → Type the given probability in the box after the P(X) box → Compute

Self-Test 4B Solutions

Each Guided Solution below summarizes key Quick Reviews (QR) steps—but not all steps—needed to complete the problem. These QRs will be useful when you are preparing for the computer components of the assignments, midterm exam, and final exam. For more detailed steps, see the relevant Computer Lab and Activity.

QRs are in italics; steps are separated by arrows →.

Problem 1. Population experiment

1a. Create the StatCrunch data file U4_ST_Q1_PopnDist.

Create a StatCrunch data file called U4_ST_Q1_PopnDist, which contains the population probability distribution for the numbers 1, 3, 5, 7, and 9. Copy and paste the probability distribution to the Word file SelfTest4B.

The pasted table is displayed below.

Guided Solution 1a

Computer Lab 1, Activity 2: Open StatCrunch.
Create the variables and enter the values that describe the population distribution:
Type X as the first column variable → Enter the five values of X → Type P(X) as the second column variable → Enter the five values of P(X) → Copy and paste the probability distribution table to the Word file under the Subheading Problem 1a → Select the probability distribution table → Ctrl C → Ctrl V to paste to the Word file.

1b. Compute the mean and standard deviation.

Use StatCrunch to display the graph and compute the mean and standard deviation for the population distribution in problem 1a above. Copy and paste the graph and mean and standard deviation from StatCrunch to the Word file SelfTest4B under the subheading Problem 1b. Your results should look the figure below.

Guided Solution 1b

Computer Lab 4A, Activity 1.Find the graph, mean and standard deviation of a population distribution:
With the X and P(X) column variables displayed in the data file:
Stat → Calculators → Custom → In the Values box, select Column variable X → In the Weights box, select the column variable P(X) → Compute

Copy and paste the probability distribution graph to a Word file:
Click right mouse button In StatCrunch Graph window → Copy Image → Paste Special → Device Independent Bitmap

1c. Use StatCrunch to generate 10,000 repetitions.

With the StatCrunch file U4_ST_Q1 data file displayed for the PopnDist, use StatCrunch to generate 10,000 repetitions of the following sampling experiment:

Drawing on the population values 1, 3, 5, 7, and 9, randomly select a sample of 3 values, with replacement, and observe the sample mean. StatCrunch will simulate this experiment 10,000 times, so that 10,000 sample means will be created in one column of the data file. Copy and paste the first five sample means (along with the variable name) from StatCrunch to the Word file SelfTest4B under the subheading Problem 1c.

Your results should look like this:

Guided Solution 1c

Computer Lab 4A, Activity 3.Approximate the graph, mean, and standard deviation of a sampling distribution of sample means through simulation.
With the Population Distribution variables X and P(X) displayed in the data file:
Data → Sample → Select X column variable → Type sample size → Type number of samples → /Click Sample with replacement → Click Sample all columns at one time → Compute statistic for each sample → Mean(“Sample (X)”)

Copy and paste the first five sample means to the Word file under the subheading problem 1c:
Select the first five sample means → Ctrl C → Ctrl V to paste to the Word file

1d. Compute the overall mean and standard deviation of the 10,000 sample means.

Use StatCrunch to compute the overall mean and standard deviation of the 10,000 sample means that you just generated in 1b above. Copy and paste the Summary Statistics to the Word file SelfTest4B under the subheading Problem 1d.

Your results should look similar to (but not identical to) the figure below.

Summary statistics:

Guided Solution 1d

Computer Lab 4A, Activity 3.With the Mean(Sample(X)) column displayed: Compute Overall mean and standard deviation of the Mean(Sample(X)) Column.
Stat → Summary Stats → Columns → Select Column variable: Mean(Sample(X)) → Select Mean and standard deviation as the statistics → Compute

To copy and paste the summary statistics table to a Word file:
With the Summary statistics table displayed:
Options → Copy → Ctrl A → Ctrl C → Ctrl V

1e. Central Limit Theorem.

Based on the Central Limit Theorem, the mean and standard deviation of the sampling distribution in problem 1d should approximate the population mean and the (population standard deviation/square root of sample size). That is, the mean of the sampling distribution should equal  5 and the standard deviation of the sampling distribution should approximate (2.8284/square root(3)) = 1.6329. As you can see, the mean and standard deviation of the 10,000 sample means displayed in Problem 1d are very close to the Central Limit values.

1f. Create a relative frequency histogram for the 10,000 sample means.

Use StatCrunch to create a relative frequency histogram for the 10,000 sample means you generated in Problem 1c above. Copy and paste the relative frequency histogram from StatCrunch to the Word file SelfTest4B under the subheading Problem 1f. Your results should look similar to (but not identical to) the figure below.

Guided Solution 1f

Computer Lab 4A, Activity 3.With the Mean(Sample(X)) column displayed, create a relative frequency histogram of the sampling distribution of the means:
Graph → Histogram → Select Column variable: Mean(Sample(X)) → In the Type box: Relative Frequency → Compute

1g. Note that the shape of the relative frequency of the 10,000 sample means approximates a normal distribution, as suggested by the Central Limit Theorem.

Computer Lab 4B,  Activity 1.Equal Variances Test: Test the hypothesis that two samples come from populations that have equal variances: Four-step P‑value approach, Step 2.
Stat → Variance Stats → Two Sample → With Data → Select Sample 1: Values In box: Spend: Where box: "Recode(Gender)" = Female → Select Sample 2: Values In box: Spend: Where box: "Recode(Gender)" = Male → Select Hypothesis Test Option → Select Two Tailed Variance Test → Compute

4d. Conduct the appropriate hypothesis test to determine if each of the two subset samples - amounts spent by the female customers and amounts spent by the male customers - comes from normal populations.

Guided Solution 4d

Computer Lab 4B, Activity 4.Hypothesis test for the assumption of normality: step two in the four-step P‑value approach:
Stat → Goodness-of-fit → Normality Test → With Data → Select Column variable: Spend → In the Group By box: Select Recode(Gender) → Select Shapiro-Wilk → Compute

Problem 5. Zoeys: Population Proportion

With the StatCrunch data file Zoeys open, use the four-step P‑value approach to test whether the population proportion of Zoeys female customers that frequently use Zoeys coupons exceeds the population proportion of Zoeys male customers that frequently use Zoeys coupons, with a level of significance of 5%.

Guided Solution 5

Computer Lab 5, Activity 4.Test hypotheses involving two population proportions with the appropriate column variable—summary data:
Stat → Proportion Stats → Two Sample → With Summary →
Sample 1 Values box: Recode(Coupon) → Success: Freq → Where: "Recode(Gender)" = Female →
Sample 1 Values box: Recode(Coupon) → Success: Freq → Where: "Recode(Gender)" = Male →
Select Hypothesis Test Option → Specify One Tailed Test → Compute

Self-Test 6B Solutions

Each Guided Solution below summarizes key Quick Reviews (QR) steps—but not all steps—needed to complete the problem. These QRs will be useful when you are preparing for the computer components of the assignments, midterm exam, and final exam. For more detailed steps, see the relevant Computer Lab and Activity.

QRs are in italics; steps are separated by arrows →.

Problem 1. Exercise 23 from Elementary Statistics, 6th edition

1a. Use StatCrunch to create a Scatterplot with Weight on the X-axis and Time on the Y-axis.

Open the eText data file Ex9_1-23.txt in StatCrunch.

Guided Solution 1a

Computer Lab 6A, Activity 1.Conduct correlation analysis.
Create a Scatterplot based on two variables:
Graph → Scatterplot → Select the X Column variable → Select the Y Column variable → Compute

1b. Does the plot suggest a positive or negative correlation between Weight and Time?

According to the Scatterplot in 1a above, there appears to be quite a strong negative linear correlation between weight and time. As the weight increases, the sprint time decreases.

1c. Correlation Coefficient, r, between Time and Maximum weight

r = −0.9745954

Guided Solution 1c

Computer Lab 6A, Activity 1.Conduct correlation analysis.
Compute the correlation coefficient between two variables:
Stat → Summary Stats → Correlation → Select the two Column variables → Compute

1d. Interpret the correlation coefficient.

A correlation coefficient equal to −0.9745954 means that there is a strong negative linear relationship between Weight and Time.

1e. Conduct the t-test.

At a 5% level of significance use StatCrunch to conduct the t-test to see if the population correlation coefficient, ρ, between Weight and Time, is significantly different from zero. Use the four-step P‑value approach.

Guided Solution 1e

Computer Lab 6A, Activity 1.Conduct correlation analysis.
Compute the P‑value for two tailed hypothesis test regarding correlation:
Stat → Summary Stats → Correlation → Select the two Column variables → Select Option: Two Sided P‑value → Compute

Problem 2. Nurses’ Salaries

The experience (in years) of 14 registered nurses and their annual salaries (in thousands of dollars) is displayed in the data file RNurse_Salaries.txt, available in the StatCrunch Group AU Math216 2020.

2a. Find the equation of the linear regression line.

Find the equation of the linear regression line with Years of experience being the independent variable and Annual salary being the dependent variable. Copy and paste the first screen of the Simple Linear Regression Results window from StatCrunch  as shown in Figure 1 below. Note that the output in Figure 1 will be used to answer Problems 2a, 2b, 2c, 2d, 2e, 2f, and 2g.

Guided Solution 2a

Computer Lab 6A, Activity 2.Conduct linear regression analysis:
Regression Equation, Fitted Plot, Coefficient of Determination, Significance Test, Prediction Intervals
Stat → Regression → Simple Linear → Select X column variable → Select Y column variable → Perform Hypothesis Test → Specify H_A: Slope greater than zero → Type X and Level for Prediction of Y → Select Fitted Line Plot option for graphs → Compute

2b. Plot the regression line along with the Scatterplot.

Copy and paste the graph created from StatCrunch in the second screen of the Simple Linear Regression Results window as shown in Figure 2 below.

2c. What does the slope of the regression line suggest about the relationship between the two variables?

The slope from the solution to 2a is 0.99758297. As the slope is positive, this implies that the regression line is upward sloping so that, as Years of Experience increases, Annual Salary increases.

2d. Compute the Coefficient of Determination, r-squared.

Based on the Simple Linear Regression Results window pasted from 2a above, the Coefficient of Determination, r-squared = 0.77395.

2e. Interpretation of Coefficient of Determination = 0.77395.

77% of the variation in annual salary of registered nurses can be explained by the variation in years of experience.

2f. At the 5% level of significance, test to see if the slope of the regression line significantly exceeds zero. Use the four-step P‑value approach.

Hypothesis Test re Slope of Regression Line: Four-step P‑value approach:

2g. Construct a 95% prediction interval for the annual salary for a registered nurse with 11 years of experience.

Based on the Simple Linear Regression Results window pasted from 2a above, the 95% prediction interval is: 44.45500 to 63.9198 in thousands.

2h. Single prediction estimate for the annual salary for a registered nurse with 11 years of work experience.

Based on the Simple Linear Regression Results window from 2a above, the single estimate for the annual salary for a registered nurse with 11 years of experience is 54.187441 thousands or $54,1874.41.

Problem 3. Funland

3a. Create a contingency table.

Use StatCrunch to create a contingency table for the Survey Responses in the Funland data file, with Marital displayed as the row variable, and Pass displayed as the column variable. Copy and paste the contingency table as displayed below. Note that the output in in the tables below will be used to answer Problems 3a, 3b, 3c, and 3d.

Contingency table results:
Rows: Recode(Marital)
Columns: Recode(Pass)

Chi-Square test:

Warning: Over 20% of cells have an expected count less than 5. Chi-Square suspect.

Guided Solution 3a

Computer Lab 6B, Activity 1.Conduct chi-square independence test.
Stat → Tables → Contingency → With Data → Select the row variable → Select the column variable → Display Row Percent → Click Chi-Square Test of Independence → Compute

3b. Determine whether the variables Marital and Pass are independent or related.

Use the contents of the Chi-Square Test Table to conduct a test of hypothesis, at a 5% level of significance to determine whether the variables Marital and Pass are independent or related. Use the four-step P‑value approach as follows, with the Chi Square Test statistic and P‑value shown in Step 2.

3c. Describe the relationship between the variables Recode(Marital) and Recode(Pass).

Based on the row percentages displayed in the second column of the contingency table, while 94.44% of the married customers surveyed said Yes to purchasing the monthly pass, only 0% of the single customers surveyed said Yes to purchasing the monthly pass. In general, married customers tend to be significantly more inclined to purchase the monthly pass than single customers.

3d. Comment on the expected frequency assumption underlying the Chi-square test used in independence test used in Problem 1b.

As displayed in the bottom of the chi-square table warning: “Over 20% of cells have an expected count less than 5”. This warning indicates that we cannot rely on the results of the Chi-Square test of independence conducted in Problem 3c.

Problem 4. Exercise 14 from Elementary Statistics, 6th edition

4a. Conduct an ANOVA test.

Open the eText data file Ex10_4-14.txt in StatCrunch. Conduct an ANOVA test of hypothesis to see if at least one city’s mean sale price is different from the mean sales prices in the other two cities surveyed, at a 10% level of significance.

Guided Solution 4a

Computer Lab 6B, Activity 2.One way analysis of variance (ANOVA).
Stat → ANOVA → One Way → Select all variables → Compute

4b. Test the assumption of equal variances at a 5% level of significance.

Test the assumption of equal variances at a 5% level of significance. Use the four‑step P‑value approach. Copy and paste the table displaying the Levene’s Test For Homogeneity of Variance under Step 2 of the test.

Guided Solution 4b

Computer Lab 6B, Activity 2.One way analysis of variance (ANOVA).
Stat → ANOVA → One Way → Select all variables → Select Option: Homogeneity of variance → Levene’s Test → Compute

4c. Determine if the samples come from normal populations.

Use the Shapiro-Wilk Test for Normality to determine if each of the three samples in the data file Ex10_4-14.txt appears to come from a normal population. Assume a 5% level of significance for this test. Use the four-step P‑value approach.

Guided Solution 4c

Stat → Goodness-of-fit → Normality Test → With Data → Select all variables → In the Test box, select Shapiro-Wilk → Compute