Stats exam #2 chapters 3-6
Why is Bayes's rule unnecessary for finding P(B|A) if events A and B are independent?
If A and B are independent, then P(B|A)=P(B), which makes using Bayes' rule unnecessary.
How do discrete and continuous random variables differ
A discrete random variable can assume a countable number of values, while a continuous random variable can assume values corresponding to any of the points contained in an interval.
What is a random variable?
A random variable is a variable that assumes numerical values associated with the random outcomes of an experiment, where one (and only one) numerical value is assigned to each sample point.
What is an experiment?
An experiment is an act or process of observation that leads to a single outcome that cannot be predicted with certainty.
Describe the shape of the sampling distribution of x.
Approximately normal
Will the sampling distribution of x always be approximately normally distributed? Explain.
No, because the Central Limit Theorem states that the sampling distribution of x is approximately normally distributed only if the sample size is large enough
How do your answers to parts b and c change when the sample size is doubled? Choose the correct answer below.
P(2.84≤x≤2.86) decreases, P(x≥2.855) decreases
Give the formula for finding P(B|A).
P(B|A)=P(A∩B)P(A), assuming P(A)≠0
Define the sample space.
The sample space of an experiment is the collection of all its sample points.
What is a sampling distribution of a sample statistic?
The sampling distribution of a sample statistic is the probability distribution of that statistic.
Describe the shape of the sampling distribution of x. Does this answer depend on the sample size? Choose the correct answer below.
The shape is that of a normal distribution and depends on the sample size.
Define the union of two events
The union of two events A and B is the event that occurs if either A or B (or both) occurs on a single performance of an experiment.
It was reported by an agency that, under its standard inspection system, one in every 100 slaughtered chickens pass inspection with fecal contamination. a. If a slaughtered chicken is selected at random, what is the probability that it passes inspection with fecal contamination? b. The probability of part a was based on a study that found that 312 of 32,055 chicken carcasses passed inspection with fecal contamination. Do you agree with the agency's statement about the likelihood of a slaughtered chicken passing inspection with fecal contamination?
a) 1/100 = answer (1 divided by the sample size) b)calculate the probability that a chicken from the study was contaminated by dividing # of contained chickens in the study by the total number of chickens in the study (327/33,039= answer) --> Yes , because approximately one in every 100 chickens in the sample of 32,055 were contaminated.
Gene expression profiling is a state-of-the-art method for determining the biology of cells. In one study, biologists reviewed several gene expression profiling methods. The biologists applied two of the methods (A and B) to data collected on proteins in human mammary cells. The probability that the protein is cross-referenced (identified) by method A is 0.35, the probability that the protein is cross-referenced by method B is 0.37, and the probability that the protein is cross-referenced by both methods is 0.32. homework 3.1-3.6 question 8
a) D b) P(AuB)=P(A)+P(B)-−P(A∩B) c) 1- answer to part b
Suppose the events B1 and B2 are mutually exclusive and complementary events, such that P(B1)=0.31 and P(B2)=0.69. Consider another event A such that P(A|B1)=0.1 and P(A|B2)=0.5. Complete parts a through e below.
a) Find PB1∩A. b) Find PB2∩A. c) Find P(A) using the results in parts a and b. d) Find PB1|A. e)Find PB2|A. question 15 in homework 3.1-3.6
A fair coin is tossed two times, and the events A and B are defined as shown below. Complete parts a through d. A: {At least one head is observed} B: {The number of heads observed is odd} homework 3.1-3.6 question 6
a) Identify the sample points in the events A, B, A∪B, Ac, and A∩B. Identify the sample points in the event A. Choose the correct answer below. answer: A:TH, HT, HH Identify the sample points in the event B. Choose the correct answer below answer: B: TH,HT Identify the sample points in the event A∪B. Choose the correct answer below. answer: A∪B: TH, HT, HH Identify the sample points in the event Ac. Choose the correct answer below. answer: Ac: TT Identify the sample points in the event A∩B. Choose the correct answer below. answer: A∩B: TH, HT
The outcomes of two variables are (Low, Medium, High) and (On, Off), respectively. An experiment is conducted in which the outcomes of each of the two variables are observed. The probabilities associated with each of the six possible outcome pairs are given in the accompanying two-way table. Consider the events below. Complete parts a through h.
a) P(A)= P(high and on)+P(high and off) b) P(B)= P(high and on)+P(high and off)+P(medium and off)+P(low and off) c) P(C)= P(on and medium) d) P(D)= P(low and on)+P(low and off) e) P(Ac)= P(A)+P(Ac)=1 f) Find P(A∪B)= P(B) g) Find P(A∩C)= A and C have no sample points in common so answer is 0 h) two events are mutually exclusive if they have no sample points in common or if the probability of their intersection is 0 Find the probability of the intersection for each pair of events. If the probability is equal to 0, then they are mutually exclusive. P(A∩B)=0.10 P(B∩C)=0 P(A∩C)=0 P(B∩D)=0.25 P(A∩D)=0 P(C∩D)=0
For two events, A and B, P(A)=.5, P(B)=.4, and P(A∩B)=.1. a. Find P(A|B). b. Find P(B|A). c. Are A and B independent events?
a) P(A|B)= P(A∩B)/P(B) b) P(B|A)= (A∩B)/P(A) c) compare P(A|B) and P(A) or P(B|A) and P(B) to determine if events A and B are independent
A table classifying a sample of 120 patrons of a restaurant according to type of meal and their rating of the service is shown to the right. Suppose we select, at random, one of the 120 patrons. Complete parts a through c below
a) P(A|B)=P(A∩B)/P(B) --> P(good|dinner)= P(good∩dinner)/P(dinner) b) question 13 in 3.1-3.6
Suppose P(B)=0.5, P(D)=0.7, and P(B∩D)=0.4. Find the probabilities below. a. P(Dc) b. P(Bc) c.P(BuD)
a) P(D)+P(Dc)=1 b) P(B)+P(Bc)=1 c) P(B)+P(D)-P(B∩D)
Psychology students at a university completed the Dental Anxiety Scale questionnaire. Scores on the scale range from 0 (no anxiety) to 20 (extreme anxiety). The mean score was 14 and the standard deviation was 3.5. Assume that the distribution of all scores on the Dental Anxiety Scale is normal with μ=14 and σ=3.5. Complete parts a through c.
a) a score b) stat crunch c) stat crunch
The random variable x has the following discrete probability distribution. x 10 --> 0.3 11 --> 0.2 12 --> 0.2 13 --> 0.2 14 --> 0.1
a) add all the values less than and equal to 12 b) add the values greater than 12 c) add the values less than or equal to 14 d) the p(x) value of 14 e) homework question 6 on 4.1-4.3
Classify the random variables below according to whether they are discrete or continuous. a. The height of a randomly selected giraffe. b. The number of statistics students now reading a book. c. The number of textbook authors now sitting at a computer. d. The weight of a T-bone steak. e. The amount of rain in City B during April.
a) continuous b) discrete c) discrete d) continuous e) continuous
Suppose x is a random variable best described by a uniform probability distribution with c=35 and d=70. a. Find f(x). b. Find the mean and standard deviation of x. c. Graph f(x) and locate μ and the interval μ±2σ on the graph. Note that the probability that x assumes a value within the interval μ±2σ is equal to 1.
a) f(x)= 1/35 (35<x<70) b) to find mean t=do (c+d/2) to find standard deviation do (d-c/square root of 12) c) question 1 on homework 5.1-5.2 d)
The random variable x has the following discrete probability distribution. Complete parts a through d. x −5 -->.1 1 --> .2 6 --> .3 7 --> .2 9 --> .2
a) list the values that x may assume: -5, 1, 6, 7, 9 b) what value of x is most probable? 6 c) what is the probability that x is greater than -3? .9 d) what is the probability that x=3? 0
Toss four fair coins and let x equal the number of tails observed. a. Identify the sample points associated with this experiment, and assign a value of x to each sample point. Then list all the possible values of x. b. Calculate p(x) for the values x=1 and x=3. c. Construct a probability histogram for p(x). d. What is P(x=2 or x=4)?
a) possible values of x: 0, 1, 2, 3, 4 b) calculate p(x) for x=1 p(1)= 1/16+1/16+1/16+1/16=.25 calculate p(x) for x=3 p(3)=.25 c) homework problem 7 on 4.1-4.3 d) P(x=2 or x=4) = .4375
The probability distribution shown here describes a population of measurements that can assume values of 2, 3, 4, and 5, each of which occurs with the same relative frequency
a) take the average of the numbers in the sample column b) 1/16 c) complete the chart d) homework problem 2 on 6.1-6.3
Consider the probability distribution shown for the random variable x found below. Complete part a through f
a) u= x(px)+x(px)... b) variance calculator or... (x-u)2(px)+(x-u)2(px)+.... c) square root of ^ d)Interpret the value you obtained for μ. Choose the correct answer below. answer: The average value of x over many trials is equal to μ. e) In this case, can the random variable x ever assume the value μ? yes f) In general, can a random variable ever assume a value equal to its expected value? yes
Suppose x is a normally distributed random variable with μ=28 and σ=7. Find a value x0 of the random variable x.
a) z= x0-u/o
Suppose x is a random variable best described by a uniform probability distribution with c=10 and d=30. Complete parts a through f.
b-a/d-c a) problem 2 in 5.1-5.3
How would the sampling distribution of x change if the sample size n were doubled from 100 to 200? Choose the correct answer below.
doesnt change, decreases by a factor of square root of 2
Almost all companies utilize some type of year-end performance review for their employees. Human Resources (HR) at a university's Health Science Center provides guidelines for supervisors rating their subordinates. For example, raters are advised to examine their ratings for a tendency to be either too lenient or too harsh. According to HR, "if you have this tendency, consider using a normal distribution—10% of employees (rated) exemplary, 20% distinguished, 40% competent, 20% marginal, and 10% unacceptable." Suppose you are rating an employee's performance on a scale of 1 (lowest) to 100 (highest). Also, assume the ratings follow a normal distribution with a mean of 48 and a standard deviation of 14. Complete parts a and b.
homework 5.3 question 4
Suppose x is a random variable best described by a uniform probability distribution with c=4 and d=12. Complete parts a through f.
homework question 3 on 5.1-5.3
n a driver-side "star" scoring system for crash-testing new cars each crash-tested car is given a rating ranging from one star to five stars; the more stars in the rating, the better is the level of crash protection in a head-on collision. A summary of the driver-side star ratings for the 98 cars is reproduced in the table. Assume that 1 of the 98 cars is selected at random, and let x equal the number of stars in the car's driver-side star rating. Find μ=E(x) for this distribution and interpret the result practically.
interpret the results practically. Choose the correct answer below: Over a very large number of trials, the average driver-side crash rating is equal to μ.
An experiment results in one of five sample points with probabilities PE1=.21, PE2=.13, PE3=.26, P(E4)=.23, and P(E5)=.17. The events shown to the right have been defined.
question 12 on 3.1-3.6
The table to the right gives a breakdown of 2,172 civil cases that were appealed. The outcome of the appeal, as well as the type of trial (judge or jury), was determined for each case. Suppose one of the cases is selected at random and the outcome of the appeal and type of trial are observed
question 9 on 3.1-3.6
Suppose the random variable x is best described by a normal distribution with μ=33 and σ=5. Find the z-score that corresponds to each of the following x-values.
to find z score; x-u/o
Consider the population described by the probability distribution shown in the table. The random variable x is observed twice. Find E(x)=μ. Then use the sampling distribution of x to find the expected value of x.
u= x(px)+x(px)... expected value of x-bar using the sampling distribution is the same as the answer you got previously
Suppose a random sample of n=100 measurements is selected from a population with mean μ and standard deviation σ. For each of the following values of μ and σ, give the values of μx
ux= the same as u ox= o/10
the number of tickets, x, is sold for a concert is of interest to the box office manager. What values can x assume? Is x a discrete or continuous random variable
what values can x assume? x=0,1,2... is x a discrete or continuous random variable? discrete
