Differentiate between permutations and combinations. How are they different? What is the formula for each?

  
MATH125: Unit 4 Submission Assignment Answer Form
Counting Techniques and Introduction to Probability
ALL questions below must be answered. Show ALL step-by-step calculation. Upload this modified Answer Form to the intellipath Unit 4 Submission lesson. Make sure you submit your work in a modified MS Word document; handwritten work will not be accepted. If you need assistance, please contact your course instructor.
Part A: Combinations & Permutations
1. Differentiate between permutations and combinations. How are they different? What is the formula for each? (15 points)
  
How are they different?
(5 pts)

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Permutation Formula
(5 pts)

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Combination Formula
(5 pts)

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2. Each state has a standard format for license plates that includes a set number of alphanumeric characters. For this assignment, you can insert a picture of your state’s non-personalized license plate or provide a sample of the format in text. (20 points total for Question 2)
  
Your State’s Name
(1 pt)

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Picture of a License Plate from Your   State
(Or a Sample)
(1 pt)

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Describe the rule for your state’s non-personalized license plates.
(1 pt)

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a. Determine the number of different license plates that can be created using this format. Assume that a license plate consists of seven alphanumeric characters using numbers (0-9) and capital letters (A-Z). Find how many unique license plates can be printed using all alphanumeric characters only once.  
  
Is   this a permutation or combination? Why?
(2   pts)

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What   formula from #1 above will you use to solve the problem?
(1   pt)

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Solution:
(4   pts)

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Show your work: 
b. You and a friend are witnesses of a car accident in your state. But you can only remember a few of the first alphanumeric characters on the license plate. 
  
How many alphanumeric characters do you   remember?
(1 pt)

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(Select a number from 2 to 5)
  
What are the characters at the beginning?
(1 pt)


  
How many license   plates start with these alphanumeric characters?
  (4 pts) 

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Show your work: 
  
How many license plates have been   eliminated?
(4 pts)

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Show your work: 
3. Your community has asked you to help the YMCA sports director organize a season of sports. You need to put together the teams. For the soccer teams, athletes signed up with three different age groups. How many different ways can you organize teams of ten for each age group? (15 points)
  
Are these a permutation or combination? Why?
(2 pt)

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What formula from #1 above will you use   to solve the problem?
(1 pt)

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How many students signed up for soccer?
(1 pt)

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(Select a multiple of 10, from 30 to 100)
  
How many kids signed up for “Little Tykes”   under the age of seven?
(1 pt)

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(Select a multiple of 10, of at least 20)
  
How many kids signed up for “Big Kids”   between 8 and 12?
(1 pt)

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(Select a multiple of 10, of at least 20)
  
How many kids signed up for “Teens”   between 13 and 18?
(1 pt)

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(Select a multiple of 10, of at least 20)
  
How many different ways can you create   teams of ten for the “Little Tykes” grade level?
(2 pts)

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Show your work: (2 pts)
  
If age levels did not matter, how many   different ways can you create teams of ten?
(2 pts)

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Show your work:  (2 pts)
Part B: Probabilities and Odds
4. For this set of exercises, you will need one standard six sided dice. If you don’t have one, you can use virtual dice: https://www.random.org/dice/,  (40 points total)
a. First, let’s differentiate between “odds” and “probability”. 
  
How   are odds and probability different?
(2   pts)

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What   is the “odds of in favor” ratio?
(3   pts)

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What   is the “probability of an event” ratio?
  (3 pts)

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What   are the odds of rolling a three? (Simplify all fraction answers.)
  (2 pts)

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What   is the theoretical probability of rolling a three? (Simplify all fraction   answers.)
  (2 pts)

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b. Please reflect on the previous question’s answer outcome. First, convert the fraction to a percent. 
  

Percent Probability
 
Theoretical   Probability   (Rounded to the nearest whole percent.)
(2 pts)

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Next, given the “Likelihood Scale” table above, what term best describes your answer?
  

Likelihood Scale
 
Term
(2 pts)

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a. What if someone challenged you to NEVER roll a 3? If you were to roll the dice 18 times, what would be the empirical probability of never getting a three? 
  

Percent Probability
 
Solution:
(Rounded to the nearest whole percent.)
(2 pts)

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Show your work: (2 pts)
b. After 18 rolls, what would be the empirical probability of getting a three on at least one of those rolls? Also, list the “Likelihood Scale” term from the table above. 
  

Percent Probability
 
Empirical   Probability 
(Rounded to the nearest whole percent.)
(2 pts)

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Likelihood Scale Term
(2 pts)

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c. What if someone challenged you to NEVER roll a 3? If you were to roll the dice 18 times, what would be the empirical probability of never getting a three? 
  

Percent Probability
 
Solution:
(Rounded to the nearest whole percent.)
(2 pts)

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Show your work:  (2 pts)
  
What   do you notice about the answers for parts c. and d. above?
(2 pts)

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d. Roll the dice 18 times and keep track of what is rolled in the table below. (2 points)
  
Roll   #

Dice

Roll   #

Dice

Roll   #

Dice
 
Roll 1

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Roll 7

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Roll 13

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Roll 2

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Roll 8

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Roll 14

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Roll 3

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Roll 9

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Roll 15

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Roll 4

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Roll 10

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Roll 16

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Roll 5

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Roll 11

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Roll 17

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Roll 6

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Roll 12

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Roll 18

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e. Based on your dice rolls, what is the experimental probability of rolling a three, out of 18 rolls? Also, list the “Likelihood Scale” term from the table above.  
  

Percent Probability
 
Experimental   Probability
(Rounded to the nearest whole percent.)
(2 pts)

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Likelihood Scale Term
(2 pts)

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Show your work:  (2 pts) 
  
In regards to the “Likelihood   Scale” terms for each, how did this differ from both the theoretical   and empirical probability?
(2 pts)

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Reasons why we choose to work with only regular parametrized curves.

 (1) There are several geometric reasons why we choose to work with only regular parametrized curves. Write a short essay to describe a few of those reasons, using examples to illustrate your points. 
(2) Show that the curve β(t) = (sin(3t) cos(t),sin(3t) sin(t), 0) is a regular curve. Then find the equation of its tangent line at the time t = π/3. 

Why is it that we need to consider reparametrizations of a curve?

 (1) Write a short essay on this topic, using examples: Why is it that we need to consider reparametrizations of a curve? Why have we chosen arclength as the “correct” parametrization? What is arclength, and how do we go about using it? How practical is reparametrization by arclength? 
(2) Reparametrize the curve below by arclength: α(t) = (e ^t cos(t), e^t sin(t), e^t )

“Correlation means Causation.” Determine whether this statement is true or false.

Debate the following statement: “Correlation means Causation.” Determine whether this statement is true or false, and provide reasoning for your determination, using the Possible Relationships Between Variables table from your textbook. 

What is regression analysis?

  
W7 Discussion “Relationship of Height and Weight”
Business Statistics
Relationship of Height and Weight
· What is regression analysis? 
· In every-day language, what is a trendline, and what is it telling us?
· What does it mean to interpolate?  What does it mean to extrapolate?
· Using the given Height and Weight data set, follow the steps in the weekly video or on pages 584-585 of the textbook for performing a regression analysis using Excel to analyze the Height and Weight Data set (assume height is the input variable x and weight is the output variable y).
· Once you have performed the analysis in Excel, state the correct simple linear regression equation and use the regression equation to predict the weight (in pounds) of a person who is 65 inches tall and the weight (in pounds) of a person who is 100 inches tall.
· Why might the regression equation you have found not be a good predication of the weight of someone who is 100 inches tall?  Are you interpolating or extrapolating when you use the trendline to predict the weight?
Download the Height and Weight Table Here.
Week 7 Assignment
Question 1
 
Assume you have noted the following prices for paperback books and the number of pages that each book contains.
Develop a least-squares estimated regression line.

1. Go to Excel.  Create a scatter diagram for the data.  Include the trendline (regression line). Make sure from Week 1 you know what a trendline is.  It is NOT simply connecting the dots.
2. Once the scatter diagram is created, right-click on the trendline itself in the graph.  Use the “format trendline” from the menu to show the equation of the trendline and the r squared value on the graph itself.
3. Provide the coefficient of determination and explain its meaning.
4. Provide the correlation coefficient between the price and the number of pages.
5. Run the regression analysis as demonstrated in the video lecture. Follow the directions of what to analyze to test and see if x and y are related. Use α = 0.10.
1. What is your null and alternative hypotheses?
2. What is the t value?
3. What are the degrees of freedom?
4. On what basis do you reject of fail to reject the null hypothesis?
 
Question 2
The following data represent a company’s yearly sales volume and its advertising expenditure over a period of 8 years.
 
1. Develop a scatter diagram of sales versus advertising and explain what it shows regarding the relationship between sales and advertising.  Make sure your axes are labeled and that you have the axes correct (notice that x and  y are above the labels).  The graph should have a title that mentions the two variables.
2. Use the method of least squares to compute an estimated regression line between sales and advertising OR just show it on the graph along with your r-squared value..
3. If the company’s advertising expenditure is $200,000, what are the predicted sales? Give the answer in dollars.  (This is found using algebra and the equation of the trendline / regression line – make sure to watch your units)
4. What does the slope of the estimated regression line indicate?
W7 Project Assignment “Project Week 7”
Business Statistics
Project Week 7
For these project assignments throughout the course you will need to reference the data in the ROI Excel spreadheet. Download it here.
 
 
Using the ROI data set:
For each of the two majors:
1. Draw the scatter diagram of Y = ‘Annual % ROI’ against X = ‘Cost’.  Include the trendline (regression line) and the r-squared value (coefficient of determination).
2. Obtain b0 and b1 of the regression equation defined as y ̂ = b0 + b1X (y intercept and slope).
3. Calculate the estimated ‘Annual % ROI’ when the ‘Cost’ (X) is $160,000.    (algebra using the equation of the line)
Test the hypothesis:   You will need to show work for the tests you are performing – t-tests, F tests…see the end of chapter 12 to make sure you are doing a comprehensive analysis.
o H0: β1 = 0
o Ha: β1 ≠ 0
o In a highlighted box, write a paragraph or more on any observations you make about the regression estimates, coefficient of determination, the plots, and the results of your hypothesis tests.  Please note the plural nature – paragraph or MORE, hypothesis testS – there is a focus on the comprehensive nature of this final part of the project.  One way to set this part of the project up is to first talk about each test – what is it, what does it test for?  Then show your work and discuss the outcome.  Repeat this for each test.  Summarize findings at the end.
   
Grading Criteria Assignments

Maximum Points
 
Meets or exceeds established   assignment criteria

40
 
Demonstrates an understanding   of lesson concepts

20
 
Clearly presents   well-reasoned ideas and concepts

30
 
Uses proper mechanics,   punctuation, sentence structure, and spelling

10
 
Total

100

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