Stat: Homework_Chapter 5 (5.1-5.4)
A pollster agency wants to estimate the proportion of citizens of the European Union who support same-sex unions. She claims that if the sample size is large enough, she does not need to worry about the method of selecting the sample. Is the pollster's statement correct? Explain.
The statement is not correct. Samples should be chosen randomly to ensure that each individual in the population has about the same probability of being chosen.
You randomly sample 7 people in your school, and none of them is a vegetarian. Does this mean that the probability of being a vegetarian for students at your school equals 0? Explain.
No. In the short run, the proportion of a given outcome can fluctuate a lot. As more people are sampled, the proportion should approach the real probability.
At the local cell phone store, the probability that a customer who walks in will purchase a new cell phone is 0.23. The probability that the customer will purchase a new cell phone protective case is 0.33. Is this information sufficient to determine the probability that a customer will purchase a new cell phone and a new cell phone protective case? If so, find the probability. If not, explain why not. (Hint: Considering the buying practices of consumers in this context, is it reasonable to assume these events are independent?)
The information is not sufficient. The events might not be independent.
A couple plans to have four children. The father notes that the sample space for the number of girls the couple can have is 0, 1, 2, 3, and 4. He goes on to say that since there are five outcomes in the sample space, and since each child is equally likely to be a boy or girl, all five outcomes must be equally likely. Therefore, the probability of all four children being girls is 1/5. Explain the flaw in his reasoning.
The outcomes are not equally likely.
A 2014 article from a business magazine discusses recidivism rates in the United States. Recidivism is defined as being reincarcerated within five years of being sent to jail initially. Among the data reported, the magazine cites that the recidivism rate for blacks is 81% compared to 73.4% among whites. Using notation, express each of these as a conditional probability.
The recidivism rate (R) for blacks (B) is 81% Express this as a conditional probability. Select the correct choice below and fill in the answer box to complete your choice. - P(R|B)=0.81 The recidivism rate (R) for whites (W) is 73.4% Express this as a conditional probability. Select the correct choice below and fill in the answer box to complete your choice. - P(R|W)=0.734
You are asked to use your best judgment to estimate the probability that there will be a nuclear war within the next 10 years. Is this an example of the relative frequency or the subjective definition of probability? Explain.
This is an example of the subjective definition because there are no previous trials in which a nuclear war occurred, so the long run probability cannot be assessed.
The accompanying table shows audit status and income of the 149,890 thousand tax returns filed in 2018. The frequencies in the table are reported in thousands. Complete parts a through c below.
a. Find the probability that a randomly selected return is in the middle income category and is not audited. - The probability is 0.0453 b. Given that a return is in the middle income category, what is the probability that it was not audited? - The probability is 0.9895 c. Given that a return is not audited, what is the probability that it was in the middle income category? - The probability is 0.0456
A survey asked subjects whether they are a member of an environmental group and whether they would be very willing to pay higher prices to protect the environment. The results are shown in the table below. For a randomly selected adult, find the probabilities specified in parts (a) through (d). Pay Higher Prices (GRNPRICE) Yes No Total Environmental Group Member Yes 31 66 97 (GRNGROUP) No 85 930 1015 Total 116 996 1112
a. Estimate the probability of being a member of an environmental group. - P(GRNGROUP)=0.087 Estimate the probability of being willing to pay higher prices to protect the environment. - P(GRNPRICE)=0.104 b. Estimate the probability of being both a member of an environmental group and very willing to pay higher prices to protect the environment. - P(GRNGROUP and GRNPRICE)=0.028 c. Given the probabilities in (a), calculate the probability in (b) if the variables were independent. - P(GRNGROUP and GRNPRICE)=0.009 d. Estimate the probability that a person is a member of an environmental group or very willing to pay higher prices to protect the environment. Do this directly using the counts in the table. - P(GRNGROUP or GRNPRICE) = 0.164 Estimate the probability that a person is a member of an environmental group or very willing to pay higher prices to protect the environment. Do this by applying the appropriate probability rule to the estimated probabilities found in (a) and (b). - P(GRNGROUP or GRNPRICE)=0.164
Part of a student opinion poll at a university asks students what they think of the quality of the existing liberal arts building on the campus. The possible responses were great, good, fair, and poor. Another part of the poll asked students how they feel about a proposed fee increase to help fund the cost of building a new liberal arts building. The possible responses to this question were in favor (I), opposed (O), and no opinion (N). a. List all potential outcomes in the sample space for someone who is responding to both questions. b. Show how a tree diagram can be used to display the outcomes listed in part a.
a. Choose the correct answer below. - Great and in favor, great and opposed, great and no opinion, good and in favor, good and opposed, good and no opinion, fair and in favor, fair and opposed, fair and no opinion, poor and in favor, poor and opposed, poor and no opinion b. Choose the correct answer below.
A couple plans on having three children. Suppose that the probability of any given child being female is 0.5, and also suppose that the genders of each child are independent events. a. Write out all outcomes in the sample space for the genders of the three children. b. What should be the probability associated with each outcome? c. Using the sample space constructed in part a, find the probability that the couple will have three girls d. Using the sample space constructed in part a, find the probability that the couple will have at least one child of each gender
a. Choose the correct answer below. - {BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG} b. The probability associated with each outcome should be 0.125. c. The probability that the couple will have three girls is 0.125. d. The probability that the couple will have at least one child of each gender is 0.75.
A standard card deck has 52 cards consisting of 26 black and 26 red cards. Three cards are dealt from a shuffled deck, without replacement. Complete parts a through c below.
a. Determine whether the following statement is true or false. The probability of being dealt three black cards is 1/2 x 1/2 x 1/2 = 1/8 If true, explain why. If false, show how to get the correct probability. - False, the probability of being dealt three black cards is 26/52 x 25/51 x 24/50 = 2/17 b. Let A=first card red and B=second card red. Are A and B independent? Explain why or why not. - A and B are not independent since being dealt a red card from the deck changes the number of red cards in the deck and the total number of cards in the deck. c. Given that the cards are replaced after being dealt determine whether the following statement is true or false. The probability of being dealt three black cards is 1/2 x 1/2 x 1/2 = 1/8 If true, explain why. If false, show how to get the correct probability. - True, since the cards are being replaced, half the cards are always black so the probability of being dealt a black card is always one half. Thus, the probability of being dealt three black cards is 1/2 x 1/2 x 1/2 = 1/8 Given that the cards are replaced after being dealt, let A=first card red and B=second card red. Are A and B independent? Explain why or why not. - A and B are independent since being dealt a red card from the deck does not change the number of red cards in the deck or the total number of cards in the deck.
Each customer of a newspaper can sign up for weekday delivery and/or weekend delivery. The customer's file records whether he or she receives each delivery as Y for yes and N for no. The probabilities of the customer receiving the newspaper are shown below. Complete parts a through d. Outcome (Weekday, Weekend) YY YN NY NN Probability 0.32 0.11 0.06 0.51
a. Display the outcomes in a contingency table, using the rows as the weekend event and the columns as the weekday event. - Weekday yes no Weekend Yes 0.32 0.06 No 0.11 0.51 b. Let W denote the event that the customer bought a newspaper during the week and S be the event that he or she got it on the weekend (S for Saturday/Sunday). Find P(W) and P(S). - P(W)=0.43 P(S)=0.38 c. Explain what the event "W and S" means and find P(W and S). - The event means the customer got a newspaper on both weekdays and the weekend. P(W and S)=0.32 d. Are W and S independent events? Explain why you would not normally expect customer choices to be independent. - The events are not independent because P(W and S) not equal P(W)P(S) Why you would not normally expect customer choices to be independent? - It makes sense because people who buy the newspaper during the week probably like to read the paper. Therefore, they are more likely to buy the paper on the weekend.
A Statistical Abstract of the United States provides information on individuals' self-described religious affiliations. The information for 2008 is summarized in the accompanying table (all numbers are in thousands). Complete parts a through c below.
a. Find the probability that a randomly selected individual is identified as a Christian. - The probability is 0.7600. b. Given that an individual identifies as Christian, find the probability that the person is catholic - The probability is 0.33 c. Given that an individual answered, find the probability the individual is identified as other non christian - The probability is 0.0166
5093 pregnant women were tested to see if their babies had Down syndrome (D). The results of the test are given in the table. Use the values in the table to answer parts a-c.
a. Given that a test result is negative (NEG), what is the probability that the fetus actually has Down syndrome? - P(D|NEG)=0.0019 b. Given that the fetus has Down syndrome, what is the probability that the test result is negative? -P(NEG|D)=0.1400 c. Is P(D|NEG) equal to P(NEG|D)? - No, because P(NEG|D) deals with a much smaller pool of fetuses than P(D|NEG).
A teacher announces a pop quiz for which the student is completely unprepared. The quiz consists of 20 true-false questions. The student has no choice but to guess the answer randomly for all 20 questions. The accompanying table gives the 20 correct answers, which were actually randomly generated. Complete parts a. and b. below.
a. How many questions can the student expect to answer correctly simply by guessing? - 10 b. What percentage of answers were true, and what percentage would you expect? Why are they not necessarily identical? - The percentage of answers that were true was 55%. The expected percentage of answers that would be true is 50%. Why are these answers not necessarily identical? - With random phenomena, the proportion of times that something happens is highly random and variable in the short run.
There are four major types of blood: A, B, AB, and O. Suppose 38%, of a certain population has Type A, 11% Type B, 4% Type AB, and 47% Type O blood. The blood type of a randomly selected person from this population is determined as part of an experiment. Complete parts a through c.
a. List all the potential outcomes in the sample space. - {A, B, AB, O} b. List the probabilities for each outcome in the sample space. Are they equally likely? - Each outcome is not equally likely; P(A)=0.38, P(B)=0.11, P(AB)=0.04, and P(O)=0.47 c. Check that the probabilities for the outcomes in the sample space are (i) between 0 and 1 and (ii) sum to 1. - Yes, because each of the probabilities are between 0 and 1. (ii) Are the probabilities for the outcomes in the sample space sum to 1? - Yes, because P(A)+P(B)+P(AB)+P(O)=1.
A teacher gives an unannounced, three-question true-false pop quiz, with two possible answers to each question. Complete parts (a)-(c) below.
a. Use a tree diagram to show the possible response patterns, in terms of whether any given response is correct or wrong. Choose the correct answer below. How many outcomes are in the sample space? - There are 8 outcomes in the sample space. b. An unprepared student guesses all the answers randomly. Find the probability the student's first and third answers are correct and his second answer is wrong. - The probability of the student's answers being CWC is 0.125. c. Refer to (b). Using the tree diagram, evaluate the probability of passing the quiz, which the teacher defines as answering no more than one question wrong. - The probability of passing the quiz by guessing randomly is 0.5.
Over a recent month, 29,453 flights took off from a certain airport. Of those, 8,525 had a departure delay of more than 5 minutes. Complete parts a through e below.
a. What is the random phenomenon of interest in the context of this example? - Whether or not a flight is delayed by more than 5 minutes b. What is a trial in the context of this example? - An individual flight c. Are the 29,453 flights a long run or short run of trials? Explain. - This can be considered a long run because the number of trials is sufficiently large in the context of departures from an airport. d. Estimate the probability of a flight being delayed by more than 5 minutes, assuming flight delays are independent of each other. - The probability of a flight being delayed by more than 5 minutes is approximately 0.289. e. Write down the definition of independence of trials in the context of this example. Might the trials be independent? - Independent trials would be where whether or not one flight is delayed by more than 5 minutes is not affected by whether any other flight is or is not delayed by more than 5 minutes. Might the trials be independent? - They would not be if, for instance, there is an overall weather pattern that delays several consecutive flight
Consider a random number generator designed for equally likely outcomes. For parts a through d below, explain why the statement is true or false.
a. When generating a random number between 1 and 8 inclusive, each number has a probability of 0.125 of being selected. Choose the correct answer below - This statement is true because this is the definition of equally likely b. When generating random numbers between 1 and 8 inclusive, each number between 1 and 8 must occur exactly twice - This statement is false because it is possible that some numbers between 1 and 8 inclusive, do not appear at all, while others appear more often than twice c. When generating a random number between 1 and 8 inclusive, the proportion of times that a 7 is generated tends to get close to 0.125 as the number of random numbers generated gets larger and larger. Choose the correct answer below. - this statement is true because this is the law of large numbers d. When generating a large set of random numbers between 1 and 8 inclusive, each number between 1 and 8 inclusive, occurs close to 12.5% of the time in this set. Choose the correct answer below. - This statement is true because this is the law of large numbers.
Your friend decides to flip a coin repeatedly to analyze whether the probability of a head-on each flip is 1/2 .He flips the coin 10 times and observes a head 7 times. He concludes that the probability of a head for this coin is 7/10 = 0.70
a. Your friend claims that the coin is not balanced, since the probability is not 0.50. What's wrong with your friend's claim? - In the short run, the proportion of a given outcome can fluctuate a lot. A long run of observations is needed to accurately calculate the probability of flipping heads b. If the probability of flipping a head is actually 1/2, what would you have to do to ensure that the cumulative proportion of heads falls very close to 1/2. - Flip the coin many times to obtain a long run of observations.
According to an article in The New Yorker (March 12, 2007), the Department of Homeland Security in the United States is experimenting with installing devices for detecting radiation at bridges, tunnels, roadways, and waterways leading into Manhattan. The New York Police Department (NYPD) has expressed concerns that the system would generate too many false alarms. Complete parts a through c below.
false alarms the NYPD fears. Complete the contingency table below. Radioactive Material Detected by Device Yes No Yes a b No. c d The cell that contains b corresponds to false alarms. c. For the diagram you sketched in part b, explain why P(A|B)=1, but P(B|A)<1. - Since A contains B, P(A|B)=1. Since B is a subset of A, P(B|A)<1