AP Statistics Unit 2: Probability, Random Variables, and Probability Distributions

Two-way tables and association between two categorical variables, probability by simulation and by rule (addition, conditional, multiplication, independence), discrete random variables with their means and standard deviations, binomial and normal distributions, and the idea that a statistic has a sampling distribution.

Topics: 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12 · free response

Key formulas and facts

• Joint, marginal and conditional relative frequencies from a two-way table: $P(B\mid A)=\dfrac{\text{count in }A\text{ and }B}{\text{count in }A}$
• Complement: $P(A^C)=1-P(A)$. Addition: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ (the last term is 0 for mutually exclusive events)
• Conditional: $P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}$. Multiplication: $P(A\cap B)=P(A)\,P(B\mid A)$
• Independent events: $P(A\mid B)=P(A)$, equivalently $P(A\cap B)=P(A)P(B)$; $P(\text{at least one})=1-P(\text{none})$
• Discrete random variable: $\mu_X=\sum x_iP(x_i)$ and $\sigma_X=\sqrt{\sum (x_i-\mu_X)^2P(x_i)}$
• Binomial (n fixed trials, independent, success probability p each time): $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$, $\mu_X=np$, $\sigma_X=\sqrt{np(1-p)}$
• Normal: $z=\dfrac{x-\mu}{\sigma}$; about 68%, 95% and 99.7% of values lie within 1, 2 and 3 standard deviations of the mean
• The standard deviation of a sample statistic shrinks like $\frac{1}{\sqrt n}$: four times the sample size halves it

Worked examples

• Conditional relative frequency: in a table where 45 of 120 seniors and 30 of 80 juniors drive to school, $P(\text{drive}\mid\text{senior})=\frac{45}{120}=0.375$ and $P(\text{drive}\mid\text{junior})=\frac{30}{80}=0.375$. They are equal, so class and driving are not associated in these data.
• Tree diagram: 60% of orders are online; 5% of online and 10% of in-store orders are returned. $P(\text{returned})=0.6(0.05)+0.4(0.10)=0.07$ and $P(\text{online}\mid\text{returned})=\frac{0.03}{0.07}\approx0.429$.
• At least one: if each of 6 independent light bulbs fails with probability 0.1, $P(\text{at least one fails})=1-0.9^6\approx0.469$.
• Expected value: a raffle ticket pays \$50 with probability 0.01 and \$0 otherwise and costs \$1, so the expected net gain is $50(0.01)-1=-\$0.50$ per ticket.
• Binomial: with $n=8$ and $p=0.4$, $P(X=3)=\binom83(0.4)^3(0.6)^5\approx0.279$, $\mu_X=3.2$ and $\sigma_X=\sqrt{1.92}\approx1.386$.
• Normal: with $\mu=64$ and $\sigma=2.5$, $P(X>68)=P(Z>1.6)\approx0.0548$, and the 90th percentile is $64+1.2816(2.5)\approx67.2$.
• Simulation: assign digits 0-3 to "win" (p = 0.4) and 4-9 to "lose"; if 37 of 100 simulated games are wins, the estimate is $\hat P(\text{win})=0.37$.

Common traps

• Judging association from counts instead of conditional relative frequencies when the groups have different sizes.
• Confusing mutually exclusive with independent: mutually exclusive events with positive probabilities are never independent.
• Mixing up $P(A\mid B)$ and $P(B\mid A)$: check which group you are dividing by.
• Multiplying probabilities for events that are not independent (for example, draws without replacement from a small group).
• Using the binomial model without checking all of its conditions (Binary outcomes, Independent trials, fixed Number of trials, Same probability).
• Interpreting an expected value as the result of one trial instead of a long-run average.
• Calling a sample proportion or sample mean a parameter: parameters describe populations, statistics describe samples.

Practice questions

96 multiple-choice questions (4 options each, as on the new exam) and 2 free-response questions worth 10 points each. A calculator is allowed throughout. Try each one before opening the answer. Each line of a scoring guide is one point, scored 0 or 1 and earned independently of the other lines, the style of the new course's 10-point free-response scoring (no E/P/I part scores). A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Topic 2.1: Two-way tables and graphs for two categorical variables

8 multiple-choice questions.

Question 1 · Multiple choice

A random sample of students at a high school were asked which device they use most for homework. The results are shown in the two-way table. What proportion of all the students in the sample are Grade 12 students who use a laptop most?

1. 0.228
2. 0.294
3. 0.410
4. 0.774

Question 2 · Multiple choice

Using the same table of devices, what proportion of the students in the sample use a tablet most?

1. 0.150
2. 0.250
3. 0.375
4. 0.450

Question 3 · Multiple choice

Using the same table, what proportion of the sample is in Grade 9?

1. 0.444
2. 0.463
3. 0.500
4. 0.556

Question 4 · Multiple choice

The segmented bar chart shows the preferred news source for random samples of adults in three age groups (counts: Under 30: 52 social media, 30 websites, 18 TV or print; 30 to 59: 35, 40, 45; 60 and over: 8, 22, 70). What proportion of the 30-to-59 group prefers websites?

1. 0.118
2. 0.333
3. 0.400
4. 0.435

Question 5 · Multiple choice

Based on the segmented bar chart of news sources, which statement is best supported?

1. No conclusion is possible without the mean age of each group.
2. There is an association, because the three groups have different sample sizes.
3. There is no association, because every age group uses all three sources.
4. There is an association between age group and preferred news source, because the conditional distributions differ a lot.

Question 6 · Multiple choice

Which graph is most useful for comparing the conditional distributions of a categorical variable across several groups of different sizes?

1. A segmented bar chart of relative frequencies with one bar per group
2. A histogram of the counts
3. A boxplot for each group
4. A single pie chart of the whole sample

Question 7 · Multiple choice

A survey of 250 adults asked whether they own a dog and whether they walk at least 30 minutes daily. How many adults in the survey walk daily?

1. 42
2. 64
3. 106
4. 144

Question 8 · Multiple choice

Using the same dog-owner table, what proportion of the adults surveyed are dog owners who do NOT walk daily?

1. 0.144
2. 0.250
3. 0.360
4. 0.400

Topic 2.2: Joint, marginal and conditional relative frequencies; association

8 multiple-choice questions.

Question 9 · Multiple choice

Using the dog-owner table, what proportion of dog owners walk daily?

1. 0.256
2. 0.424
3. 0.604
4. 0.640

Question 10 · Multiple choice

Using the dog-owner table, what proportion of the adults who walk daily are dog owners?

1. 0.256
2. 0.396
3. 0.604
4. 0.640

Question 11 · Multiple choice

Using the dog-owner table, what is the difference between the proportion of dog owners who walk daily and the proportion of non-owners who walk daily (owners minus non-owners)?

1. 0.09
2. 0.22
3. 0.24
4. 0.36

Question 12 · Multiple choice

Do the data in the dog-owner table suggest an association between owning a dog and walking daily for these adults?

1. No: more non-owners than owners were surveyed.
2. No: 106 walk daily and 144 do not, so walking is not common.
3. Yes: 64% of dog owners walk daily compared with 28% of non-owners.
4. Yes: 64 is greater than 42.

Question 13 · Multiple choice

Students were asked whether they prefer morning or evening study sessions. Is there an association between class and preference for these students?

1. No: the classes have different sizes.
2. Yes: most students prefer evenings.
3. No: the proportion preferring mornings is 0.30 in both classes.
4. Yes: more seniors than juniors prefer mornings.

Question 14 · Multiple choice

Using the morning/evening table, what proportion of the students who prefer evening sessions are seniors?

1. 0.385
2. 0.450
3. 0.550
4. 0.700

Question 15 · Multiple choice

A college recorded whether students in a statistics course passed, by course type. What proportion of the online students passed?

1. 0.240
2. 0.345
3. 0.600
4. 0.758

Question 16 · Multiple choice

Using the course-type table, what proportion of the students who did not pass took the course online?

1. 0.160
2. 0.400
3. 0.475
4. 0.525

Topic 2.3: Estimating probabilities by simulation

8 multiple-choice questions.

Question 17 · Multiple choice

To estimate the probability that a family with 4 children has at least 3 girls, a student simulated 200 families. In 63 of the simulated families there were at least 3 girls. What is the estimated probability?

1. 0.315
2. 0.460
3. 0.630
4. 0.750

Question 18 · Multiple choice

A school raffle claims that 1 in 5 scratch tickets wins. A student simulated buying 20 tickets 30 times, assuming the claim is true; the dotplot shows the number of winning tickets in each trial. Based on the simulation, estimate the probability of getting 7 or more winning tickets out of 20.

1. 0.07
2. 0.10
3. 0.23
4. 0.35

Question 19 · Multiple choice

Using the raffle simulation, a student who bought 20 tickets won 8. Treating a simulated probability below 5% as unusual, is this result convincing evidence that the true win rate is greater than 1 in 5?

1. No: 8 or more winners happened often in the simulation.
2. Yes: 8 or more winners happened in only 1 of 30 simulated trials (about 0.033), which is unusual if the claim is true.
3. No: the simulation used only 30 trials, so nothing can be concluded.
4. Yes: 8 is greater than the expected 4, so the claim must be false.

Question 20 · Multiple choice

A basketball player makes 70% of her free throws. Which assignment of random digits correctly simulates one free throw?

1. Digits 1 to 7 = make, 0, 8 and 9 = miss, skipping repeats
2. Even digits = make, odd digits = miss
3. Digits 0 to 6 = make, 7 to 9 = miss
4. Digits 0 to 7 = make, 8 to 9 = miss

Question 21 · Multiple choice

What does the law of large numbers say about a simulation of tossing a fair coin?

1. As the number of tosses increases, the proportion of heads gets closer to 0.5.
2. After several tails in a row, heads becomes more likely.
3. In every 10 tosses there will be exactly 5 heads.
4. The number of heads minus the number of tails gets closer to 0.

Question 22 · Multiple choice

A simulation of a carnival game was run 500 times and the player won 115 times. Based on the simulation, about how many wins should a player expect in 40 plays?

1. 4.6
2. 9.2
3. 11.5
4. 23.0

Question 23 · Multiple choice

To simulate choosing 3 different students at random from a class of 25, a teacher labels the students 01 to 25 and reads two-digit groups from a random digit table. Which rule must the teacher follow?

1. Read one digit at a time and allow repeats.
2. Ignore 00, groups above 25, and any label already chosen.
3. Use every two-digit group, including those above 25.
4. Stop after reading three two-digit groups, whatever they are.

Question 24 · Multiple choice

Each dot shows the proportion of heads in a simulated set of 20 tosses of a fair coin (25 sets). What proportion of the simulated sets had a proportion of heads of 0.60 or more?

1. 0.12
2. 0.24
3. 0.40
4. 0.60

Topic 2.4: Probability basics: sample spaces and complements

8 multiple-choice questions.

Question 25 · Multiple choice

Two fair six-sided dice are rolled. What is the probability that the sum is 8?

1. 0.091
2. 0.111
3. 0.139
4. 0.167

Question 26 · Multiple choice

A fair coin is tossed 3 times. What is the probability of getting at least one head?

1. 0.125
2. 0.375
3. 0.500
4. 0.875

Question 27 · Multiple choice

The probability that a randomly chosen adult in a town has a library card is 0.62. What is the probability that a randomly chosen adult does NOT have a library card?

1. 0.26
2. 0.38
3. 0.50
4. 0.62

Question 28 · Multiple choice

A restaurant offers 4 appetizers, 6 main dishes and 3 desserts. How many different meals of one appetizer, one main dish and one dessert are possible?

1. 13
2. 24
3. 36
4. 72

Question 29 · Multiple choice

A bag holds the marbles shown in the table. One marble is drawn at random. What is the probability that it is not green?

1. 0.35
2. 0.40
3. 0.60
4. 1.50

Question 30 · Multiple choice

Two fair six-sided dice are rolled. What is the probability that the sum is at least 10?

1. 0.083
2. 0.167
3. 0.273
4. 0.333

Question 31 · Multiple choice

A weather app says the probability of rain tomorrow is 0.3. What is the best interpretation?

1. It will rain in 30% of the area tomorrow.
2. If it rains today, it will not rain tomorrow.
3. It will rain for 30% of tomorrow.
4. In the long run, on days with conditions like tomorrow's, it rains on about 30% of them.

Question 32 · Multiple choice

A spinner has 5 sections with probabilities 0.15, 0.25, 0.20, 0.30 and an unknown value for the fifth section. What is the probability of the fifth section?

1. 0.10
2. 0.20
3. 0.25
4. 0.90

Topic 2.5: Mutually exclusive events and the addition rule

8 multiple-choice questions.

Question 33 · Multiple choice

At a high school, the probability that a randomly chosen student is a freshman is 0.28 and the probability that the student is a sophomore is 0.26. What is the probability that the student is a freshman or a sophomore?

1. 0.0728
2. 0.4600
3. 0.4672
4. 0.5400

Question 34 · Multiple choice

For two events, P(A) = 0.45, P(B) = 0.35 and P(A and B) = 0.12. What is P(A or B)?

1. 0.16
2. 0.68
3. 0.80
4. 0.92

Question 35 · Multiple choice

Using the dog-owner table (dog owner or not; walks daily or not), are the events "is a dog owner" and "walks daily" mutually exclusive?

1. No: 64 adults are both dog owners and daily walkers.
2. No: the events are independent.
3. Yes: the proportion of dog owners is less than 0.5.
4. Yes: being a dog owner and walking daily are different variables.

Question 36 · Multiple choice

In a survey, 40% of customers bought coffee, 25% bought a pastry and 55% bought coffee or a pastry. What percent bought both? Give the answer as a probability.

1. 0.10
2. 0.15
3. 0.30
4. 0.65

Question 37 · Multiple choice

Which pair of events is mutually exclusive when one card is drawn from a standard 52-card deck?

1. The card is red; the card is a heart.
2. The card is a face card; the card is a king.
3. The card is a heart; the card is a king.
4. The card is a heart; the card is a club.

Question 38 · Multiple choice

Using the dog-owner table, what is the probability that a randomly chosen adult from the survey is a dog owner or walks daily?

1. 0.256
2. 0.568
3. 0.824
4. 0.832

Question 39 · Multiple choice

Events A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.5. What is P(A and B)?

1. 0.00
2. 0.15
3. 0.20
4. 0.80

Question 40 · Multiple choice

A fair die is rolled. Let A = "roll an even number" and B = "roll a number greater than 4." What is P(A or B)?

1. 0.167
2. 0.500
3. 0.667
4. 0.833

Topic 2.6: Conditional probability and tree diagrams

8 multiple-choice questions.

Question 41 · Multiple choice

In a club, P(plays chess) = 0.40 and P(plays chess and plays piano) = 0.12. What is the probability that a member plays piano, given that the member plays chess?

1. 0.048
2. 0.120
3. 0.300
4. 0.520

Question 42 · Multiple choice

A test for a disease is used in a population where 4% of people have the disease. The test is positive for 92% of people with the disease and for 6% of people without it. What is the probability that a randomly chosen person tests positive?

1. 0.0368
2. 0.0576
3. 0.0944
4. 0.9800

Question 43 · Multiple choice

Using the disease test above (4% prevalence, positive for 92% with the disease and 6% without), what is the probability that a person who tests positive actually has the disease?

1. 0.037
2. 0.040
3. 0.390
4. 0.920

Question 44 · Multiple choice

A bag holds 6 red and 4 blue tokens. Two tokens are drawn without replacement. What is the probability that both are red?

1. 0.300
2. 0.333
3. 0.360
4. 0.600

Question 45 · Multiple choice

Using the course-type table (online or in person; passed or not), what is the probability that a randomly chosen student passed, given that the student took the course in person?

1. 0.455
2. 0.600
3. 0.655
4. 0.758

Question 46 · Multiple choice

At a factory, Machine 1 makes 70% of the parts and Machine 2 makes 30%. Of Machine 1's parts, 2% are defective; of Machine 2's, 5% are defective. What proportion of all parts are defective?

1. 0.014
2. 0.029
3. 0.035
4. 0.070

Question 47 · Multiple choice

At the same factory, Machine 1 makes 70% of the parts (2% defective) and Machine 2 makes 30% (5% defective). A part is found to be defective. What is the probability it came from Machine 2?

1. 0.050
2. 0.300
3. 0.483
4. 0.517

Question 48 · Multiple choice

Which statement about conditional probability is correct?

1. P(A | B) always equals P(B | A).
2. P(A | B) = P(A) + P(B).
3. P(A | B) and P(B | A) are usually different.
4. P(A | B) is always smaller than P(A).

Topic 2.7: Independent events and unions

8 multiple-choice questions.

Question 49 · Multiple choice

A traffic light on Elm Street is green on arrival 45% of the time and an independent light on Oak Street is green 60% of the time. What is the probability that both are green when a driver arrives?

1. 0.15
2. 0.27
3. 0.78
4. 1.05

Question 50 · Multiple choice

A website's login fails 3% of the time, independently from attempt to attempt. What is the probability of at least one failure in 10 attempts?

1. 0.030
2. 0.263
3. 0.300
4. 0.737

Question 51 · Multiple choice

Events C and D have P(C) = 0.6, P(D) = 0.35 and P(C and D) = 0.21. Are C and D independent?

1. It cannot be determined.
2. No, because P(C and D) is not 0.
3. No, because P(C) is not equal to P(D).
4. Yes, because P(C and D) = P(C) · P(D).

Question 52 · Multiple choice

In the morning/evening study table, is the event "is a senior" independent of "prefers morning sessions"?

1. No: there are more seniors than juniors.
2. No: 33 is not equal to 27.
3. Yes: the events are mutually exclusive.
4. Yes: P(morning | senior) = 0.30 = P(morning).

Question 53 · Multiple choice

In the dog-owner table, are "is a dog owner" and "walks daily" independent events?

1. No: the events are mutually exclusive.
2. Yes: 64 is less than 100.
3. Yes: the two variables were measured on the same people.
4. No: P(walks daily | dog owner) = 0.64 but P(walks daily) = 0.424.

Question 54 · Multiple choice

A quiz has 4 multiple-choice questions, each with 4 options. A student guesses at random on every question. What is the probability that the student gets all 4 correct?

1. 0.0039
2. 0.0625
3. 0.2500
4. 1.0000

Question 55 · Multiple choice

Two independent events have P(A) = 0.3 and P(B) = 0.6. What is P(A or B)?

1. 0.18
2. 0.28
3. 0.72
4. 0.90

Question 56 · Multiple choice

A smoke detector works 95% of the time. A home has three detectors that work independently. What is the probability that at least one works in a fire?

1. 0.850000
2. 0.857375
3. 0.950000
4. 0.999875

Topic 2.8: Discrete random variables and their probability distributions

8 multiple-choice questions.

Question 57 · Multiple choice

The table shows the probability distribution of X, the number of pets in a randomly chosen household in a town. What is the missing probability for X = 2?

1. 0.20
2. 0.25
3. 0.40
4. 0.60

Question 58 · Multiple choice

Using the completed pets distribution (P(X = 2) = 0.40), what is P(X ≥ 2)?

1. 0.25
2. 0.40
3. 0.65
4. 0.75

Question 59 · Multiple choice

Using the completed pets distribution, what is P(X < 2)?

1. 0.10
2. 0.25
3. 0.35
4. 0.75

Question 60 · Multiple choice

A loaded die has the probability distribution shown. What is the probability of rolling an odd number?

1. 0.42
2. 0.47
3. 0.50
4. 0.58

Question 61 · Multiple choice

Which of the following could be a probability distribution for a discrete random variable with values 1, 2 and 3?

1. P(1) = 0.3, P(2) = 0.3, P(3) = 0.3
2. P(1) = 0.6, P(2) = 0.5, P(3) = −0.1
3. P(1) = 0.4, P(2) = 0.4, P(3) = 0.4
4. P(1) = 0.2, P(2) = 0.5, P(3) = 0.3

Question 62 · Multiple choice

Using the loaded die distribution, what is P(2 ≤ X ≤ 4)?

1. 0.30
2. 0.35
3. 0.45
4. 0.57

Question 63 · Multiple choice

Let X be the number of customers who complain on a randomly chosen hour at a help desk, with the distribution shown. What is the cumulative probability P(X ≤ 1)?

1. 0.30
2. 0.40
3. 0.70
4. 0.90

Question 64 · Multiple choice

Using the help-desk distribution, what is the probability that there is at least one complaint in a randomly chosen hour?

1. 0.10
2. 0.30
3. 0.40
4. 0.70

Topic 2.9: Mean and standard deviation of a discrete random variable

8 multiple-choice questions.

Question 65 · Multiple choice

Using the pets distribution (X = 0, 1, 2, 3, 4 with probabilities 0.10, 0.25, 0.40, 0.20, 0.05), what is the expected number of pets, E(X)?

1. 0.20
2. 1.50
3. 1.85
4. 2.00

Question 66 · Multiple choice

Using the same pets distribution, what is the standard deviation of X?

1. 0.500
2. 1.014
3. 1.028
4. 1.581

Question 67 · Multiple choice

Using the help-desk distribution (X = 0, 1, 2, 3 complaints with probabilities 0.30, 0.40, 0.20, 0.10), what is the mean number of complaints per hour?

1. 0.25
2. 1.00
3. 1.10
4. 1.50

Question 68 · Multiple choice

A carnival game costs \$2 to play. A player wins nothing (net −\$2) with probability 0.70, wins \$5 (net \$3) with probability 0.25, or wins \$20 (net \$18) with probability 0.05. The table shows the net gain X. What is the expected net gain per play?

1. 0.25
2. 0.75
3. 1.50
4. 6.33

Question 69 · Multiple choice

In the carnival game above, the expected net gain is \$0.25. What does this mean?

1. A player who plays 4 times will win exactly \$1.
2. Most plays result in a gain of \$0.25.
3. Over many plays, a player's average net gain will be about \$0.25 per play.
4. Each play wins exactly \$0.25.

Question 70 · Multiple choice

A random variable X takes the values 10, 20 and 30 with probabilities 0.2, 0.5 and 0.3. What is the variance of X?

1. 7.00
2. 10.00
3. 21.00
4. 49.00

Question 71 · Multiple choice

Using the same distribution (values 10, 20, 30 with probabilities 0.2, 0.5, 0.3), what is the standard deviation of X?

1. 7.000
2. 8.165
3. 21.000
4. 49.000

Question 72 · Multiple choice

X is the number of heads in two tosses of a fair coin, with the distribution shown. What is the standard deviation of X?

1. 0.250
2. 0.500
3. 0.707
4. 1.000

Topic 2.10: Binomial distributions

8 multiple-choice questions.

Question 73 · Multiple choice

A basketball player makes 80% of her free throws, and shots are independent. She takes 10 free throws. What is the probability she makes exactly 8?

1. 0.168
2. 0.302
3. 0.678
4. 0.800

Question 74 · Multiple choice

Using the same free-throw shooter (p = 0.8, n = 10 shots), what is the probability she makes at least 9?

1. 0.107
2. 0.268
3. 0.376
4. 0.624

Question 75 · Multiple choice

About 12% of people are left-handed. In a random sample of 25 people, what is the mean number of left-handed people?

1. 1.62
2. 2.64
3. 3.00
4. 12.00

Question 76 · Multiple choice

Using the same setting (n = 25, p = 0.12), what is the standard deviation of the number of left-handed people?

1. 0.325
2. 1.625
3. 2.640
4. 3.000

Question 77 · Multiple choice

A multiple-choice quiz has 12 questions with 4 options each. A student guesses at random on every question. What is the probability the student gets at most 2 correct?

1. 0.232
2. 0.391
3. 0.500
4. 0.649

Question 78 · Multiple choice

Which situation describes a binomial random variable?

1. The number of heads in 15 tosses of a fair coin
2. The number of red cards when 5 cards are drawn without replacement from a 10-card hand with 4 red cards
3. The number of tosses of a coin until the first head
4. The total weight of 15 randomly chosen apples

Question 79 · Multiple choice

A shipping company says 95% of packages arrive on time. A store receives 20 randomly selected packages (assume independence). What is the probability that all 20 arrive on time?

1. 0.050
2. 0.358
3. 0.736
4. 0.950

Question 80 · Multiple choice

In a large city, 30% of adults have a library card. For a random sample of 15 adults, what is the probability that between 3 and 6 (inclusive) have a library card?

1. 0.425
2. 0.500
3. 0.742
4. 0.873

Topic 2.11: Normal distributions

8 multiple-choice questions.

Question 81 · Multiple choice

The lengths of a species of fish in a lake are approximately normal with mean 32 cm and standard deviation 4 cm. What proportion of these fish are shorter than 27 cm?

1. 0.1056
2. 0.1250
3. 0.2500
4. 0.8944

Question 82 · Multiple choice

SAT-style scores on a test are approximately normal with mean 500 and standard deviation 100. What proportion of scores are above 650?

1. 0.0228
2. 0.0668
3. 0.1500
4. 0.9332

Question 83 · Multiple choice

The fill weights of cereal boxes are approximately normal with mean 368 g and standard deviation 6 g. What proportion of boxes weigh between 360 g and 375 g?

1. 0.5000
2. 0.7871
3. 0.8783
4. 0.9088

Question 84 · Multiple choice

Reaction times for a video game task are approximately normal with mean 250 ms and standard deviation 30 ms. What reaction time marks the fastest 10% (the 10th percentile)?

1. 200.7
2. 211.6
3. 240.0
4. 288.4

Question 85 · Multiple choice

Adult heights in a population are approximately normal with mean 170 cm and standard deviation 8 cm. Using the empirical rule, about what proportion of adults are between 154 cm and 186 cm?

1. 0.47
2. 0.68
3. 0.95
4. 1.00

Question 86 · Multiple choice

Battery lifetimes are approximately normal with mean 40 hours and standard deviation 5 hours. A manufacturer wants to guarantee a lifetime that 98% of batteries exceed. What lifetime should it guarantee?

1. 29.7
2. 30.0
3. 35.1
4. 50.3

Question 87 · Multiple choice

A student scored 84 on a test whose scores are approximately normal with mean 76 and standard deviation 5. What proportion of scores are below 84 (her percentile, as a proportion)?

1. 0.0548
2. 0.8400
3. 0.8849
4. 0.9452

Question 88 · Multiple choice

Daily water use per household in a town is approximately normal with mean 300 liters and standard deviation 40 liters. What is the third quartile ($Q_3$) of daily water use?

1. 324.0
2. 327.0
3. 330.0
4. 340.0

Topic 2.12: The idea of a sampling distribution

8 multiple-choice questions.

Question 89 · Multiple choice

A poll of 1,000 randomly chosen voters finds that 46% favor a ballot measure. Which statement is correct?

1. 46% is a parameter because it describes the voters polled.
2. 46% is the proportion of all voters who favor the measure.
3. 46% is a statistic that estimates the parameter: the proportion of all voters who favor the measure.
4. 46% is a parameter, and the proportion of all voters is a statistic.

Question 90 · Multiple choice

What is a sampling distribution?

1. The distribution of the values of a statistic over all possible samples of the same size from the same population
2. A list of the samples a researcher chose
3. The distribution of the values in one sample
4. The distribution of the values in the population

Question 91 · Multiple choice

The standard deviation of a sample statistic is proportional to $\frac{1}{\sqrt n}$. If the sample size is increased from 100 to 400, by what factor is the standard deviation multiplied?

1. 0.25
2. 0.50
3. 2.00
4. 4.00

Question 92 · Multiple choice

A researcher wants to cut the standard deviation of a sample mean to $\frac{1}{3}$ of its value with samples of 50. What sample size is needed?

1. 150
2. 300
3. 450
4. 900

Question 93 · Multiple choice

The dotplot shows the sample means from 20 simulated random samples of size 25 from a population. What proportion of the simulated sample means are 3.9 or greater?

1. 0.10
2. 0.15
3. 0.20
4. 0.39

Question 94 · Multiple choice

Two simulated sampling distributions of a sample proportion are made, one for samples of 30 and one for samples of 300, from the same population. How do they compare?

1. The distribution for samples of 30 is less spread out.
2. Both are centered near the population proportion, but the distribution for samples of 300 is much less spread out.
3. The distribution for samples of 300 is centered closer to 0.5.
4. Both have the same spread, but different centers.

Question 95 · Multiple choice

Why do two different random samples from the same population usually give different sample means?

1. The population mean changed between samples.
2. One of the samples must be biased.
3. Sampling variability: each random sample contains different individuals.
4. The samples were too large.

Question 96 · Multiple choice

In a simulation, 200 random samples of size 40 were taken from a population, and the sample proportions had standard deviation 0.078. About what standard deviation would you expect for samples of size 160 from the same population?

1. 0.0195
2. 0.0390
3. 0.0780
4. 0.1560

Free response

Write a complete answer, in context, as you would type it on the exam. The scoring guides are ours, written in the new course's style: 10 points per question, each point scored 0 or 1 and earned independently (no E/P/I part scores). They are not College Board scoring guidelines.

Question 97 · Free response (10 points)

A music service surveyed 400 randomly selected people about whether they stream music daily. The results by age group are shown in the table.

1. (a) What proportion of the people surveyed stream music daily? [1 point]
2. (b) What proportion of people aged 18-34 stream music daily? [2 points]
3. (c) Are the events "aged 18-34" and "streams music daily" independent? Justify your answer. [3 points]
4. (d) Suppose the proportion of all people in the population who stream daily is 0.628. Three people are chosen at random from this large population. Find the probability that at least one of them does NOT stream music daily. [2 points]
5. (e) Does the table suggest an association between age group and daily streaming? Explain using the table. [2 points]

Question 98 · Free response (10 points)

A bakery's sourdough loaves are approximately normally distributed in weight with mean 820 grams and standard deviation 15 grams. A loaf is labeled "underweight" if it weighs less than 800 grams.

1. (a) Find the probability that a randomly chosen loaf is underweight. Show your work. [2 points]
2. (b) A store receives 12 randomly chosen loaves. Explain why the number of underweight loaves can be modeled by a binomial distribution. [2 points]
3. (c) Find the probability that at least 2 of the 12 loaves are underweight. [2 points]
4. (d) Find the mean and standard deviation of the number of underweight loaves in a shipment of 12, and interpret the mean. [4 points]