
Concept explainers
(a)
The explanatory variable and response variable.
(a)

Answer to Problem 121E
Solution: “Age (years)” of the candidates is the explanatory variable and “Rejected (Yes/No)” is the response variable.
Explanation of Solution
In the study, there are two variables. One is an explanatory variable and the other is a response variable. An explanatory variable is a variable that affects the value of response variable and a response variable measures the result of the study. In the study, “Age (years)” is considered as the explanatory variable and the “Rejected (Yes/No)” is the response variable.
(b)
To find: The joint distribution.
(b)

Answer to Problem 121E
Solution: The joint distribution is provided below:
Explanation of Solution
Calculation: The joint distribution is computed by dividing the cell element by the total observation. The obtained joint distribution for under 20 can be calculated as follows:
Similarly, for other age group, the joint distribution can be calculated by using the above formula.
The joint distribution of the rejection data for the candidates classified by age is provided below:
From the table above, each cell shows a probability. For each cell, proportion is computed by dividing the cell entry by the total
(c)
To find: The two marginal distributions.
(c)

Answer to Problem 121E
Solution: The marginal distribution of “Rejected” is given below:
Marginal distribution of “Rejected” |
|
Yes |
No |
The marginal distribution of “Age” is given below:
Marginal distribution of “Age” |
|||||
Under 20 |
20 |
25 |
30 |
35 |
Over 40 |
Explanation of Solution
Calculation: Now, the marginal distribution is computed by dividing the row or column totals by the overall total. The marginal distribution of “Rejected” is given below:
Marginal distribution of “Rejected” |
|
Yes |
No |
The marginal distribution of “Age” is given below:
Marginal distribution of “Age” |
|||||
Under 20 |
20–25 |
25–30 |
30–35 |
35–40 |
Over 40 |
It is clearer to present these distributions as percent of the table total. The marginal distribution of the “Rejection” and “Age” is provided below:
Marginal distribution of “Rejected” |
|
Yes |
No |
3% |
97% |
Marginal distribution of “Age” |
|||||
Under 20 |
20–25 |
25–30 |
30–35 |
35–40 |
Over 40 |
17.63% |
23.52% |
16.96% |
13.69% |
15.09% |
13.30% |
Interpretation: This distribution does not provide any information about the relationship between the variables. The sum of the proportion should be 1.
(d)
The conditional distribution that will give the better explanation between the variables.
(d)

Answer to Problem 121E
Solution: The conditional distribution of “Rejection given Age” will give the better explanation between the variables.
Explanation of Solution
(e)
To find: The conditional distribution.
(e)

Answer to Problem 121E
Solution: The conditional distribution is provided below:
Explanation of Solution
Calculation: The conditional distribution is obtained by dividing the row or column elements by the sum of that row or column observation. The conditional distribution of “Rejection” for given age can be calculated as follows:
Similarly, the conditional distribution for the others can be calculated by using the above formula.
The conditional distribution of “Rejection for given age” is provided below:
Interpretation: When the value of one variable is conditioned in calculating the distribution of the other variable, we obtain a conditional distribution. From the above table, as the age increases, the number of rejected candidates also increases. This happens due to weak teeth in the older age.
Want to see more full solutions like this?
Chapter 2 Solutions
Introduction to the Practice of Statistics 9E & LaunchPad for Introduction to the Practice of Statistics 9E (Twelve-Month Access)
- In this problem, we consider a Brownian motion (W+) t≥0. We consider a stock model (St)t>0 given (under the measure P) by d.St 0.03 St dt + 0.2 St dwt, with So 2. We assume that the interest rate is r = 0.06. The purpose of this problem is to price an option on this stock (which we name cubic put). This option is European-type, with maturity 3 months (i.e. T = 0.25 years), and payoff given by F = (8-5)+ (a) Write the Stochastic Differential Equation satisfied by (St) under the risk-neutral measure Q. (You don't need to prove it, simply give the answer.) (b) Give the price of a regular European put on (St) with maturity 3 months and strike K = 2. (c) Let X = S. Find the Stochastic Differential Equation satisfied by the process (Xt) under the measure Q. (d) Find an explicit expression for X₁ = S3 under measure Q. (e) Using the results above, find the price of the cubic put option mentioned above. (f) Is the price in (e) the same as in question (b)? (Explain why.)arrow_forwardProblem 4. Margrabe formula and the Greeks (20 pts) In the homework, we determined the Margrabe formula for the price of an option allowing you to swap an x-stock for a y-stock at time T. For stocks with initial values xo, yo, common volatility σ and correlation p, the formula was given by Fo=yo (d+)-x0Þ(d_), where In (±² Ꭲ d+ õ√T and σ = σ√√√2(1 - p). дго (a) We want to determine a "Greek" for ỡ on the option: find a formula for θα (b) Is дго θα positive or negative? (c) We consider a situation in which the correlation p between the two stocks increases: what can you say about the price Fo? (d) Assume that yo< xo and p = 1. What is the price of the option?arrow_forwardWe consider a 4-dimensional stock price model given (under P) by dẴ₁ = µ· Xt dt + йt · ΣdŴt where (W) is an n-dimensional Brownian motion, π = (0.02, 0.01, -0.02, 0.05), 0.2 0 0 0 0.3 0.4 0 0 Σ= -0.1 -4a За 0 0.2 0.4 -0.1 0.2) and a E R. We assume that ☑0 = (1, 1, 1, 1) and that the interest rate on the market is r = 0.02. (a) Give a condition on a that would make stock #3 be the one with largest volatility. (b) Find the diversification coefficient for this portfolio as a function of a. (c) Determine the maximum diversification coefficient d that you could reach by varying the value of a? 2arrow_forward
- Question 1. Your manager asks you to explain why the Black-Scholes model may be inappro- priate for pricing options in practice. Give one reason that would substantiate this claim? Question 2. We consider stock #1 and stock #2 in the model of Problem 2. Your manager asks you to pick only one of them to invest in based on the model provided. Which one do you choose and why ? Question 3. Let (St) to be an asset modeled by the Black-Scholes SDE. Let Ft be the price at time t of a European put with maturity T and strike price K. Then, the discounted option price process (ert Ft) t20 is a martingale. True or False? (Explain your answer.) Question 4. You are considering pricing an American put option using a Black-Scholes model for the underlying stock. An explicit formula for the price doesn't exist. In just a few words (no more than 2 sentences), explain how you would proceed to price it. Question 5. We model a short rate with a Ho-Lee model drt = ln(1+t) dt +2dWt. Then the interest rate…arrow_forwardIn this problem, we consider a Brownian motion (W+) t≥0. We consider a stock model (St)t>0 given (under the measure P) by d.St 0.03 St dt + 0.2 St dwt, with So 2. We assume that the interest rate is r = 0.06. The purpose of this problem is to price an option on this stock (which we name cubic put). This option is European-type, with maturity 3 months (i.e. T = 0.25 years), and payoff given by F = (8-5)+ (a) Write the Stochastic Differential Equation satisfied by (St) under the risk-neutral measure Q. (You don't need to prove it, simply give the answer.) (b) Give the price of a regular European put on (St) with maturity 3 months and strike K = 2. (c) Let X = S. Find the Stochastic Differential Equation satisfied by the process (Xt) under the measure Q. (d) Find an explicit expression for X₁ = S3 under measure Q. (e) Using the results above, find the price of the cubic put option mentioned above. (f) Is the price in (e) the same as in question (b)? (Explain why.)arrow_forwardThe managing director of a consulting group has the accompanying monthly data on total overhead costs and professional labor hours to bill to clients. Complete parts a through c. Question content area bottom Part 1 a. Develop a simple linear regression model between billable hours and overhead costs. Overhead Costsequals=212495.2212495.2plus+left parenthesis 42.4857 right parenthesis42.485742.4857times×Billable Hours (Round the constant to one decimal place as needed. Round the coefficient to four decimal places as needed. Do not include the $ symbol in your answers.) Part 2 b. Interpret the coefficients of your regression model. Specifically, what does the fixed component of the model mean to the consulting firm? Interpret the fixed term, b 0b0, if appropriate. Choose the correct answer below. A. The value of b 0b0 is the predicted billable hours for an overhead cost of 0 dollars. B. It is not appropriate to interpret b 0b0, because its value…arrow_forward
- Using the accompanying Home Market Value data and associated regression line, Market ValueMarket Valueequals=$28,416+$37.066×Square Feet, compute the errors associated with each observation using the formula e Subscript ieiequals=Upper Y Subscript iYiminus−ModifyingAbove Upper Y with caret Subscript iYi and construct a frequency distribution and histogram. LOADING... Click the icon to view the Home Market Value data. Question content area bottom Part 1 Construct a frequency distribution of the errors, e Subscript iei. (Type whole numbers.) Error Frequency minus−15 comma 00015,000less than< e Subscript iei less than or equals≤minus−10 comma 00010,000 0 minus−10 comma 00010,000less than< e Subscript iei less than or equals≤minus−50005000 5 minus−50005000less than< e Subscript iei less than or equals≤0 21 0less than< e Subscript iei less than or equals≤50005000 9…arrow_forwardThe managing director of a consulting group has the accompanying monthly data on total overhead costs and professional labor hours to bill to clients. Complete parts a through c Overhead Costs Billable Hours345000 3000385000 4000410000 5000462000 6000530000 7000545000 8000arrow_forwardUsing the accompanying Home Market Value data and associated regression line, Market ValueMarket Valueequals=$28,416plus+$37.066×Square Feet, compute the errors associated with each observation using the formula e Subscript ieiequals=Upper Y Subscript iYiminus−ModifyingAbove Upper Y with caret Subscript iYi and construct a frequency distribution and histogram. Square Feet Market Value1813 911001916 1043001842 934001814 909001836 1020002030 1085001731 877001852 960001793 893001665 884001852 1009001619 967001690 876002370 1139002373 1131001666 875002122 1161001619 946001729 863001667 871001522 833001484 798001589 814001600 871001484 825001483 787001522 877001703 942001485 820001468 881001519 882001518 885001483 765001522 844001668 909001587 810001782 912001483 812001519 1007001522 872001684 966001581 86200arrow_forward
- MATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th...StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C...StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage Learning
- Elementary Statistics: Picturing the World (7th E...StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman





