Exam 2

What are the two requirements for a discrete probability​ distribution?

0(greater than equal to) P(x)(greater than equal to)1 EP(x)=1

Determine μx and σx from the given parameters of the population and sample size. μ=79​, σ=20​, n=39

79, 3.203

What is a random​ variable?

A random variable is a numerical measure of the outcome of a probability experiment.

A simple random sample of size n=49 is obtained from a population with μ=83 and σ=14. ​(a) Describe the sampling distribution of x. ​(b) What is P x>86.1​? ​(c) What is P x≤78.3​? ​(d) What is P 81.1<x<86​?

A) The distribution is approximately normal. 83, 2 B) P(x>86.1)= .0606 C) P (x<78.3) .0094 D) P (81.1<x<86).7621

Describe what an unusual event is. Should the same cutoff always be used to identify unusual​ events? Why or why​not?

An event is unusual if it has a low probability of occurring. The same cutoff should not always be used to identify unusual events. Selecting a cutoff is subjective and should take into account the consequences of incorrectly identifying an event as unusual.

Describe the sampling distribution of p. Assume the size of the population is 30,000. n=900​ p=0.7 Choose the phrase that best describes the shape of the sampling distribution of p below. Determine the mean of the sampling distribution of p. Determine the standard deviation of the sampling distribution of p.

Approximately normal because n≤0.05N and np(1−p)≥10. .7 .15

Describe how the value of n affects the shape of the binomial probability histogram.

As n​ increases, the binomial distribution becomes more bell shaped.

Explain the Law of Large Numbers. How does this law apply to gambling​ casinos?

As the number of repetitions of a probability experiment​ increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome. This applies to casinos because they are able to make a profit in the long run because they have a small statistical advantage in each game.

What is the formula for the expected number of successes in a binomial experiment with n trials and probability of success​ p?

E(X)=np

To cut the standard error of the mean in​ half, the sample size must be doubled.

False. The sample size must be increased by a factor of four to cut the standard error in half.

Suppose that a single card is selected from a standard​ 52-card deck. What is the probability that the card drawn is a king​? Now suppose that a single card is drawn from a standard​ 52-card deck, but it is told that the card is court (jack, queen, or king). What is the probability that the card drawn is a king​?

.077 .333

The distribution of the sample​ mean, x​, will be normally distributed if the sample is obtained from a population that is normally​distributed, regardless of the sample size.

true

The mean of the sampling distribution of p^ is p.

true

Suppose a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ. The sampling distribution of x has mean μx=​______ and standard deviation

μ, a/squared n

Find the value of zα. z0.49

.03

ind the​ z-score such that the area under the standard normal curve to the left is 0.37.

-0.33 is the​ z-score such that the area under the curve to the left is 0.37

Find the​ z-scores that separate the middle 77​% of the distribution from the area in the tails of the standard normal distribution.

-1.20, 1.20

State the criteria for a binomial probability experiment.

(1) The experiment consists of a fixed​ number, n, of trials, (2) Each trial has two possible mutually exclusive​ outcomes: success and failure, (3) The probability of​ success, p, remains constant for each trial of the experiment, (4) The trials are independent.

Draw a normal curve with μ=62 and σ=19. Label the mean and the inflection points.

(43, 62, 81)

In a national survey college students were​ asked, "How often do you wear a seat belt when riding in a car driven by someone​ else?" The response frequencies appear in the table to the right.​ (a) Construct a probability model for​ seat-belt use by a passenger.​ (b) Would you consider it unusual to find a college student who never wears a seat belt when riding in a car driven by someone​ else? Response Frequency Never 113 Rarely 339 Sometimes 559 Most of the time 1020 Always 2568

(a) Response Prob. Never .025 Rarely .074 Sometimes .122 Most .222 Always .558 Yes, because P(never)<0.05.

Suppose the following data represent the ratings​ (on a scale from 1 to​ 5) for a certain smart phone​ game, with 1 representing a poor rating. Complete parts​ (a) through​ (d) below. Stars Frequency 1 2,537 2 2,414 3 4,875 4 4,201 5 11,422 ​(a) Construct a discrete probability distribution for the random variable x. ​(c) Compute and interpret the mean of the random variable x. ​(d) Compute the standard deviation of the random variable x.

(a) Stars(x) P(x) 1 .100 2 .095 3 .192 4 .165 5 .449 (b) (1, .1) (2,.1) (3,.2) (4,.15) (5,.45) (c) 3.7 As the number of experiments​ increases, the mean of the observations will approach the mean of the random variable. (d) 1.4

Use n=6 and p=0.4 to complete parts​ (a) through​ (d) below. ​(a) Construct a binomial probability distribution with the given parameters. ​(b) Compute the mean and standard deviation of the random variable using μX=∑[x•P(x)]and σX=∑x2•P(x)−μ2X. ​(c) Compute the mean and standard​ deviation, using μX=np and σX=np(1−p). ​(d) Draw a graph of the probability distribution and comment on its shape.

(a) x P(x) 0 .0467 1 .1866 2 .3110 3 .2765 4 .1382 5 .0369 6 .0041 (b) 2.4 1.20 (c) 2.4 1.20 (d) skewed right

Suppose Ari loses 37​% of all air hockey games. ​(a) What is the probability that Ari loses two air hockey games in a​ row? ​(b) What is the probability that Ari loses four air hockey games in a​ row? ​(c) When events are​ independent, their complements are independent as well. Use this result to determine the probability that Ari loses four air hockey games in a​ row, but does not lose five in a row.

(a) .1369 (b) .0187 (c) .0118

A test to determine whether a certain antibody is present is 99.8​% effective. This means that the test will accurately come back negative if the antibody is not present​ (in the test​ subject) 99.8​% of the time. The probability of a test coming back positive when the antibody is not present​ (a false​ positive) is 0.002. Suppose the test is given to five randomly selected people who do not have the antibody. ​(a) What is the probability that the test comes back negative for all five ​people? ​(b) What is the probability that the test comes back positive for at least one of the five ​people?

(a) .9900 (b) .01

According to an​ almanac, 80​% of adult smokers started smoking before turning 18 years old. ​(a) Compute the mean and standard deviation of the random variable​ X, the number of smokers who started before 18 in 400 trials of the probability experiment. ​(b) Interpret the mean. ​(c) Would it be unusual to observe 360 smokers who started smoking before turning 18 years old in a random sample of 400 adult​ smokers? Why?

(a) 320 ? (b) It is expected that in a random sample of 400 adult​ smokers, 320 will have started smoking before turning 18 (c) Yes ​,because 360 is greater than h+2o

Determine whether the probabilities below are computed using the classical​ method, empirical​ method, or subjective method. Complete parts ​(a) through ​(d) below. (a) The probability of having eight girls in an eight​-child family is 0.00390625. ​(b) On the basis of a survey of 1000 families with eight ​children, the probability of a family having eight girls is 0.0064. (c) According to a sports​ analyst, the probability that a football team will win the next game is 0.46. (d) On the basis of clinical​ trials, the probability of efficacy of a new drug is 0.78.

(a) Classical method (b) Empirical (c) Subjective (d) Empirical

Determine whether the events E and F are independent or dependent. Justify your answer. ​(a) E: A person having an at-fault accident. ​F: The same person being prone to road rage. ​(b) E: A randomly selected person coloring her hair black. ​F: A different randomly selected person coloring her hair blond. (c) E: The war in a major oil-exporting country. ​F: The price of gasoline.

(a) E and F are dependent because being prone to road rage can affect the probability of a person having an at-fault accident. (b) E cannot affect F and vice versa because the people were randomly​ selected, so the events are independent. (c) The war in a major oil-exporting country could affect the price of gasoline​, so E and F are dependent.

Suppose that the probability that a passenger will miss a flight is 0.0902. Airlines do not like flights with empty​ seats, but it is also not desirable to have overbooked flights because passengers must be​ "bumped" from the flight. Suppose that an airplane has a seating capacity of 55 passengers. ​(a) If 57 tickets are​ sold, what is the probability that 56 or 57 passengers show up for the flight resulting in an overbooked​ flight? ​(b) Suppose that 61 tickets are sold. What is the probability that a passenger will have to be​ "bumped"? ​(c) For a plane with seating capacity of 52 ​passengers, what is the largest number of tickets that can be sold to keep the probability of a passenger being​ "bumped" below 10​%?

(a) The probability of an overbooked flight is .0304. (b) The probability that a passenger will have to be bumped is .5245. ​(c) The largest number of tickets that can be sold while keeping the probability of a passenger being​ "bumped" below 10​% is 54

Determine whether the random variable is discrete or continuous. In each​ case, state the possible values of the random variable. ​(a) The number of points scored during a basketball game. ​(b) The time it takes for a light bulb to burn out.

(a) The random variable is discrete. The possible values are x=​0,1, 2,... (b) The random variable is continuous. The possible values are t>0.

Determine whether the random variable is discrete or continuous. In each​ case, state the possible values of the random variable. ​(a) The number of fish caught during a fishing tournament. ​(b) The time required to download a file from the Internet.

(a) The random variable is discrete. The possible values are x=​0, 1, 2,... (b) The random variable is continuous. The possible values are t>0.

In the probability distribution to the​ right, the random variable X represents the number of marriages an individual aged 15 years or older has been involved in. Complete parts​ (a) through​ (f) below. x P(x) 0 .0263 1 .578 2 .130 3 .025 4 .003 5 .001 ​(a) Verify that this is a discrete probability distribution. ​(b) Draw a graph of the probability distribution. Describe the shape of the distribution. ​(c) Compute and interpret the mean of the random variable X. Which of the following interpretations of the mean is​ correct? ​(d) Compute the standard deviation of the random variable X. ​(e) What is the probability that a randomly selected individual 15 years or older was involved in two​ marriages? ​(f) What is the probability that a randomly selected individual 15 years or older was involved in at least two ​marriages?

(a) This is a discrete probability distribution because all of the probabilities are between 0 and 1​, inclusive, and the sum of the probabilities is 1. (b) (0,.25) (1,.6) (2,.1.25) (3,.025) (4,0) (5,0) The distribution has one mode and is skewed right. (c) .93 If many individuals aged 15 year or older were​ surveyed, one would expect the mean number of marriages to be the mean of the random variable. (d) .7 (e) .130 (f) .159

A probability experiment is conducted in which the sample space of the experiment is S={3,4,5,6,7,8,9,10,11,12,13,14}. Let event E={4,5,6,7,8,9} and event F={8,9,10,11}. List the outcomes in E and F. Are E and F mutually​ exclusive?

(a) {8,9} (b) No. Have outcomes in common.

According to an​ airline, flights on a certain route are on time 75​% of the time. Suppose 10 flights are randomly selected and the number of​ on-time flights is recorded. ​(a) Explain why this is a binomial experiment. ​(b) Find and interpret the probability that exactly 6 flights are on time. ​(c) Find and interpret the probability that fewer than 6 flights are on time. ​(d) Find and interpret the probability that at least 6 flights are on time. ​(e) Find and interpret the probability that between 4 and 6 ​flights, inclusive, are on time.

(c) ? In 100 trials of this​ experiment, it is expected about 88 to result in fewer than 6 flights being on time. (d) The probability that at least 6 flights are on time is .9219. In 100 trials of this​ experiment, it is expected about 92 to result in at least 6 flights being on time. (e) The probability that between 4 and 6 flights, inclusive, are on time is .2206. In 100 trials of this​ experiment, it is expected about 2222 to result in between 4 and 6 flights, inclusive, being on time.

In the game of​ roulette, a player can place a $9 bet on the number 36 and have a 138 probability of winning. If the metal ball lands on 36​, the player gets to keep the $9 paid to play the game and the player is awarded an additional $315. Otherwise, the player is awarded nothing and the casino takes the​ player's $9. What is the expected value of the game to the​ player? If you played the game 1000​ times, how much would you expect to​ lose?

-.47 470

A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n=15​, p=0.9​, x=12

.1285

The graph to the right is the uniform probability density function for a friend who is x minutes late. ​(a) Find the probability that the friend is between 25 and 30 minutes late. ​(b) It is 10 A.M. There is a 50​% probability the friend will arrive within how many​ minutes?

.167 15

A survey of 300 randomly selected high school students determined that 60 play organized sports. ​(a) What is the probability that a randomly selected high school student plays organized​ sports? ​(b) Interpret this probability.

.2 If​ 1,000 high school students were​ sampled, it would be expected that about 200of them play organized sports.

Suppose that events E and F are​ independent, ​P(E)=0.4​, and P(F)=0.8. What is theP(E and F)​?

.32

A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n=13​, p=0.4​, x<(equal to)4

.3530

Let the sample space be S={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Suppose the outcomes are equally likely. Compute the probability of the event E={3, 5, 7, 8}.

.4

A binomial probability experiment is conducted with the given parameters. Use technology to find the probability of x successes in the n independent trials of the experiment. n=9​, p=0.4​, x<4

.4826

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 $40,000. 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 $40,000. You contact an economist who believes there is a 30​% probability the economy will remain​ strong, a 60​% 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?

14,000

Suppose a life insurance company sells a ​$290,000 one-year term life insurance policy to a 21​-year-old female for ​$290. The probability that the female survives the year is 0.999583. Compute and interpret the expected value of this policy to the insurance company.

169.07 The insurance company expects to make an average profit of $169.07 on every 21-year-old female it insures for 1 year.

Assume the random variable X is normally distributed with mean μ=50 and standard deviation σ=7. Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded. P(X>34)

34+ .9889

Complete parts (a​) through ​(d) for the sampling distribution of the sample mean shown in the accompanying graph (390, 400, 410) (a) What is the value of μx​? (b) What is the value of σx​? (c) If the sample size is n=9​,what is likely true about the shape of the​ population? ​(d) If the sample size is n=9​, what is the standard deviation of the population from which the sample was​ drawn?

400 10 The shape of the population is approximately normal. 30

Suppose the monthly charges for cell phone plans are normally distributed with mean μ=$64 and standard deviation σ=​$16. ​(a) Draw a normal curve with the parameters labeled. ​(b) Shade the region that represents the proportion of plans that charge less than ​$48. ​(c) Suppose the area under the normal curve to the left of X=​$48 is 0.1587. Provide an interpretation of this result.

48, 64, 80 48, 64, 80 The probability is 0.1587 that a randomly selected cell phone plan in this population is less than ​$48 per month.

Assume the random variable X is normally distributed with mean μ=50 and standard deviation σ=7. Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded. ​P(54≤X≤70​)

54-70 .2817

Assume the random variable X is normally distributed with mean μ=50 and standard deviation σ=7. Find the 77th percentile.

55.17

A man has nine shirts and seven ties. Assuming that they all​ match, how many different shirt​-and-tie combinations can he wear?

63

One graph in the figure represents a normal distribution with mean μ=11 and standard deviation σ=2. The other graph represents a normal distribution with mean μ=16 and standard deviation σ=2. Determine which graph is which and explain how you know. (Left A Red, Right B Blue, 11, 16)

Graph A has a mean of μ=11 and graph B has a mean of μ=16 because a larger mean shifts the graph to the right.

Fill in the blanks below to make a true statement.

In a binomial experiment with n trials and probability of success​ p, if np(1-p)≥10, the binomial random variable X is approximately normal with μX=np and σX=Root over np(1-p)

A simple random sample of size n=58 is obtained from a population with μ=31 and σ=2. Does the population need to be normally distributed for the sampling distribution of x to be approximately normally​ distributed? Why? What is the sampling distribution of x​?

No because the Central Limit Theorem states that regardless of the shape of the underlying​ population, the sampling distribution of x becomes approximately normal as the sample​ size, n, increases. The sampling distribution of x is normal or approximately normal with μx=31 and σx=.263.

You suspect a​ 6-sided die to be loaded and conduct a probability experiment by rolling the die 400 times. The outcome of the experiment is listed in the following table. Do you think the die is​loaded? Why?

No, equal chance of occurring.

Find the probability of the indicated event if ​P(E)=0.30 and ​P(F)=0.45. Find​ P(E or​ F) if​ P(E and ​F)=0.10.

P(E or F)= .65

If E and F are not disjoint​ events, then​ P(E or F)=​________.

P(E)+P(F)-P(E and F)

Compute​ P(X) using the binomial probability formula. Then determine whether the normal distribution can be used to estimate this probability. If​ so, approximate​ P(X) using the normal distribution and compare the result with the exact probability. n=64​, p=0.4​, and X=

P(X)= .0522 ​Yes, the normal distribution can be used because np(1−p)≥10. By how much do the exact and approximated probabilities​ differ? Select the correct choice below and fill in any answer boxes in your choice. .001

Assume that the probability of the binomial random variable will be approximated using the normal distribution. Describe the area under the normal curve that will be computed. Find the probability that there are exactly 14 defective parts in a shipment.

The area between 13.5 and 14.5

Describe the difference between classical and empirical probability.

The empirical method obtains an approximate empirical probability of an event by conducting a probability experiment. The classical method of computing probabilities does not require that a probability experiment actually be performed.​ Rather, it relies on counting​ techniques, and requires equally likely outcomes.

In a certain card​ game, the probability that a player is dealt a particular hand is 0.4. Explain what this probability means. If you play this card game 100​ times, will you be dealt this hand exactly 40 times? Why or why​ not?

The probability 0.4 means that approximately 40 out of every 100 dealt hands will be that particular hand.​ No, you will not be dealt this hand exactly 40 times since the probability refers to what is expected in the​long-term, not​ short-term.

The graph of a normal curve is given. Use the graph to identify the value of μ and σ. (522, 524, 526, 528, 530, 532, 534, 536, 538)

The value of μ is 530. The value of σ is 2.

Why is the following not a probability model? Color Probability Red 0.2 Green -0.3 Blue 0.2 Brown 0.3 Yellow 0.2 Orange 0.4

This is not a probability model because at least one probability is less than 0.

Determine if the following statement is true or false. In the binomial probability distribution​ function, nCx represents the number of ways of obtaining x successes in n trials.

True

Determine if the following statement is true or false. Probability is a measure of the likelihood of a random phenomenon or chance behavior.

True

True or False​: In a probability​ model, the sum of the probabilities of all outcomes must equal 1.

True

When can the Empirical Rule be used to identify unusual results in a binomial​ experiment? Why can the Empirical Rule be used to identify results in a binomial​ experiment?

When the binomial distribution is approximately bell​ shaped, about​ 95% of the outcomes will be in the interval from μ−2σ to μ+2σ. The Empirical Rule can be used to identify results in binomial experiments when np(1−p)≥10.

The notation zα is the​ z-score that the area under the standard normal curve to the right of zα is​ _______.

a

Suppose that the lifetimes of light bulbs are approximately normally​ distributed, with a mean of 56 hours and a standard deviation of 3.3 hours. With this​ information, answer the following questions. ​(a) What proportion of light bulbs will last more than 61 ​hours? ​(b) What proportion of light bulbs will last 51 hours or​ less? ​(c) What proportion of light bulbs will last between 57 and 62 ​hours? ​(d) What is the probability that a randomly selected light bulb lasts less than 46 ​hours?

a) .0649 b) .0649 c) .3464 d) .0012

The reading speed of second grade students in a large city is approximately​ normal, with a mean of 92 words per minute​ (wpm) and a standard deviation of 10 wpm. Complete parts​ (a) through​ (f). (a) What is the probability a randomly selected student in the city will read more than 98 words per​ minute? Interpret this probability. Select the correct choice below and fill in the answer box within your choice.

a) .2743 If 100 independent samples of n=10 students were chosen from this​ population, we would expect 3 sample(s) to have a sample mean reading rate of more than 98 words per minute.

The time required for an automotive center to complete an oil change service on an automobile approximately follows a normal​distribution, with a mean of 17 minutes and a standard deviation of 2 minutes. ​(a) The automotive center guarantees customers that the service will take no longer than 20 minutes. If it does take​ longer, the customer will receive the service for​ half-price. What percent of customers receive the service for​ half-price? ​(b) If the automotive center does not want to give the discount to more than 5​% of its​ customers, how long should it make the guaranteed time​ limit?

a) 6.68% b) 21

Suppose a simple random sample of size n=125 is obtained from a population whose size is N=10,000 and whose population proportion with a specified characteristic is p=0.8. Complete parts ​(a) through​ (c) below. (a) Describe the sampling distribution of p. Choose the phrase that best describes the shape of the sampling distribution below. b) What is the probability of obtaining x=105 or more individuals with the​ characteristic? That​ is, what is ​P(p≥0.84​)? (c) What is the probability of obtaining x=90 or fewer individuals with the​ characteristic? That​ is, what is ​P(p≤0.72​)?

a) Approximately normal because n≤0.05N and np(1−p)≥10. .8 .035777 b)P(p≥0.84​)=.1318 c)P(p≤0.72​)=.0127

Suppose Jack and Diane are each attempting to use a simulation to describe the sampling distribution from a population that is skewed right with mean 60 and standard deviation 10. Jack obtains 1000 random samples of size n=5 from the​ population, finds the mean of the​ means, and determines the standard deviation of the means. Diane does the same​ simulation, but obtains 1000 random samples of size n=35 from the population. Complete parts​ (a) through​ (c) below. (a) Describe the shape you expect for Jack​'s distribution of sample means. Describe the shape you expect for Diane​'s distribution of sample means. Choose the correct answer below. (b) What do you expect the mean of Jack​'s distribution to​ be? What do you expect the mean of Diane​'s distribution to​ be? ​(c) What do you expect the standard deviation of Jack​'s distribution to​ be? What do you expect the standard deviation of Diane​'s distribution to​ be?

a) Jack​'s distribution is expected to be skewed right​, but not as much as the original distribution. Diane​'s distribution is expected to be approximately normal. b) Jack's ​'s distribution is expected to have a mean of 60. Diane​'s distribution is expected to have a mean of 60. c) Jack​'s distribution is expected to have a standard deviation of 4.47. Diane​'s distribution is expected to have a standard deviation of 1.69.

A simple random sample of size n=37 is obtained from a population with μ=64 and σ=14. ​(a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample​ mean? Assuming that this condition is​ true, describe the sampling distribution of x. ​(b) Assuming the normal model can be​ used, determine ​P(x<68.1​). ​(c) Assuming the normal model can be​ used, determine ​P(x≥65.5​).

a) Since the sample size is large enough, the population distribution does not need to be normal. Approximately normal​, with μx=64 and σx=14/root 37 b) P(x<68.1​)=. 9627 c) P(x≥65.5​)=. 2571.

Suppose the lengths of human pregnancies are normally distributed with μ=266 days and σ=16 days. Complete parts ​(a) and​(b) below. (a) The figure to the right represents the normal curve with μ=266 days and σ=16 days. The area to the left of X=235 is 0.0263. Provide two interpretations of this area. Provide one interpretation of the area using the given values. Select the correct choice below and fill in the answer boxes to complete your choice.

a. The proportion of human pregnancies that last less than 235 days is .0263. The probability that a randomly selected human pregnancy lasts less than 235 days is .0263. b. The proportion of human pregnancies that last between 285 and 305 days is .1101. The probability that a randomly selected human pregnancy lasts between 285 and 305 days is .1101.

A bag of 100 tulip bulbs purchased from a nursery contains 35 red tulip​ bulbs, 40 yellow tulip​ bulbs, and 25 purple tulip bulbs. ​(a) What is the probability that a randomly selected tulip bulb is​ red? ​(b) What is the probability that a randomly selected tulip bulb is​ purple? ​(c) Interpret these two probabilities.

a.) .35 b.) .25 c.) If 100 tulip bulbs were sampled with​ replacement, one would expect about 35 of the bulbs to be red and about 25 of the bulbs to be purple.

The graph of a normal curve is given on the right. Use the graph to identify the values of μ and σ. (-6, -2, 2, 6, 10, 14, 18)

h=6 σ=4

Determine μx and σx from the given parameters of the population and sample size. μ=89​, σ=27​, n=81

hx=89 σx=3

List all the combinations of four objects m, l, n, and k taken two at a time. What is 4C2​?

ml, mn, mk, ln, lk, nk 6

List all the permutations of five objects m, l, n, k, and p taken two at a time without repetition. What is 5P2​?

ml, mn, mk, mk, lm, lk, lp, nm, nl, nk, np, km, kl, kn, kp, pm, pl. pn, pk 20

The​ _____ _____, denoted p​, is given by the formula p=​_____, where x is the number of individuals with a specified characteristic in a sample of n individuals.

sample, proportion, x/n

The standard deviation of the sampling distribution of x​, denoted σx​, is called the​ _____ _____ of the​ _____.

standard, error, mean

Determine the area under the standard normal curve that lies to the left of ​(a) Z=−0.34, ​(b) Z=−1.65​, ​(c) Z=1.56​, and​ (d) Z=1.11.

​(a) The area to the left of Z=−0.34 is .3669. ​(b) The area to the left of Z=−1.65 is .0495. (c) The area to the left of Z=1.56 is .9406. (d) The area to the left of Z=1.11 is .8665.

Determine the area under the standard normal curve that lies to the right of ​(a) Z=0.24, ​(b) Z=−0.17, ​(c) Z=−0.23, and ​(d) Z=−1.71.

​(a) The area to the right of Z=0.24 is .4052. (b) The area to the right of Z=−0.17 is .5675. ​(c) The area to the right of Z=−0.23 is .5909. ​(d) The area to the right of Z=−1.71 is .9564.

Fourteen jurors are randomly selected from a population of 4 million residents. Of these 4 million​ residents, it is known that 45​% are of a minority race. Of the 14 jurors​ selected, 2 are minorities. ​(a) What proportion of the jury described is from a minority​ race? ​(b) If 14 jurors are randomly selected from a population where 45​% are​ minorities, what is the probability that 2 or fewer jurors will be​ minorities? ​(c) What might the lawyer of a defendant from this minority race​ argue?

​(a) The proportion of the jury described that is from a minority race is .14. ​(b) The probability that 2 or fewer out of 14 jurors are​ minorities, assuming that the proportion of the population that are minorities is 45​%, is .0170. (c) ​(b) The probability that 2 or fewer out of 14 jurors are​ minorities, assuming that the proportion of the population that are minorities is 45​%, is .0170.

Without doing any​ computation, decide which has a higher​ probability, assuming each sample is from a population that is normally distributed with μ=100 and σ=15. Explain your reasoning. ​(a)​ P(90≤x≤​110) for a random sample of size n=10 ​(b)​ P(90≤x≤​110) for a random sample of size n=20

​P(90≤x≤​110) for a random sample of size n=20 has a higher probability. As n​increases, the standard deviation decreases.

Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. An experimental drug is administered to 170 randomly selected​ individuals, with the number of individuals responding favorably recorded.

​Yes, because the experiment satisfies all the criteria for a binomial experiment.

Is the following a probability​ model? What do we call the outcome "green​"? Color Probability Red 0.35 Green 0 Blue 0.2 Brown 0.1 Yellow 0.25 Orange 0.1

​Yes, because the probabilities sum to 1 and they are all greater than or equal to 0 and less than or equal to 1. Impossible event

Determine whether the distribution is a discrete probability distribution. x P(x) 0 .22 1 .17 2 .18 3 .23 4 .20

​Yes, because the sum of the probabilities is equal to 1 and each probability is between 0 and​ 1, inclusive.