STA2023 STATISTICS MODULE 2 TEST(Units 4 & 5) - Test#2
Freshman, Sophomore, Junior, Senior Total Satisfied 50, 47, 62, 59 218 Neutral 30, 10, 16, 10 66 Not satisfied 22, 19, 11, 20 72 Total 102, 76, 89, 89 356
(a) If a survey participant is selected at random, what is the probability that he or she is satisfied with student government? P(satisfied)=0.612 218/356 = 0.6123 (b) If a survey participant is selected at random, what is the probability that he or she is a junior? P(junior)=0.25 89/356 = 0.25 (c) If a survey participant is selected at random, what is the probability that he or she is satisfied and a junior? P(satisfied and junior)=0.174 62/356 = 0.1742 (d) If a survey participant is selected at random, what is the probability that he or she is satisfied or a junior? P(satisfied or junior)=0.688 P(E or F)=P(E)+P(F)−P(E and F). 0.612 + 0.25 - 0.174 = 0.688
Find the probability that when a couple has three children, at least one of them is a girl. (Assume that boys and girls are equally likely.)
1 - (1/2)^3 = 1 - 0.125 = 0.875 The probability is 0.875 that at least one of the three children is a girl.
Assuming boys and girls are equally likely, find the probability of a couple having a baby boy when their third child is born, given that the first two children were both boys.
1/2 = 0.5 The probability is 0.5
In an ice hockey game, a tie at the end of one overtime period leads to a "shootout" with three shots taken by each team from the penalty mark. Each shot must be taken by a different player. How many ways can 3 players be selected from the 5 eligible players? For the 3 selected players, how many ways can they be designated as first, second, and third?
3 players can be selected from the 5 eligible players in 10 (5 nCr 3) different ways. Out of those 3 players that areselected, they can be designated as first, second, and third in 6 (3 nPr 3) different ways.
Purchased Gum, Kept the Money Students Given Four Quarters 34 , 12 Students Given a $1 Bill 13 , 32 a. Find the probability of randomly selecting a student who spent the money, given that the student was given four quarters. b. Find the probability of randomly selecting a student who kept the money, given that the student was given four quarters.
34/(34+12) = 34 + 46 = 0.7391 a. The probability is 0.740 1-0.740 = 0.260 b. The probability is 0.260 c. A student given four quarters is more likely to have spent the money.
A statistics class consists of 5 males and 7 females. Two of those students are randomly selected with replacement. What is the probability that both of the selected students are females?
5+7 = 12 PA = 7/12
An investment counselor calls with a hot stock tip. He believes that if the economy remains strong, the investment will result in a profit of $60,000. If the economy grows at a moderate pace, the investment will result in a profit of $10,000. However, if the economy goes into recession, the investment will result in a loss of $60,000. You contact an economist who believes there is a 20% probability the economy will remain strong, a 70% probability the economy will grow at a moderate pace, and a 10% probability the economy will slip into recession. What is the expected profit from this investment?
60,000(0.2) + 10,000(0.7) + (-60,000*0.1) = 13000 The expected profit is $13000
Assume that 700 births are randomly selected and 347 of the births are girls. Use subjective judgment to describe the number of girls as significantly high, significantly low, or neither significantly low nor significantly high.
700/2 = 350 = mean The number of girls is neither significantly low nor significantly high.
Probability
A. If P(A)=0, then the probability of the complement of A is 1 B.If the probability of an event occurring is 0, then it is impossible for that event to occur. C. Probability can never be a negative value. D. The probability of an impossible event is 0. E. The probability of an event that is certain to occur is 1. F. The probability of any event is between 0 and 1 inclusive.
In horse racing, a trifecta is a bet that the first three finishers in a race are selected, and they are selected in the correct order. Does a trifecta involve combinations or permutations? Explain.
Because the order of the first three finishers does make a difference, the trifecta involves permutations.
Which word is associated with multiplication when computing probabilities?
And
Assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.75 probability that a pea has green pods. Assume that the offspring peas are randomly selected in groups of 30.
Complete parts (a) through (c) below. a. The value of the mean is μ = 22.5 peas. (30*0.75 = 22.5) The value of the standard deviation is σ = 2.4 peas. (square root of (30)(0.75)(0.25))=2.4 b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high. Values of 17.7 peas or fewer are significantly low. (22.5-2(2.4))=17.7) = (mean - 2* standard deviation) Values of 27.3 peas or greater are significantly high. (22.5+2(2.4))=27.7) = (mean + 2* standard deviation)
Suppose a life insurance company sells a $180,000 one-year term life insurance policy to a 21-year-old female for $190. The probability that the female survives the year is 0.999549.
Compute and interpret the expected value of this policy to the insurance company. The expected value is $108.82 190 - ( 1 - 0.999549 )* 180,000 = 108.82
A statistics class consists of 8 males and 12 females. Two of those students are randomly selected without replacement. Let A be the event of getting a male on the first selection, and let B be the event of getting a female on the second selection. Which of the following is true?
Events A and B are dependent.
Let A be the event of rolling a die and getting 6, and let B be the event of tossing a coin and getting heads. Which one of the following is true?
Events A and B are independent.
x , P(x) 0 , 0.026 1 , 0.153 2 , 0.321 3 , 0.321 4 , 0.153 5 , 0.026
Find the mean (μ) of the random variable x. [x * P(x) = μ] 0*0.026 + 1*0.153 +2*0.321 + 3*0.321 + 4*0.153 + 5*0.026 = 2.5 Find the standard deviation of the random variable x. [square root of (x-μ)^2 * P(x)] squareroot of [(0-2.5)^2*0.026 + (1-2.5)^2*0.153 +(2-2.5)^2*0.321 + (3-2.5)^2*0.321 + (4-2.5)^2*0.153 + (5-2.5)^2*0.026] = 1.08..
The peak shopping time at a home improvement store is between 8:00 a.m. and 11:00 a.m. on Saturday mornings. Management at the store randomly selected 150 customers last Saturday morning and decided to observe their shopping habits. They recorded the number of items that each of the customers purchased as well as the total time the customers spent in the store.
Identify the types of variables recorded by the home improvement store: number of items, discrete; total time, continuous
Confusion of the inverse occurs when we incorrectly believe _______.
P(B|A)=P(A|B)
What are the two requirements for a discrete probability distribution?
P(x) = 1 0 <= P(x) <= 1
In a state pick 4 lottery game, a bettor selects four numbers between 0 and 9 and any selected number can be used more than once. Winning the top prize requires that the selected numbers match those and are drawn in the same order. Do the calculations for this lottery involve the combinations rule or either of the two permutations rules? Why or why not? If not, what rule does apply?
The combination and permutations rules do not apply because repetition is allowed and numbers are selected with replacement. The multiplication counting rule applies to this problem.
A thief steals an ATM card and must randomly guess the correct six-digit pin code from a 5-key keypad. Repetition of digits is allowed. What is the probability of a correct guess on the first try?
The number of possible codes is 15625 (5^6) The probability that the correct code is given on the first try is 1/15625
Seventeen of the 100 digital video recorders(DVRs) in an inventory are known to be defective. What is the probability that a randomly selected item is defective?
The probability is 0.17 17/100 = 0.17
In a certain test, there is a multiple choice question with the possible answers a, b, c, d, and e. By randomly guessing, the chances of getting the correct answer are stated as "1 in 5." Express the indicated degree of likelihood as a probability value between 0 and 1 inclusive.
The probability is 0.2 1/5 = 0.2
In a certain country, the true probability of a baby being a girl is 0.476. Among the next five randomly selected births in the country, what is the probability that at least one of them is a boy?
The probability is 0.976 (1-(0.476^5) = 0.976)
What does P(B|A) represent?
The probability of event B occurring after it is assumed that event A has already occurred
In a test of a sex-selection technique, results consisted of 213 female babies and 11 male babies. Based on this result, what is the probability of a female being born to a couple using this technique? Does it appear that the technique is effective in increasing the likelihood that a baby will be a female?
The probability that a female will be born using this technique is approximately 0.951 213/(213+11) = 0/95089 Yes
Drove When Drinking Alcohol? Yes , No Texted While Driving 705 , 3059 No Texting While Driving 187 , 4543 If four different high school drivers are randomly selected, find the probability that they all drove when drinking alcohol.
The probability that four randomly selected high school drivers all drove when drinking alcohol is 0.000122 [(705 + 187)/(705 + 187 + 3059 + 4543)]^4 = 0.000122
A research center poll showed that 79% of people believe that it is morally wrong to not report all income on tax returns. What is the probability that someone does not have this belief?
The probability that someone does not believe that it is morally wrong to not report all income on tax returns is 0.21 79/100 = 0/79 1-0.79 = 0.21
Let event A=subjectis telling the truth and event B=polygraph test indicates that the subject is lying. Use your own words to translate the notation P(B|A) into a verbal statement.
The probability that the polygraph indicates lying given that the subject is actually telling the truth.
The accompanying table lists probabilities for the corresponding numbers of unlicensed software packages. What is the random variable, what are its possible values, and are its values numerical? Number of unlicensed software packages x, P(x) 0 , 0.125 1 , 0.375 2 , 0.375 3 , 0.125
The random variable is x, which is the number of unlicensed software packages. The possible values of x are 0, 1, 2, and 3. The values of the random value x are numerical.
Based on a survey, assume that 32% of consumers are comfortable having drones deliver their purchases. Suppose that we want to find the probability that when six consumers are randomly selected, exactly two of them are comfortable with delivery by drones. Identify the values of n, x, p, and q.
The value of n is 6 The value of x is 2 The value of p is 0.32 The value of q is 0.68 (p + q = 1)
Rewrite the following statement so that the likelihood of rain is expressed as a value between 0 and 1. A weather forecasting website indicated that there was a 15% chance of rain in a certain region.
There is a 0.15 probability that it will rain somewhere in the region at some point during the day.
A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 46 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. If one shipment of 6000 aspirin tablets actually has a 4% rate of defects, what is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected?
Use stat crunch n = 46 p = 0.04 p (x <= 3) = 0.4460 The probability that this whole shipment will be accepted is 0.4460 The company will accept 44.6% of the shipments and will reject 55.4% of the shipments, so many of the shipments will be rejected.
Assume that when human resource managers are randomly selected, 57% say job applicants should follow up within two weeks. If 5 human resource managers are randomly selected, find the probability that exactly 3 of them say job applicants should follow up within two weeks.
Use stat crunch n = 5 p = 0.57 p (x = 3) = 0.34242186
Assume that when human resource managers are randomly selected, 58% say job applicants should follow up within two weeks. If 5 human resource managers are randomly selected, find the probability that at least 3 of them say job applicants should follow up within two weeks.
Use stat crunch n = 5 p = 0.58 p (x >= 3) = 0.64745966
40.3% of consumers believe that cash will be obsolete in the next 20 years. Assume that 6 consumers are randomly selected. Find the probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years.
Use stat crunch n = 6 p = 0.403 p (x < 3) = 0.53809939
Let A denote the event of placing a $1 straight bet on a certain lottery and winning. Suppose that, for this particular lottery, there are 4,032 different ways that you can select the four digits (with repetition allowed) in this lottery, and only one of those four-digit numbers will be the winner.
What is the value of P(A)? P(A)=0.00025 1/4032 = 0.000248 What is the value of P(-A)? P(-A) = 0.99975 1 - 0.00025 = 0.99975
Drive-thru Restaurant A , B , C , D Order Accurate 340 , 269 , 244 , 145 Order Not Accurate 30 , 50 , 37 , 14 If two orders are selected, find the probability that they are both from Restaurant D.
a. Assume that the selections are made with replacement. Are the events independent? The probability of getting two orders from Restaurant D is 0.0198. (145+14) / (340 + 269 + 244 + 145 + 30 + 50 + 37 + 14) = (159/1131)(159/1131) = 0.01976 The events are independent because choosing the first order does not affect the probability of the choice of the second order. b. Assume that the selections are made without replacement. Are the events independent? The probability of getting two orders from Restaurant D is 0.0197 (159/1131)(158/1130) The events are not independent because choosing the first order affects the probability of the choice of the second order.
Requirements of the Permutations Rule, nPr=n!(n−r)!, for items that are all different?
a. Exactly r of the n items are selected (without replacement). b. There are n different items available. c. Order is taken into account (rearrangements of the same items are considered to be different).
Requirements of the Combinations Rule, nCr=n!r!(n−r)!, for items that are all different?
a. Exactly r of the n items are selected (without replacement). b. There are n different items available. c. That order is not taken into account (consider rearrangements of the same items to be the same).
A corporation must appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO). It must also appoint a planning committee with five different members. There are 14 qualified candidates, and officers can also serve on the committee. Complete parts (a) through (c) below.
a. How many different ways can the officers be appointed? There are 24024 different ways to appoint the officers. (14 nPr 4 = 24024) b. How many different ways can the committee be appointed? There are 2002 different ways to appoint the committee. (14 nCr 5 = 2002) c. What is the probability of randomly selecting the committee members and getting the five youngest of the qualified candidates? P(getting the five youngest of the qualified candidates)= 1/2002
Drive-thru Restaurant A , B , C , D Order Accurate 336 , 264 , 238 , 136 Order Not Accurate 31 , 58 , 40 , 12 a. If one order is selected, find the probability of getting an order from Restaurant A or an order that is accurate. b. Are the events of selecting an order from Restaurant A and selecting an accurate order disjoint events?
a. The probability of getting an order from Restaurant A or an order that is accurate is 0.901 (336 + 264 + 238 + 136 + 31 ) / (336 + 264 + 238 + 136 + 31 + 58 + 40 + 12) = 0.9013 b. The events are not disjoint because it is possible to receive an accurate order from Restaurant A.
a. Find the probability that when a single six-sided die is rolled, the outcome is 2. b. Find the probability that when a coin is tossed, the result is tails. c. Find the probability that when a six-sided die is rolled, the outcome is 17.
a. The probability of rolling a 2 on a six-sided die is 0.167 1 / 6 = 1.666667 b. The probability of a result of tails when tossing a coin is 0.5 1 / 2 = 0.5 c. The probability of rolling a 17 on a six-sided die is 0
Based on a poll, 67% of Internet users are more careful about personal information when using a public Wi-Fi hotspot. a. What is the probability that among four randomly selected Internet users, at least one is more careful about personal information when using a public Wi-Fi hotspot? b. How is the result affected by the additional information that the survey subjects volunteered to respond?
a. The probability that at least one of them is careful about personal information is 0.988 1 - (0.33^4) = 0.988 b. It is very possible that the result is not valid because the sample may not be representative of the people who use public Wi-Fi.
Use the following results from a test for marijuana use, which is provided by a certain drug testing company. Among 142 subjects with positive test results, there are 25 false positive results. Among 153 negative results, there are 3 false negative results. Complete parts (a) through (c) a. How many subjects were included in the study? b. How many subjects did not use marijuana? c. What is the probability that a randomly selected subject did not use marijuana?
a. The total number of subjects in the study was 295 142+153 = 295 b. A total of 175 subjects did not use marijuana. 153 + 25 - 3 = 175 c. The probability that a randomly selected subject did not use marijuana is 0.593 175 / 295 = 0.59322
There are 63 members on the board of directors for a certain non-profit institution. a. If they must elect a chairperson, first vice chairperson, second vice chairperson, and secretary, how many different slates of candidates are possible? b. If they must form an ethics subcommittee of four members, how many different subcommittees are possible?
a. There are 14295960 different slates of candidates possible. 63 nPr 4 = 14295960 b. There are 595665 different ethics subcommittees possible. 63 nCr 4 = 595665
The principle of redundancy is used when system reliability is improved through redundant or backup components. Assume that a student's alarm clock has a 6.9% daily failure rate. Complete parts (a) through (d) below.
a. What is the probability that the student's alarm clock will not work on the morning of an important final exam? 0.069 6.9 / 100 = 0.069 b. If the student has two such alarm clocks, what is the probability that they both fail on the morning of an important final exam? 0.00476 0.069^2 = 0.004761 c. What is the probability of not being awakened if the student uses three independent alarm clocks? 0.00033 0.069^3 = 0.000328 d. Do the second and third alarm clocks result in greatly improved reliability? Yes, because total malfunction would not be impossible, but it would be unlikely.
Is the following a discrete random variable, a continuous random variable, or not a random variable: a. number of textbook authors now sitting at a computer b. number of people in a restaurant that has a capacity of 100 c. the gender of college students d. amount of rain in City B during April e. number of points scored during a basketball game f. time it takes for a light bulb to burn out
a. discrete random variable b. discrete random variable c. not a random variable d. continuous random variable e. discrete random variable f. continuous random variable
Does the table show a probability distribution?
a. the sum of all the probabilities is equal to 1. b. every probability is between 0 and 1 inclusive. c. the random variable x's number values are associated with probabilities. d. the random variable x is numerical instead of categorical.
When using the _______ always be careful to avoid double-counting outcomes.
addition rule
The conditional probability of B given A can be found by _______.
assuming that event A has occurred, and then calculating the probability that event B will occur
A _______ probability of an event is a probability obtained with knowledge that some other event has already occurred.
conditional
Events that are _______ cannot occur at the same time.
disjoint
The classical approach to probability requires that the outcomes are _______.
equally likely
A discrete random variable
has either a finite or a countable number of values.
A continuous random variable
has infinitely many values associated with measurements.
Selections made with replacement are considered to be _______.
independent
Two events A and B are _______ if the occurrence of one does not affect the probability of the occurrence of the other.
independent
A random variable
is a variable that has a single numerical value, determined by chance, for each outcome of a procedure.
The expression for calculating the mean of a binomial distribution.
mean = np
Assume that random guesses are made for 4 multiple-choice questions on a test with 2 choices for each question, so that there are n=4 trials, each with probability of success (correct) given by p=0.50. Find the probability of no correct answers.
n = 4, x = 4, p = 0.5 The probability of no correct answers is 0.0620.062.
The complement of "at least one" is _______.
none
The expected value
of a discrete random variable represents the mean value of the outcomes.
"At least one" is equivalent to _______.
one or more
If the order of the items selected matters, then we have a
permutation problem.
P(-A)+PA= 1 is one way to express the_______.
rule of complementary events.
The _______ for a procedure consists of all possible simple events or all outcomes that cannot be broken down any further.
sample space
For the binomial distribution, which formula finds the standard deviation?
square root of npq
P(A or B) indicates_______.
the probability that in a single trial, event A occurs, event B occurs, or they both occur.
Multiple-choice questions each have five possible answers (a, b, c, d, e), one of which is correct. Assume that you guess the answers to three such questions.
P(CWW) = (1/5)(4/5)(4/5) = 0.128 what is the probability of getting exactly one correct answer when three guesses are made? P(CWW) / P(WCW)/ P(WWC) = 3 * 0.128 = 0.384
A "combination" lock is opened with the correct sequence of three numbers between 1 and 59 inclusive. (A number can be used more than once.) What is the probability of guessing those three numbers and opening the lock with the first try?
P(first guess opens lock)= 1/205379 59^3 = 205379
A certain lottery is won by selecting the correct five numbers from 1, 2, ..., 36. The probability of winning that game is 1/376,992. What is the probability of winning if the rules are changed so that in addition to selecting the correct five numbers, you must now select them in the same order as they are drawn?
P(winning)= 1/45239040 36 nPr 5 = 45239040
Select the six winning numbers from 1, 2, . . . , 44. (In any order. No repeats.)
P(winning)= 1/7059052 44 nCr 6 = 7059052
