Lesson_03_sol_prob
4 Pages

Lesson_03_sol_prob

Course Number: STAT 200, Fall 2010

College/University: Penn State

Word Count: 1277

Rating:

Document Preview

Probability - Solutions 1 Situation: The High School and Beyond data is from a large-scale longitudinal study conducted by the National Opinion Research Center (1980) under contract with the National Center for Education Statist ics. Below is a table representing a sample of 100 students from this data that includes the students gender and whether the high school they attended was public or private. Let A denote...

Unformatted Document Excerpt
Coursehero >> Pennsylvania >> Penn State >> STAT 200

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

- Probability Solutions 1 Situation: The High School and Beyond data is from a large-scale longitudinal study conducted by the National Opinion Research Center (1980) under contract with the National Center for Education Statist ics. Below is a table representing a sample of 100 students from this data that includes the students gender and whether the high school they attended was public or private. Let A denote the subset of these 100 subjects that the student is Female, so the event A = {Female}. Similarly define event B as the student attended a Public high school, so the event B = {Public}.. The following 2x2 table classifies each student according to gender and school type. Female Male Total Public 38 46 84 Private 7 9 16 Total 45 55 100 For example, the table reveals that 38 students were both Female and attended a Public high school, so: P(A B) = 38/100 = 0.38 (read the probability of A and B is 0.38)). a. Fill in the marginal totals of the table (the row and column totals). From these totals determine the probability that the student is Female and also the probability the student attended a Public high school. Answer the following placing the proper event notation (e.g. A, B) in the ( ). P(Female) = P( A ) = .45 P(Public) = P( B ) = .84 b. Translate the following events into set notation using the symbols A and B, complement, union, intersection. Also give the probability of the event as determined from the table above. Fill in the table below with these values. {If you cannot construct the symbol , just use & or copy and paste} Lastly, draw a Venn diagram (on scratch paper, but you need not submit it) showing the events of Female, Public, and both Female and Public. The first one is completed for you. Event in words Event in set notation Probability Female and Public A B P(A B)=.38 c Female and Private A B P(A Bc)=.07 Public and Male AcB P(Ac B)=.46 Neither Female nor Public AcBc P(Ac Bc)=.09 c. Determine the probability that the student is Male. Do the same for student attending a Private high school. Use proper event notation remember that A is Female and B is Public. P(Male) = P( Ac ) = 0.55 P(Public) = P( Bc ) = 0.16 . d. If you had not been given the table, but instead had merely been told that P(A)=.45 and P(B)=.84, would you have been able to calculate P(Ac) and P(Bc)? Explain how. Yes we could compute the probability of the complement of an event using the probability of the event by subtracting the value of the probability of the event from one: P(Male) = P(Ac) = 1- P(A)=1-.45 = .55 P(Private) =P(Bc) =1- P(B) = 1-.84= .16 e. Calculate the probability that at least either the student is Female or the student attended a Public high school. Using the addition rule: P(A B) = P(A) + P(B) P(A B)=.45+.84-.38=.91 f. Another way to calculate P(A B) in part e is to use the complement of the event. First, in words what is the complement to the event described in e and then find this probability using the complement. Does this match your answer in e? The complement of the event {Female or Publick} is the event {neither Female nor Public}, or in other words {Male and Private}. In symbols we can describe this event as (A B)c which equals Ac Bc. Thus P(A B)= 1 - P(Ac Bc) = 1 - 0.09 = 0.91 g. If you had not been given the table but instead had merely been told P(A)=.45 and P(B)=.84 and were asked to calculate P(A B), you might first consider P(A)+P(B). Calculate this sum and compare the result to your answer for e. Are they the same? Is this even a legitimate answer? P(A)+P(B)=1.29 This value does not equal the value of P(A B) computed above. It is also not a legitimate since answer the probability be a number in the interval [0,1]. h. Given the knowledge that the student is Female, what is the conditional probability that the student attended a Public high school? [Hint: Restrict your consideration to Females, and ask yourself what fraction of those female students attended a Public high school?] This is asking to find P(B|A) = P(A B)/P(A) = 0.38/0.45 = 0.844 i. How does this conditional probability of the student attending a Public high school given that the student is Female compare with the unconditional probability of the student attending a Public high school? Does the knowledge that the student is Female make it more or less likely (or neither) that the student attended a Public high school? The conditional probability (0.844) is approximately equal to than the unconditional probability (0.84) that the student attended a Public high school. Since these are so close in probability, knowing that the student is Female does not improve the probability that the student attended a Public high school. [Note: later in the course we will discuss a procedure that tests whether a relationship exists between two categorical variables such as the ones used in this activity. This test will allow us to statistically show whether 0.844 is statistically different from 0.84, thus removing the personal judgment involved in saying are these close enough?] 2 Graduate School Admissions Suppose that you apply to two graduate schools A and B, and that you believe your probability of acceptance by A to be 0.7, your probability of acceptance by B to be 0.6, and your probability of acceptance by both to be 0.5. a. Are the events {acceptance by A} and {acceptance by B} independent? Explain. {Hint: look at the definition for independence from the lecture notes or in the Probability Rules table on page 223.} If two events are independent the probability that both events occur equals the product of their individual probabilities. That is, P(A B) = P(A)*P(B). In this case, P(A)*P(B) = 0.7*0.6 = 0.42, and P(A B) = 0.5. Since these are not equal, then A and B are not independent. b. Determine the conditional probability of acceptance by B given acceptance by A? How does it compare to the (unconditional) probability of acceptance by B? P(B|A) = P(A B)/P(A) = 0.5/0.7 = 0.71 c. What is the probability that you are accepted by at least one of the two schools? P(A B) = P(A) + P(B) - P(A B) = 0.7 + 0.6 0.5 = 0.8 3 Just for fun! Back in the Seventies there was a game show called Lets Make a Deal. The rules were simple: a contestant was selected from the crowd and offered the chance to open one of 3 doors. Behind one door was a prize (e.g. money, new car), while behind the other two doors were gag prizes (e.g. a donkey, barrel of popcorn). After making their choice of door the game show host would open one of the non-selected doors which was always one that did not contain the prize. The contestant was then asked if he or she would like to switch from their first choice to the remaining unopened door or stay with their original choice. If you were the contestant, what choice would you make? That is, given that you now have two doors unopened, one with a prize behind it and the other without, which gives you a better chance of winning: switching doors or staying with your original cho ice? After you have thought about this, please visit the following website and try it out! {You may have to click the refresh button after opening the page} http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html You should switch! The website provides a nice explanation as to why.

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Penn State - STAT - 200
Probability Distributionshttp:/onlinecourses.science.psu.edu/stat200/book/export/html/34Probability DistributionsIntroductionLearning objectives for this lesson Upon completion of this lesson, you should be able to: distinguish between discrete and co
Penn State - STAT - 200
Situation I: Find P(X x) 1. Convert x to z-score by z = (x mean)/SD 2. Now find cumulative probability P(Z z) using either the table or Minitab 3. This cumulative probability is the answerSituation II: Find P(X x) 1. Convert x to z-score by z = (x mean)/
Penn State - STAT - 200
1. Suppose that a student needs to buy 6 books for her history course. The number of books that she will be able to find used is a binomial random variable X with n = 6 and p = 0.30. What is the probability that she will find more than 2 used books? P(X>2
Penn State - STAT - 200
Use software for the following problem. 1. Suppose that a student needs to buy 6 books for her history course. The number of books that she will be able to find used is a binomial random variable X with n = 6 and p = 0.30. In other words, the probability
Penn State - STAT - 200
Probability Distributions - Solutions1 In each part, indicate, (1) whether the variable is discrete or continuous AND (2) whether it is binomial or not AND (3) if it is binomial, give values for n and p. a. Number of times a head is flipped in 10 flips o
Penn State - STAT - 200
1 part d: This was discrete but not binomial. There are two outcomes (win/lose), the probability of success is the same for each ticket (or trial) 1/10, and this is independent form one trial to the next (the probability of winning is same whether you won
Penn State - STAT - 200
1 Suppose that State College police department want to estimate the true proportion of all underage drinking among all PSU students. They have come to the statistics department for help with this study. To make the estimation, we randomly asked n = 400 PS
Penn State - STAT - 200
Sampling Distributionshttp:/onlinecourses.science.psu.edu/stat200/book/export/html/42Sampling DistributionsIntroductionLearning objectives for this lesson Upon completion of this lesson, you should be able to: determine the standard error for the samp
Penn State - STAT - 200
Sampling Distributions Solutions1 Suppose that medical researchers want to estimate the true proportion of all teenagers with high blood pressure whose blood pressure would decrease if they took calcium supplements. To test this, they plan a clinical tri
Penn State - STAT - 200
Confidence Intervalshttp:/onlinecourses.science.psu.edu/stat200/book/export/html/46Confidence IntervalsIntroductionLearning objectives for this lesson Upon completion of this lesson, you should be able to: Correctly interpret the meaning of confidence
Penn State - STAT - 200
Confidence Intervals Proportion and One Mean1 The term sampling frame refers to the group that actually had a chance to get into the sample. Ideally, this is the same as the population of interest, but sometimes it isnt. In the following situation, descr
Penn State - STAT - 200
3 6.Use the Standard Normal Table to find the following probability for X = Student Heights for which the mean is 66 inches and the standard deviation is 2 inches: P(X > 69) A) 0.9332 B) - 0.0668 < > C) 0.0668 D) 0.8531 E) 0.1469 F)-0.1469 A lternate Met
Penn State - STAT - 200
Unit Quiz 01https:/cms.psu.edu/Section/Assessment/Question/GradeDelivery.aspx?entryId=433E57EB0D76.Points Awarded Points Missed Percentage8.00 2.00 80.0%1. Decide if the sample is representative (or not) of the population for the question of interest.
Penn State - STAT - 200
Unit Quiz 01https:/cms.psu.edu/Section/Assessment/Question/GradeDelivery.aspx?entryId=433E57EB0D76.Points Awarded Points Missed Percentage8.00 2.00 80.0%1. Decide if the sample is representative (or not) of the population for the question of interest.
Penn State - STAT - 200
Unit Quiz 02https:/cms.psu.edu/Section/Assessment/Question/GradeDelivery.aspx?entryId=C2134DB024A.Points Awarded Points Missed Percentage17.00 1.00 94.4%1. Click the following link Minitab Data or Excel Data. From this data, use software to answer the
Penn State - STAT - 200
Unit Quiz 02https:/cms.psu.edu/Section/Assessment/Question/GradeDelivery.aspx?entryId=C2134DB024A.Points Awarded Points Missed Percentage16.00 2.00 88.9%1. Click the following link Minitab Data or Excel Data. From this data, use software to answer the
Penn State - STAT - 200
Unit Quiz 03https:/cms.psu.edu/Section/Assessment/Question/GradeDelivery.aspx?entryId=74F41B758B0.Points Awarded Points Missed Percentage12.00 3.00 80.0%1. Decide if the probability described is a subjective (personal) probability or a relative freque
Penn State - STAT - 200
Unit Quiz 03https:/cms.psu.edu/Section/Assessment/Question/GradeDelivery.aspx?entryId=74F41B758B0.Points Awarded Points Missed Percentage13.00 2.00 86.7%1. In the past five years, only 5% of pre-school children did not improve their swimming skills af
Penn State - STAT - 200
Unit Quiz 04https:/cms.psu.edu/Section/Assessment/Question/GradeDelivery.aspx?entryId=FA12CE6394B.Points Awarded Points Missed Percentage15.00 1.00 93.8%1. Suppose that a student needs to buy 10 books for her history course. The number of books that s
Penn State - STAT - 200
Unit Quiz 04https:/cms.psu.edu/Section/Assessment/Question/GradeDelivery.aspx?entryId=FA12CE6394B.Points Awarded Points Missed Percentage12.00 4.00 75.0%1. Suppose that a student needs to buy 6 books for her history course. The number of books that sh
Penn State - STAT - 200
Unit Quiz 05https:/cms.psu.edu/Section/Assessment/Question/GradeDelivery.aspx?entryId=A60882BCEC1.Points Awarded Points Missed Percentage10.00 1.00 90.9%1. When a random sample is to be taken from a population and a statistic is to be computed, the st
Penn State - STAT - 200
Unit Quiz 05https:/cms.psu.edu/Section/Assessment/Question/GradeDelivery.aspx?entryId=A60882BCEC1.Points Awarded Points Missed Percentage8.00 3.00 72.7%1. Which of the following statements is true about a parameter and a statistic for samples taken fr
Penn State - STAT - 200
Unit Quiz 06https:/cms.psu.edu/Section/Assessment/Question/GradeDelivery.aspx?en.Submitted by LUNBURG, ERIK (exl5087) on 10/7/2010 12:57:06 AM Points Awarded Points Missed Percentage 10.00 5.00 66.7%1. Which theorem or law is correct for supporting the
Penn State - STAT - 200
Unit Quiz 06https:/cms.psu.edu/Section/Assessment/Question/GradeDelivery.aspx?ent.Points Awarded Points Missed Percentage13.00 2.00 86.7%1. Which theorem or law is correct for supporting the notion that as sample size increases the distribution of the
Penn State - STAT - 200
Complement, Independence, Mutually Exclusive Complement and Mutually Exclusive Consider a course where, if completed, you can receive one of the following letter grades: A, B, C, D, or F and each has an equal probability of occurring. That is, P(A) read p
Penn State - STAT - 200
All confidence intervals follow the same formula: Sample Statistic +/- Multiplier*Standard Error For example in problem 7.25 on page 343 since we are talking about a 95% confidence interval to estimate a population mean the above formula would consist of:
Penn State - STAT - 462
Finding P-values For hypothesis testing for proportions were we calculate a Z test statistic (Ill call it Zstat) and for testing means we use a t test statistic (Ill call it tstat). Finding the p-value (or probability value) is based on the alternative hy
Penn State - STAT - 500
Probability Independence and Test of Two Categorical Variables The idea of independence between two categorical variables may sound familiar as we spoke of independence when we discussed probability. Recall that in that lesson we stated that two events, c
Penn State - STAT - 500
Variances, N vs N-1 (1995) Seal.<- file 95varqn.html -> . divide by N or N-1 for variance? Seal =David Seal, 06 Nov 1995=ssm, .From: dseal@armltd.co.uk (David Seal)Newsgroups: sci.math.num-analysis,sci.math,sci.stat.mathSubject: Re: What's Standard D
Penn State - STAT - 200
SOLUTIONS TO ACTIVITY SET 8Activity 8.1 Suppose the amount students at PSU spent on textbooks this semester is a normal random variable with mean = $360 and standard deviation = $90.a. Use the empirical rule for bell-shaped data to determine intervals t
Penn State - STAT - 462
Solution to Homework 1 The following data were collected for a class of 20 students on reading readiness stanines (X) at the end of kindergarten and reading achievement stanines (Y) at the end of first grade. You want to examine how well the readiness sco
Penn State - STAT - 462
Solution - Homework 2 Use the data from Homework 1 to complete this assignment and regress Y on X and store the residuals. 1. Create boxplots for both X and Y. Are there any outliers? No outliers identified. See boxplot below.Box pl ot of X, Y0 X 2 4 Y
Penn State - STAT - 462
Homework 3 - Solutions Covers Chapters 5 and 6 Matrix Methods 1. Using the data for Homework 1 and 2, write in proper matrix form, the matrices for Y, and X. 3 1 1 3 5 4 7 6 7 8 Y = 5 2 7 6 9 8 4 6 9 7 1 1 1 1 1 1 1 1 1 1 X = 1 1 1 1 1 1 1 1 1 1 2 2 1 1 3
Penn State - STAT - 462
Homework 4 Covers Chapters 7 Use the Project Talent data set. 1. Perform a multiple regression by regressing Math on Gender, SES, Sociability, Reading, and Mechanical Reasoning entering the predictors in this order. a. What is the interpretation of the t-
Penn State - STAT - 462
Homework 5 Covers Chapter 8 Use the Project Talent data set. The variable School Size is interpreted as follows: 1 = number of students is less than 100 2 = number of students is from 100 to 399 3 = number of students is 400 or more 1. A lack-of-fit test
Penn State - STAT - 462
Homework 6 Covers Chapters 9 and 10 Use the High School and Beyond (HSB) data set. The data is explained in the HSB Read Me file. USE MATH AS PREDICTOR 1. Some researchers feel an interaction exists between Gender and Writing ability. Create an Gender*Wri
Penn State - STAT - 500
STAT500 HW#2_solutionsSolutions to Homework 2 1) (fifth) a) A=cfw_6 n(A) / n(S) = 1/6 b) B=cfw_2,4,6 n(B) / n(S) = 3/6=1/2 c) C=cfw_3,4,5,6 n(C) / n(S) = 4/6=2/3 d) D=cfw_4,6 n(D) / (S)=2/6=1/3 (sixth) a) A=cfw_6 n(A) / n(S) = 1/6 b) B=cfw_1,3,5 n(B) / n
Penn State - STAT - 500
Stat 500 Homework 3 SolutionsSTAT500 HW#3 Solutions1) Random sample of 25 generated without replacement from Minitab (Will most likely differ from yours):2 240 125 409 783 526 219 364 584 789 84 341 296 652 134 708 104 155 21 562 378 522 86 667 4282)
Penn State - STAT - 500
STAT500 HW#4_solutionsSTAT500 HW#4 Solutions1) a) 95% T Confidence IntervalsVariable speed 10.304) N 20 Mean 9.100 StDev 2.573 SE Mean 0.575 95.0 % CI (7.896,b) The normal probability plot for reading speed suggests no reason to believe that the data
Penn State - STAT - 500
STAT500 HW#5_solutionsSTAT500 HW#5 Solutions1) Chicago Title Company problem T 0 = 0.831; because nT 0 = 2544*(0.831) = 2114.1 > 5, n*(1 - T 0) = 2544*(1 0.831) = 429.9 > 5, thus the one-proportion z-test can be used. Ho: = 0.831; Ha: 0.831 = 0.024; two
Penn State - STAT - 500
STAT500 HW#6_solutionsHomework 6 Solutions1. a. This poses an interesting question since we never discussed intuitive decisions prior to performing an analysis. So here is an intuitive thought process. Since one assumption is that the each population fo
Penn State - STAT - 500
Solution to Homework 1 The following data were collected for a class of 20 students on reading readiness stanines (X) at the end of kindergarten and reading achievement stanines (Y) at the end of first grade. You want to examine how well the readiness sco
Penn State - STAT - 500
Solution to Homework 1 The following data were collected for a class of 20 students on reading readiness stanines (X) at the end of kindergarten and reading achievement stanines (Y) at the end of first grade. You want to examine how well the readiness sco
Penn State - STAT - 200
Solutions Displaying Data2.3 Identify the variable type: a) quantitative b) categorical c) categorical d) quantitative 2.6 Discrete or continuous?: a) continuous b) discrete c) continuous d) discrete 2.8 Number of children:a) The variable, number of chi
Penn State - STAT - 200
Solutions Gathering Data 4.5 School testing for drugs : Although this study found similar levels of drug use in schools that used drug testing and schools that did not, lurking variables might have affected the results. For example, it is possible that sc
Penn State - STAT - 200
Solutions - Probability Distributions 6.3 Boston Red Sox hitting: a) The probabilities give a legitimate probability distribution because each one is between 0 and 1 and the sum of all of them is 1. b) = 0P(0) + 1P(1) + 2P(2) + 3P(3) + 4P(4) = 0(0.718) +
Penn State - STAT - 200
Solutions - Sampling Distributions 7.6 Exit poll and n: a) The interval of values within the sample proportion will almost certainly fall within three standard errors of the mean: 0.53 to 0.59. b) Based on the interval calculated in (a), it would be unusu
Penn State - STAT - 200
Solutions - Hypothesis Testing9.4 Iowa GPA: H 0 : = 2.80Ha :2.80In the above hypotheses, H 0 : is the notation for the null hypothesis, H a : is the notation for the alternative hypothesis, and is the parameter, the mean GPA of the population, about w
Penn State - STAT - 200
Solutions - Categorical Variables11.3 FBI statistics: a) These distributions refer to those of x at given categories of y. RACE OF VICTIM RACE OF MURDERER Blacks WhitesBlacks Whites91% 9%17% 83%b) x and y are dependent because the probability of a mu
Penn State - STAT - 200
Confidence Intervals Proportion and One Mean1 The term sampling frame refers to the group that actually had a chance to get into the sample. Ideally, this is the same as the population of interest, but sometimes it isnt. In the following situation, descr
Penn State - PSYCH - 412
Growth Spurt: Muscle Mass and Body Fat Rapid acceleration in height and weight; significant boost in muscle and body fat By the end of puberty, the muscle to fat ratio is: Boys = 3:1 Girls = 5:4 Boys gain muscle at a faster rate than girls Girls gain fat
Penn State - PSYCH - 412
Changes in Cognition Logical Thinking Transductive Thinking: connects two particular events into a cause-effect relationship simply because they occurred close in time Inductive Thinking: we make inferences about the world based on a limited set of experi
Penn State - PSYCH - 412
Brain Development neuron: single cell unit of the central nervous system nerve: single cell unit of the peripheral nervous system dendrites: top of the neuron; extensions, branches; gather incoming information to pass it along to be processed soma (or cel
Penn State - PSYCH - 412
Social Redefinition a period when the individual is being redefined by their society, from that of a child towards that of an adult (increased privileges and responsibilities) social redefinition tends to be more pronounced in traditional societies compar
Penn State - PSYCH - 412
The Generation Gap occurs when older and younger people fail to understand one another due to their different experiences, opinions, habits and overall behaviorParent-Adolescent Conflict G. Stanley Hall (1904) believed in storm and stress Anna Freud (194
Penn State - PSYCH - 412
Adolescents and their Peers high school students spend twice as much time with their peers as their parents for boys, time spent with family is replaced with time spent alone for girls, time spent with family is replaced with time with friendsOrigins of
Penn State - PSYCH - 238
What do we know when we know a person? What we know about someone depends on how well we know them. However, you can never truly know everything about someone even ones own spouse or child. We often develop first impressions about individuals based on the
Penn State - PSYCH - 238
1. Say that you want to create a self-judgment questionnaire to measure extroversion. You know from reading the Lesson Commentary that research by Dr. Johnson identified seven kinds of statements in self-judgment personality questionnaires. Which of the s
Penn State - PSYCH - 238
1. Name a personality trait that you think would be more validly measured with I-data than Sdata and explain why.Validity is the extent to which a measurement actually reflects or measures what you think it does. Likability is one personality trait that
Penn State - PSYCH - 238
L ist four personality t raits that you think describe you well. Make sure that one t rait is a behavioral t rait, one, an emotional t rait, one, a cognitive t rait, and one, a social imp ression. Label what kind of t rait each of your personality t raits
Penn State - PSYCH - 238
A r ebellious person is someone who defies or resists established r ules and a uthority. Write an item for an S-data personality test of rebelliousness and d escribe examples of L-data and B-data-not to be collected by self-judgment-that should be predict