Chapter 11

Introduction to Statistical Investigations Test Bank

Note: TE = Text entry TE-N = Text entry - Numeric

Ma = Matching MS = Multiple select

MC = Multiple choice TF = True-False

DD = Drop-down

CHAPTER 11 LEARNING OBJECTIVES

11.1: Understand probabilities through simulations of random processes and be able to recognize equally likely outcomes.

11.2: Calculate probabilities of events using probability tables, Venn diagrams, complement rule and addition rule; and understand when events are mutually exclusive.

11-3: Use the multiplication rule and tree diagrams to calculate conditional probabilities and understand whether events are independent.

11-4: Understand aspects of a discrete random variable including: probability distribution, expected value, variance, and standard deviation.

11.5: Recognize a linear transformation of a random variable and be able to calculate its expected value, variance, and standard deviation.

11.6: Recognize binomial and geometric distributions and be able to calculate probabilities, expected values, and variances from these distributions.

11.7: Understand and calculate probabilities and z-scores from a normal distribution.

11.8: Apply the normal distribution as an approximate model for sampling distributions.

Section 11.1: Basics of Probability

11.1-1: Describe how to conduct simulation analyses of random processes.

11.1-2: Use simulation results to approximate probabilities.

11.1-3: List outcomes in the sample space of a random process.

11.1-4: Determine whether outcomes of a random process are equally likely.

11.1-5: Calculate probabilities for random processes with equally likely outcomes.

11.1-6: Interpret a calculated probability and explain how it depends on assumptions made.

1. Your brother says he is better than you at playing some video game. You play 6 games and he wins 4 of them. He says that this proves he is a better player. You have studied statistics and you want to determine the probability of anyone winning at least 4 games out of 6 just by chance if you are both equally as skilled. Which of the following would provide an accurate estimate of that probability?
   1. Take 6 cards and write “win” on 4 of them and “lose” on 2 of them. Shuffle the cards and place them in two piles and determine if one pile has more “win” cards than the other. Repeat this many times. Calculate the proportion of times one pile had more wins than the other.
   2. Toss a six-sided die many times and calculate the proportion of times the number 4 or more lands face up.
   3. Flip a coin 6 times and count the number of heads. Repeat this many times. Calculate the proportion of your simulated results that gave 4 or more heads.

Questions 2 through 6: There are ten Academic Senate faculty in the Statistics Department at the University of California, Irvine: three females and seven males. Two faculty are to be selected without replacement to serve on a committee where one of those selected will serve as the head of the committee.

1. If we record the sex of the chosen committee members (male or female) and their roles (head or not), how many possible outcomes are in the sample space?
   1. 1
   2. 2
   3. 3
   4. 4

LO: 11.1-3; Difficulty: Medium; Type: MC

1. True or False: The events in the sample space are all equally likely.
2. How could you simulate this random process?
   1. Roll a six-sided die twice. If the first roll is a one, then the committee head is female; if the second roll is a one, then the other committee member is also female.
   2. Write male on seven cards and female on three cards. Select one card to be the committee head, and another card to be the other committee member.
   3. Flip a coin twice. If the first flip lands on heads, then the committee head is female; if the second coin lands on heads, then the other committee member is also female.
3. Now, instead of recording the sex and role of the selected committee members, you only record the number of females selected. How many possible outcomes are in this new sample space?
   1. 1
   2. 2
   3. 3
   4. 4
4. A simulation of this random process was run 1,000 times, and the number of females selected for the committee was recorded on each simulation. A bar plot of the results follows.

Use the bar plot to estimate the probability that zero females are selected for the committee.

LO: 11.1-2; Difficulty: Medium; Type: TE-N

Questions 7 through 12: An online psychic (psi) ability or extrasensory perception (ESP) test^[1] shows the participant five face-down cards on the screen and asks the participant to “click on the one card that has a picture on the other side.” After clicking a card, the test shows you the card to determine if your guess was correct.

1. How many possible outcomes are in this sample space?
   1. 2
   2. 3
   3. 4
   4. 5

LO: 11.1-3; Difficulty: Medium; Type: MC

1. If a person does not have ESP, what is the probability the person’s guess is correct?
   1. 0.10
   2. 0.20
   3. 0.50
   4. 0.80
2. How could you simulate one play of the game, assuming the participant did not have ESP abilities?
   1. Flip a coin five times and count the number of heads that appear.
   2. Spin a spinner where 1/5^th of the spinner is red and 4/5^ths are white. Record whether you land on the red portion of the spinner or not.
   3. Take five cards where one card is black and the other four are white. Shuffle the cards and select one card from the five. Record whether you chose a white card or not.
3. The ESP test website states: “Keep in mind that your results should only be considered suggestive, because high scores in these tests can be due to chance as well as to actual abilities. Only repeated testing can distinguish between luck and genuine psi abilities.” Suppose you take the test five times. How would you simulate the probability of choosing the correct card in at least three of the trials, assuming you do not have ESP?
   1. Flip a coin five times and count the number of heads that appear. Repeat this many times. Calculate the proportion of your simulated results that gave 3 or more heads.
   2. Spin a spinner five times, where 1/5^th of the spinner is red and 4/5^ths are white. Record the number of times the spinner lands on white. Repeat this many times. Calculate the proportion of your simulated results where the spinner landed on white at least three times.
   3. Take five cards where one card is black and the other four are white. Shuffle the cards and choose five cards with replacement. Record the number of black cards chosen. Repeat this many times. Calculate the proportion of your simulated results where you chose least three black cards.
4. A simulation of this random process was run 1,000 times, assuming the participant did not have ESP, and the number of correct guesses was recorded on each simulation. A bar plot of the results follows.

Using the bar plot, estimate the probability of three or more correct guesses.

• 1. 0.01
  2. 0.04
  3. 0.07
  4. 0.60

1. Based on the simulation results in the bar plot, if someone guessed correctly four times, would you consider this evidence that the person has ESP?
   1. Yes, since the probability the person does not have ESP is less than 1%.
   2. Yes, since if they did not have ESP, the probability of four or more correct guesses is less than 1%.
   3. No, since they did not guess all five correctly.
   4. No, since it’s still possible for a person to guess correctly four times if they don’t have ESP.
2. According to my neighbor, the probability that the tomato plants she planted last month will actually survive to produce fruit is only 0.6. True or False: This means her tomato plants will survive to produce fruit in the next six out of 10 growing seasons.
3. True or False: The probability of an event is always equal to the number of outcomes in the event divided by the number of outcomes in the sample space.
4. According to Krantz in his book What the Odds Are (1992, p. 161), the probability that a randomly selected American will be injured by lightning in a given year is 1/685000. What does this probability mean?
   1. Every 1 out of 685,000 Americans will be injured by lightning in a given year.
   2. If we were to randomly select an American many many times, then the proportion of times that selected American would be injured by lightning in a given year is 1/685000.
   3. If we were to randomly select 100 Americans, none of them would be injured by lightning in a given year.
   4. Out of 685,000 years, no one will get injured by lightning in 684,999 of those years.

Section 11.2: Probability Rules

11.2-1: Express events using set notation.

11.2-2: Use the complement rule and addition rule to calculate probabilities of events.

11.2-3: Use probability tables and Venn diagrams to calculate probabilities of events.

11.2-4: Determine whether events are mutually exclusive.

1. Assume that, for events and , and . Are and are mutually exclusive?
   1. Yes
   2. No
   3. We do not have enough information to determine whether and are mutually exclusive.

LO: 11.2-4; Difficulty: Hard; Type: MC

1. John is taking Statistics and Biology this semester. The probability John will get an A in Statistics is 0.4. The probability John will get an A in Biology is 0.3. The probability John will get an A in both Statistics and Biology is 0.15. What is the probability that John will get at least one A between his Statistics and Biology courses?
   1. 0.12
   2. 0.15
   3. 0.55
   4. 0.7

LO: 11.2-1; Difficulty: Medium; Type: MC

Questions 18 through 26: In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events

= student lives in a campus residence hall

= student participates in a campus meal program

1. Write the probability 0.42 in probability notation.
2. How would you write the event that a college student “does not participate in a campus meal program” in probability notation?
3. How would you write the event that a college student “lives on campus and does not participate in a campus meal program” in probability notation?
4. Find .

LO: 11.2-2; Difficulty: Easy; Type: TE-N

1. Find .

LO: 11.2-2; Difficulty: Easy; Type: TE-N

1. Use the addition rule to find .

LO: 11.2-2; Difficulty: Medium; Type: TE-N

1. Are the events and mutually exclusive?
   1. Yes
   2. No
   3. Not enough information provided

LO: 11.2-4; Difficulty: Easy; Type: MC

1. Fill in the probability table below to represent this scenario.

LO: 11.2-3; Difficulty: Hard; Type: TE-N

1. Suppose there are 100 students in this population. Fill in the Venn diagram below to represent the number of students in each category for this scenario. Hint: The four numbers should sum to 100!

(4)

LO: 11.2-3; Difficulty: Hard; Type: TE-N

Questions 27 through 31: Suppose you roll a six-sided die once. Define events

A = roll an even number

B = roll a two

C = roll an odd number

D = roll a number greater than three

1. Are events A and B mutually exclusive?
   1. Yes
   2. No
2. Are events B and C mutually exclusive?
   1. Yes
   2. No
3. Find .

LO: 11.2-2; Difficulty: Easy; Type: TE-N

1. Find .
2. Find .

LO: 11.2-2; Difficulty: Medium; Type: TE-N

Section 11.3: Conditional Probability and Independence

11.3-1: Calculate conditional probabilities from a table and from the definition.

11.3-2: Determine whether events are independent.

11.3-3: Use the general multiplication rule to calculate probabilities.

11.3-4: Use the multiplication rule to calculate probabilities for independent events.

11.3-5: Use tree diagrams to represent and calculate probabilities.

1. A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.39, and winning is independent from one bottle to the next. You buy six bottles. What is the probability that none of the six bottles reveals a prize?
   1. 0.39
   2. 0.61
   3. 0.052
   4. 0.004
2. Assume that events and are independent, where and . What is ?
   1. 0.6
   2. 0.3
   3. 0.18
   4. 0
3. Assume that events and are mutually exclusive, where and . What is ?
   1. 0.6
   2. 0.3
   3. 0.18
   4. 0

Questions 35 through 37: In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events

= student lives in a campus residence hall

= student participates in a campus meal program

1. Are the events and independent?
   1. Yes
   2. No
   3. Not enough information provided

LO: 11.3-2; Difficulty: Hard; Type: MC

1. What is the appropriate expression for the probability that a randomly chosen college student who lives in a campus residence hall participates in a campus meal program?
2. Find .

LO: 11.3-1; Difficulty: Medium; Type: TE-N

Questions 38 and 39: Suppose that 70% of the seniors at a large university have taken calculus, and 30% of the seniors have taken physics. Of the seniors who have taken calculus, 40% have taken physics. A student who is a senior at this university is randomly selected. Define the following events.

= The student has taken calculus

= The student has taken physics

1. Which of the following is true about the events and ?
   1. and are mutually exclusive.
   2. and are complements.
   3. and are independent.
   4. None of the above.
2. Find
   1. 0.70 + 0.30 – 0.40 = 0.6
   2. (0.70)(0.30) = 0.21
   3. (0.70)(0.40) = 0.28
   4. (0.30)(0.40) = 0.12

Questions 40 and 41: Data on sex and opinion on the death penalty for respondents in the 2008 General Social Survey in which a random sample of 1,902 adults in the U.S. was surveyed are below.

Suppose we choose a respondent of this survey at random.

1. What is the probability that the respondent will oppose the death penalty, given that the respondent is male?
   1. 254/1902
   2. 639/1902
   3. 254/639
   4. 254/885
2. What is the probability that the respondent is female and opposes the death penalty?
   1. 385/1902
   2. 385/639
   3. 385/1017
   4. 639/1902

Questions 42 through 47: Suppose that only 8% of a large population has a certain disease. A diagnostic test has been developed which is 90% accurate for people with the disease (90% of people with the disease test positive), and 85% accurate for people without the disease (85% of people without the disease test negative). Define the following events:

= person has the disease

= person tests positive on the diagnostic test

1. For each of the given probabilities, state whether the probability is a conditional probability or an unconditional probability:

0.08?

0.90?

0.85?

1. How would you express the probability 0.90 in terms of and ?
2. Fill in the values below to create a hypothetical table of 1,000 randomly selected individuals in this population.

LO: 11.3-1; Difficulty: Hard; Type: TE-N

1. Fill in the tree diagram below to represent this scenario.

LO: 11.3-5; Difficulty: Hard; Type: TE-N

1. What is the probability that a randomly selected person tests positive on the diagnostic test?

LO: 11.3-5; Difficulty: Medium; Type: TE-N

1. Given that a person tests positive on the diagnostic test, what is the probability he or she has the disease?

LO: 11.3-5; Difficulty: Medium; Type: TE-N

Section 11.4: Discrete Random Variables

11.4-1: Determine the probability distribution of a discrete random variable.

11.4-2: Calculate the expected value of a discrete random variable.

11.4-3: Interpret the expected value of a discrete random variable.

11.4-4: Calculate the variance and standard deviation of a discrete random variable.

11.4-5: Interpret the variance and standard deviation of a discrete random variable.

1. Which of the following variables is a discrete random variable?
   1. Time it takes one email to travel between a sender and receiver
   2. Number of letters in the last name of a randomly chosen student
   3. Weight a dieter will lose after following a two-week weight loss program
   4. High temperature in Irvine on a randomly chosen day
2. Which of the following variables is a discrete random variable?
   1. Time it takes a randomly selected student to finish this exam
   2. Body temperature of a randomly selected adult
   3. Distance a randomly selected adult walks in a day
   4. Number of texts a randomly selected college student receives in a day
3. The standard deviation of a random variable Y is the…
   1. range of possible values for Y.
   2. most common value over a large number of observations of Y.
   3. approximate average distance from the mean that one would see in a large number of observations of Y.
   4. approximate mean value over a large number of observations of Y.

Questions 51 through 55: Let X be the amount in claims (in dollars) that a randomly chosen policyholder collects from an insurance company this year. The probability distribution function of X is given below.

1. What is the probability that a randomly chosen policy holder claims more than $100?
   1. 0.07
   2. 0.13
   3. 0.20
   4. 0.87
2. Calculate the expected value of X.

$_________

LO: 11.4-2; Difficulty: Medium; Type: TE-N

1. Let $D be the expected value of X. Which of the following is a correct interpretation of this value?
   1. We would expect a randomly chosen policyholder to collect $D in claims this year.
   2. The most likely value for the amount in claims is $D.
   3. If we were to observe a large number of policyholders, the average amount claimed would be around $D.
   4. On average, the amount claimed is about $D away from the mean.
2. Calculate the standard deviation of X.

$________

LO: 11.4-4; Difficulty: Medium; Type: TE-N

1. Let $S be the standard deviation of X. Which of the following is a correct interpretation of this value?
   1. We would expect a randomly chosen policyholder to collect $S in claims this year.
   2. The most likely value for the amount in claims is $S.
   3. If we were to observe a large number of policyholders, the average amount claimed would be around $S.
   4. On average, the amount claimed is about $S away from the mean.

Questions 56 through 61: An “instant lottery” is played by buying a ticket and scratching off a coating to reveal whether or not you have won a prize, and if so, how much. Suppose an instant lottery pays $5 with probability 0.05 and $100 with probability 0.006. Otherwise it pays nothing. Define

X = amount won for a single ticket. (You can ignore the cost of the ticket.)

1. Determine the probability distribution of X:

LO: 11.4-1; Difficulty: Easy; Type: TE-N

1. Calculate E(X).

$_______

LO: 11.4-2; Difficulty: Medium; Type: TE-N

1. Let $D be the expected value of X. Which of the following is a correct interpretation of this value?
   1. We would expect a randomly chosen ticket to result in a win of $D.
   2. The most likely value for the amount won for a single ticket is $D.
   3. The amount won for a single ticket will deviate, on average, from the expected amount won by about $D.
   4. If you were to play the instant lottery many times, the average amount won per ticket would be around $D.
2. Fill in the blank with the correct dollar amount:

If the lottery agency does not want to lose money over the long run, they must charge at least

$_______ per ticket.

LO: 11.4-3; Difficulty: Hard; Type: TE-N

1. Calculate the standard deviation of X.

$________

LO: 11.4-4; Difficulty: Medium; Type: TE-N

1. Let $S be the standard deviation of X. Which of the following is a correct interpretation of this value?
   1. We would expect a randomly chosen ticket to result in a win of $D.
   2. The most likely value for the amount won for a single ticket is $D.
   3. The amount won for a single ticket will deviate, on average, from the expected amount won by about $D.
   4. If you were to play the instant lottery many times, the average amount won per ticket would be around $D.

Questions 62 through 64: Based on her past experience, a professor knows that the probability distribution function for X = number of students who come to her office hours on any given Wednesday is given below.

1. What is the value of P(X = 4) (the question mark in the table)?
   1. 0.05
   2. 0.10
   3. 0.25
   4. 1

LO: 11.4-1; Difficulty: Medium; Type: MC

1. The expected value of X is 1.85. Interpret this value in the context of the problem.
   1. If you were to observe many students, the average number of students that visit office hours is 1.85.
   2. If you were to observe many Wednesdays, the average number of students that visit office hours is 1.85.
   3. On average, the number of students that visit office hours will deviate from the mean by 1.85.
   4. Next Wednesday, we expect 2 students to visit office hours.
2. The standard deviation of X is 0.96. Interpret this value in the context of the problem.
   1. If you were to observe many students, the average number of students that visit office hours is 0.96.
   2. If you were to observe many Wednesdays, the average number of students that visit office hours is 0.96.
   3. On average, the number of students that visit office hours will deviate from the mean by about one student.
   4. Next Wednesday, we expect one student to visit office hours.
3. Which of the following has a higher expected earning?

Option 1: A gift of $240, guaranteed.

Option 2: A 25% chance to win $1,000, and a 75% chance of getting nothing.

• 1. Option 1
  2. Option 2
  3. The two expected earnings are equal.

LO: 11.4-2; Difficulty: Medium; Type: MC

1. Which of the following has a larger expected loss?

Option 1: A sure loss of $740.

Option 2: A 25% chance to lose nothing, and a 75% chance of losing $1000.

• 1. Option 1
  2. Option 2
  3. The two expected earnings are equal.

LO: 11.4-2; Difficulty: Medium; Type: MC

Section 11.5: Random Variable Rules

11.5-1: Recognize situations in which rules for expected values and variance apply.

11.5-2: Calculate expected values of linear transformations of random variables.

11.5-3: Calculate variance and standard deviation of linear transformations of random variables.

Questions 67 through 73: Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player’s net profit (amount won – amount paid to play).

1. Calculate E(X).

LO: 11.5-1; Difficulty: Medium; Type: TE-N

1. Express Y as a linear transformation of X.

Y = ­­­___(1)____ X + ____(2)_____

LO: 11.5-1; Difficulty: Hard; Type: TE-N

1. Use the rules for expected value for linear transformations to find E(Y).

­­$________

LO: 11.5-2; Difficulty: Medium; Type: TE-N

1. Based on the expected value, is it worth the $5 to enter?
   1. No, since the expected profit is also $5.
   2. Yes, since in the long-run, your average net profit is greater than zero.
2. Calculate .

LO: 11.5-1; Difficulty: Medium; Type: TE-N

1. Use the rules for expected value for linear transformations to find .

LO: 11.5-3; Difficulty: Medium; Type: TE-N

1. Find .

LO: 11.5-3; Difficulty: Medium; Type: TE-N

Questions 74 through 76: Let be the amount in claims (in dollars) that a randomly chosen policyholder collects from an insurance company this year. From past data, the insurance company has determined that = $72, and = $60.

Suppose the insurance company decides to offer a discount to attract new customers. They will pay the new customer $50 for joining, and offer a 5% “cash back” offer for all claims paid. Let be the amount in claims (in dollars) for a randomly chosen new customer. Then .

1. Find .

LO: 11.5-2; Difficulty: Medium; Type: TE-N

1. Find .

LO: 11.5-3; Difficulty: Medium; Type: TE-N

1. Find .

LO: 11.5-3; Difficulty: Medium; Type: TE-N

Questions 77 through 79: Four friends are contemplating joining a local bowling league. Let be the score of the first, second, third, and fourth friend, respectively, on a randomly chosen game. From past experience, the friends know that: , , , and . Additionally, , ,, and. Define their total score on a randomly chosen game as . Assume the four players’ scores are independent.

1. The friends decide it would only be worth it to join the bowling league if they could average a total score of 500. Should they join the league?
   1. Yes, since their expected total score is greater than 500.
   2. No, since their expected total score is less than 500.
   3. Yes, since it is possible for their total score to be greater than 500 on a randomly chosen game.
   4. No, since it is not possible for their total score to be greater than 500 on a randomly chosen game.
2. Calculate the variance of their total score on a randomly chosen game.

LO: 11.5-3; Difficulty: Hard; Type: TE-N

1. Calculate .

LO: 11.5-3; Difficulty: Hard; Type: TE-N

Questions 80 and 81: Suppose is a random variable with and . Define .

1. Calculate the expected value of .

LO: 11.5-2; Difficulty: Easy; Type: TE-N

1. Calculate the variance of .

LO: 11.5-3; Difficulty: Medium; Type: TE-N

Section 11.6: Binomial and Geometric Random Variables

11.6-1: Recognize situations for which binomial or geometric distributions apply.

11.6-2: Determine parameter values for binomial or geometric distributions from the description of a random process.

11.6-3: Perform probability calculations from binomial and geometric distributions.

11.6-4: Calculate expected values and variances related to binomial and geometric distributions.

1. Consider a random process that involves repeatedly rolling a rubber pig. If the pig lands on its feet, a person scores 10 points. Otherwise, a person scores zero points. Which of the following is a binomial random variable?
   1. The number of times the pig lands on its feet when the pig is rolled 20 times.
   2. The number of rolls until the pig lands on its feet for the first time.
   3. The total number of points a person scores in five rolls.
   4. The time the pig spends in the air on one roll.
2. Consider a random process that involves repeatedly rolling a rubber pig. If the pig lands on its feet, a person scores 10 points. Otherwise, a person scores zero points. Which of the following is a geometric random variable?
   1. The number of times the pig lands on its feet when the pig is rolled 20 times.
   2. The number of rolls until the pig lands on its feet for the first time.
   3. The total number of points a person scores in five rolls.
   4. The time the pig spends in the air on one roll.
3. There are ten Academic Senate faculty in the UCI Statistics Department, three females and seven males. Two faculty are to be selected without replacement to serve on a committee and the X = the number of females will be recorded. Is X a binomial random variable?
   1. Yes, since all conditions are met.
   2. No, since there is not a fixed number of trials.
   3. No, since there are more than two possible outcomes on each trial.
   4. No, since the trials are not independent of each other.
   5. No, since the probability of success on each trial is not the same.
   6. Both B and C.
   7. Both D and E.

Questions 85 through 92: A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.39, and winning is independent from one bottle to the next. You buy six bottles. Let X be the number of prizes you win.

1. What is the distribution of X?
   1. Binomial
   2. Geometric
2. Identify the parameter values for the distribution of X.

=

=

LO: 11.6-2; Difficulty: Easy; Type: TE-N

1. What is the expected value of the number of prizes in six bottles?
   1. 2
   2. 2.34
   3. 2.56
   4. 3

LO: 11.6-4; Difficulty: Medium; Type: MC

1. What is the standard deviation of the number of prizes in six bottles?

LO: 11.6-4; Difficulty: Medium; Type: MC

1. Calculate .

LO: 11.6-3; Difficulty: Medium; Type: TE-N

1. Again buy six bottles, but now define the random variable Y = the number of bottles with no prize. Identify the parameter values for the distribution of X.

=

=

LO: 11.6-2; Difficulty: Medium; Type: TE-N

1. Define Y as in question 90. Find .

LO 11.6-4; Difficulty: Easy; Type: TE-N

1. True or False: .

Questions 93 through 97: A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.26, and winning is independent from one bottle to the next. You plan to keep buying bottles until you win a prize. Let X be the number of bottles you purchase.

1. What is the distribution of X?
   1. Binomial
   2. Geometric
2. Identify the parameter value for the distribution of X.

=

LO: 11.6-2; Difficulty: Easy; Type: TE-N

1. What is the expected value of the number of bottles you will need to purchase until you win a prize?
   1. 1.56
   2. 2
   3. 3.85
   4. 4

LO: 11.6-4; Difficulty: Medium; Type: MC

1. Calculate .

LO: 11.6-3; Difficulty: Medium; Type: TE-N

1. Calculate .

LO: 11.6-3; Difficulty: Hard; Type: TE-N

Section 11.7: Continuous Random Variables and Normal Distributions

11.7-1: Use the empirical rule to approximate probabilities from a normal distribution.

11.7-2: Calculate z-scores from a normal distribution.

11.7-3: Interpret a z-score in context.

11.7-4: Calculate probabilities from a normal distribution.

11.7-5: Calculate percentiles from a normal distribution.

11.7-6: Calculate parameter values and probabilities related to linear combinations of normal distributions.

1. Suppose we randomly select a healthy adult from a large population and measure X = body temperature in degrees Fahrenheit. An appropriate probability model for the distribution of X is a
   1. binomial distribution.
   2. normal distribution.
   3. geometric distribution.

Questions 99 through 105: A tire manufacturer designed a new tread pattern for its all-weather tires. Repeated tests were conducted on cars of approximately the same weight traveling at 60 miles per hour. The tests showed that the new tread pattern enables the cars to stop completely in an average distance of 125 feet with a standard deviation of 6.5 feet and that the stopping distances are approximately normally distributed.

1. Fill in the two blanks: About 95% of cars of approximately the same weight traveling at 60 miles per hour will stop completely in a distance of between

____(1)_____ and _____(2)_____ feet.

LO: 11.7-1; Difficulty: Easy; Type: TE-N

1. One car tested had a stopping distance of 140 feet. Calculate the z-score for this observation.

LO: 11.7-2; Difficulty: Easy; Type: TE-N

1. One car tested had a stopping distance of 112 feet. How many standard deviations was this car away from the mean stopping distance?
   1. Thirteen standard deviations above the mean
   2. Thirteen standard deviations below the mean
   3. Two standard deviations above the mean
   4. Two standard deviations below the mean

LO: 11.7-3; Difficulty: Medium; Type: MC

1. Use the Normal Probability Calculator applet to find the 70^th percentile of the distribution of stopping distances?

LO: 11.7-5; Difficulty: Medium; Type: TE-N

1. Use the Normal Probability Calculator applet to find the probability that one randomly selected car in the study will stop completely in a distance that is greater than 135 feet.

LO: 11.7-4; Difficulty: Medium; Type: TE-N

1. Now suppose you randomly select three cars from the study, and let X = the number of cars out of three randomly selected cars that will stop completely in a distance that is greater than 135 feet. What is the distribution of X?
   1. Binomial
   2. Geometric
   3. Normal
   4. None of the above
2. Define X as in question 103. Calculate P(X = 0).

LO: 11.7-4; Difficulty: Hard; Type: TE-N

Questions 106 through 114: The volume in a can of soda is normally distributed with mean 358 mililiters (ml) and standard deviation 6 ml.

1. You buy a can of soda that has 364 ml of soda. How many standard deviations away from the mean volume is this can of soda?
   1. Six standard deviations above the mean
   2. Six standard deviations below the mean
   3. One standard deviations above the mean
   4. One standard deviations below the mean

LO: 11.7-3; Difficulty: Medium; Type: MC

1. Fill in the two blanks: About 68% of soda cans will contain between

____(1)_____ and _____(2)_____ ml of soda.

LO: 11.7-1; Difficulty: Easy; Type: TE-N

1. Find the z-score for a can filled with 349 ml of soda.

LO: 11.7-2; Difficulty: Easy; Type: TE-N

1. Use the Normal Probability Calculator applet to calculate the probability that a randomly chosen can of soda is filled with less than 350 ml of soda.

LO: 11.7-4; Difficulty: Medium; Type: TE-N

1. Use the Normal Probability Calculator applet to calculate the probability that a randomly chosen can of soda is filled with at least 360 ml of soda.

LO: 11.7-4; Difficulty: Medium; Type: TE-N

1. Fill in the blank: Approximately _____(1)_____ % of soda cans will be filled with less than 362 ml of soda.

LO: 11.7-4; Difficulty: Medium; Type: TE-N

1. If the company wanted to ensure that 90% of soda cans were filled to the volume advertised on the can or more, what volume should they advertise? Round to the next highest integer.

LO: 11.7-5; Difficulty: Hard; Type: TE-N

1. Let be the total volume in a six-pack of soda cans. Assume the six cans are filled independently. Then has a normal distribution with what mean and what standard deviation?

Mean:____(1)____

SD: ____(2)____

LO: 11.7-6; Difficulty: Hard; Type: TE-N

1. What is the probability that a six-pack of soda cans have a total volume of at least 2150 ml? Assume the six cans are filled independently.

LO: 11.7-6; Difficulty: Hard; Type: TE-N

Section 11.8: Revisiting Theory-Based Approximations of Sampling Distributions

11.8-1: Apply the normal distribution as an approximate model for the sampling distribution of a sample proportion.

11.8-2: Apply the normal distribution as an approximate model for the sampling distribution of a sample mean.

11.8-3: Apply the normal distribution as an approximate model for the sampling distribution of the difference between two sample proportions.

11.8-4: Apply the normal distribution as an approximate model for the sampling distribution of the difference between two sample means.

Questions 115 through 119: A rental car company has noticed that the distribution of the number of miles customers put on rental cars per day is skewed to the right, with some occasional high outliers. The distribution has a mean of 80 miles and a standard deviation of 50 miles.

1. Which of the following most represents the shape of the distribution of the number of miles customers put on rental cars per day?

1. Which of the following most represents the shape of the distribution of the average number of miles per day put on a typical one of the company’s rental car in a year (365 days)?

1. Let = the average number of miles per day put on a typical one of the company’s rental car in a year (365 days). What is ?
2. Let = the average number of miles per day put on a typical one of the company’s rental car in a year (365 days). What is the standard deviation of ?

LO: 11.8-2; Difficulty: Medium; Type: TE-N

1. Use the Normal Probability Calculator applet to approximate the probability that one of the company’s rental cars will average over 85 miles per day.

LO: 11.8-2; Difficulty: Hard; Type: TE-N

1. Suppose half of all newborns are girls and half are boys. Hospital A, a large city hospital, records an average of 50 births a day. Hospital B, a small, rural hospital, records an average of 10 births per day. On a particular day, which hospital is less likely to record 80% or more female births?
   1. Hospital A (with 50 births a day), because the more births you see, the closer the proportions will be to 0.5.
   2. Hospital B (with 10 births a day), because with fewer births there will be less variability.
   3. The two hospitals are equally likely to record such an event, because the probability of a boy does not depend on the number of births.
   4. We are not given enough information to determine which hospital is less likely to record such an event.
2. A random sample of 25 college statistics textbook prices was collected and the mean price was found to be $91. To determine the probability of finding a sample mean of $91 or more extreme, you would need to refer to
   1. the population distribution of all college statistics textbook prices.
   2. the distribution of prices for this sample of college statistics textbooks.
   3. the distribution of sample mean textbook prices for all samples of 25 textbooks from this population.
   4. You don’t need to refer to anything since if the sample of 25 was randomly selected, then the sample mean of $91 must be close to the population mean textbook price.

Questions 122 through 124: Math SAT scores for students admitted to University A are bell-shaped with a mean of 520 and a standard deviation of 60. Math SAT scores for students admitted to University B are also bell-shaped, but with a mean of 580 and a standard deviation of 45. Suppose you plan to take a simple random sample of 15 students from University A and 25 students from University B. Let be the difference in sample mean math SAT scores between the two samples (B – A).

1. What is ?

LO: 11.8-4; Difficulty: Easy; Type: TE-N

1. If you were to take a very large number of samples, about how far away would you expect to fall from its mean?
   1. 24.49 points
   2. 17.92 points
   3. 12.61 points
   4. 6.49 points
2. Fill in the blanks: Approximately 95% of differences in sample mean math SAT scores between the two universities will be between ____(1)____ and ____(2)____ points.

LO: 11.8-4; Difficulty: Hard; Type: TE-N

Questions 125 through 127: Suppose that 45% of customers at the local farmer’s market prefer plastic bags to paper bags, whereas 62% of customers at the local grocery store prefer plastic bags to paper bags. You plan to take a simple random sample of 40 farmer’s market customers and 40 grocery story customers and survey their bag preferences. Define as the difference in sample proportions that prefer plastic bags to paper bags between the two locations (farmer’s market – grocery store).

1. Can we apply the normal approximation to the distribution of ?
   1. Yes, because there are two samples.
   2. Yes, because 0.4540, 0.5540, 0.6240, and 0.3840 are all greater than 10.
   3. No, because whether a person prefers plastic bags to paper bags is a categorical variable.
   4. No, because we are modeling a difference in proportions rather than a single proportion.
2. Calculate .

LO: 11.8-4; Difficulty: Medium; Type: TE-N

1. Calculate .

LO: 11.8-4; Difficulty: Medium; Type: TE-N

Questions 128 through 130: A lightbulb manufacturing plant’s production line has determined that the probability a lightbulb will be defective is 0.04. The quality control team decides to take a random sample of 500 lightbulbs and measure the proportion of those lightbulbs that are defective.

1. Can we apply the normal approximation to the distribution of the sample proportion of defective lightbulbs?
   1. No, because there is only one sample.
   2. No, because 0.04 is less than 0.10.
   3. Yes, because 0.04500 and 0.96500 are both greater than 10.
   4. Yes, because it is a random sample.
2. On average, how far away should the quality control team expect the sample proportion of defective lightbulbs to be from 0.04?

LO: 11.8-1; Difficulty: Hard; Type: TE-N

1. Use the Normal Probability Calculator applet to approximate the probability the quality control team sees at least 25 defective items in their sample.

LO: 11.8-1; Difficulty: Hard; Type: TE-N

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