AP Statistics Unit 2 Probability

[Maximum number: 4]

A local elementary school decided to sell bottles printed with the school district's logo as a fund-raiser. The students in the elementary school were asked to sell bottles in three different sizes (small, medium, and large). The relative frequencies of the number of bottles sold for each size by the elementary school were 0.5 for small bottles, 0.3 for medium bottles, and 0.2 for large bottles.
A local middle school also decided to sell bottles as a fund-raiser, using the same three sizes (small, medium, and large). The middle school students sold three times the number of bottles that the elementary school students sold. For the middle school students, the proportion of bottles sold was equal for all three sizes.

(a)

Complete the segmented bar graphs representing the relative frequencies of the number of bottles sold for each size by students at each school.

(b)

An administrator at the elementary school concluded that the elementary school students sold more small bottles than the middle school students did. Is the elementary school administrator's conclusion correct? Explain your response.

No, the segment for small bottles for the elementary school is wider than the segment for small bottles for the middle school; however, the middle school students sold three times as many bottles as the elementary school students. So, if the elementary school sold x number of bottles, the middle school sold 3 x number of bottles.

For example, if the elementary students sold 100 bottles total, then they sold 0.5(100) or 50 small bottles. However, because the middle school students sold three times the total number of bottles as the elementary students, they would have sold 300 bottles total and $(0. \overline{3})(300)=100$ small bottles. Because 100 > 50, the middle school sold more small bottles, and the elementary school's administrator is not correct.

Scoring
Essentially correct (E) if the response satisfies the following three components:

States that the elementary school's administrator is incorrect

Provides correct mathematical support verifying the middle school sold more small bottles than the elementary school, consistent with the response to component 1

Includes context

Partially correct (P) if the response satisfies only two of the three components required for E.

Incorrect (I) if the response does not meet the criteria for E or P.

Additional Notes:

- A response that shows mathematical support by showing the inequalities $0.5 x<0. \overline{3}(3 x), 0.5 x<x$ , $0.5(1)<0. \overline{3}(3)$ , or $(0.5)\left(\frac{1}{4}\right)<(0. \overline{3})\left(\frac{3}{4}\right)$ may satisfy component 2.

- In order to satisfy component 3, a response must include "elementary," "middle," and "bottles."

- Part (b) should be scored consistent with the response to part (a).

(c)

Two high schools are also selling the bottles and are competing to see which one sold more large bottles.

A mosaic plot for the distribution of the number of bottles sold by each of the high schools is shown here.

(i)

Which of the two high schools sold a greater proportion of large bottles? Justify your answer.

(ii)

Which of the two high schools sold a greater number of large bottles? Justify your answer.

[Maximum number: 4]

Baseball cards are trading cards that feature data on a player's performance in baseball games. Michelle is at a national baseball card collector's convention with approximately 20,000 attendees. She notices that some collectors have both regular cards, which are easily obtained, and rare cards, which are harder to obtain. Michelle believes that there is a relationship between the number of months a collector has been collecting baseball cards and whether the majority of the cards (cards appearing more often) in their collection are regular or rare. She obtains information from a random sample of 500 baseball card collectors at the convention and records how many full months they have been collecting baseball cards and whether the majority of the cards in their card collection are regular or rare. Her results are displayed in a two-way table.

	Fewer Than 6 Months	6-10 Months	11-15 Months	16-20 Months	21 or More Months	Total
Has a Majority of Regular Baseball Cards	80	84	71	76	112	423
Has a Majority of Rare Baseball Cards	11	16	9	6	35	77
Total	91	100	80	82	147	500

Majority Type of Baseball Cards and Months of Collecting Baseball Cards

(a)

If one collector from the sample is selected at random, what is the probability that the collector has been collecting baseball cards for 11 or more months and has a majority of regular baseball cards? Show your work.

Model Solution
$P(\text{11+ months and majority regular})=\frac{71+76+112}{500}=\frac{259}{500}=0.518.$

Scoring components
1. Provides the correct probability.
2. Shows how the numerator 259 is computed.

(b)

Given that a randomly selected collector from the sample has been collecting baseball cards for fewer than 6 months, what is the probability the collector has a majority of regular baseball cards? Show your work.

$\begin{aligned} P(\text { majority regular cards } \mid \text { fewer than } 6 \text { months }) =\frac{P(\text { majority regular cards } \cap \text { fewer than } 6 \text { months })}{P(\text { fewer than } 6 \text { months })} =\frac{\frac{80}{500}}{\frac{91}{500}} =\frac{80}{91} =0.879 \end{aligned}$

Essentially correct (E) if the response satisfies the following two components:

Provides the correct probability

Shows work for the correct probability

Partially correct (P) if the response satisfies only one of the two components required for E.

Incorrect (I) if the response does not meet the criteria for E or P.

Additional Notes:

- An arithmetic or transcription error in a response can be ignored if correct work is shown.

- A response of $\frac{80}{91}$ satisfies both components 1 and 2.

- A response that satisfies components 1 and 2 for the probability that given a randomly selected collector has been collecting baseball cards for fewer than six months, they have a majority of rare baseball cards may be scored P.

- A specific probability statement is not required, but if correctly given should be considered a positive in holistic scoring.

[Maximum number: 4]

Nine sales representatives, 6 men and 3 women, at a small company wanted to attend a national convention. There were only enough travel funds to send 3 people. The manager selected 3 people to attend and stated that the people were selected at random. The 3 people selected were women. There were concerns that no men were selected to attend the convention.

(a)

Calculate the probability that randomly selecting 3 people from a group of 6 men and 3 women will result in selecting 3 women.

Part (a):
The probability that all 3 people selected are women can be calculated using the multiplication rule, as follows:

\begin{aligned} & P(\text { all three selected are women }) \\ & =P(\text { first is a woman }) \times P(\text { second is a woman } \mid \text { first is a woman }) \times P(\text { third is a woman } \mid \text { first two are women }) \\ & =\frac{3}{9} \times \frac{2}{8} \times \frac{1}{7} \approx 0.012 \end{aligned}

(b)

Based on your answer to part (a), is there reason to doubt the manager's claim that the 3 people were selected at random? Explain.

(c)

An alternative to calculating the exact probability is to conduct a simulation to estimate the probability. A proposed simulation process is described below.

Each trial in the simulation consists of rolling three fair, six-sided dice, one die for each of the convention attendees. For each die, rolling a 1, 2, 3, or 4 represents selecting a man; rolling a 5 or 6 represents selecting a woman. After 1,000 trials, the number of times the dice indicate selecting 3 women is recorded.

Does the proposed process correctly simulate the random selection of 3 women from a group of 9 people consisting of 6 men and 3 women? Explain why or why not.

Part (c):

No, the process does not correctly simulate the random selection of three women from a group of nine people of whom six are men and three are women. The random selection of three people among nine is done without replacement. However, in the simulation with the dice, the three dice rolls in any given trial are independent of one another, indicating a selection process that is done with replacement.

Scoring

Parts (a), (b), and (c) were scored as essentially correct (E), partially correct (P), or incorrect (I).
Part (a) is scored as follows:
Essentially correct (E) if the response correctly computes the probability of selecting the three women, and shows how the probability was computed.

Partially correct (P) if the response shows only one of the following:
Gives the correct probability of $\frac{1}{84}$ (0.012 or 0.011 is acceptable) but does not show how it was computed;

OR
Correctly shows how the probability should be computed, but does not carry the computation through correctly;

OR
Correctly computes (showing work) only the numerator, or only the denominator of the correct answer. (For example, $\frac{1}{9} \times \frac{1}{8} \times \frac{1}{7} \approx 0.002$ , or $\frac{3}{9} \times \frac{2}{9} \times \frac{1}{9} \approx 0.008$ , or $\frac{3}{9} \times \frac{3}{8} \times \frac{3}{7} \approx 0.054$ );

OR
Mistakenly assumes independence and calculates (showing work) the binomial probability $\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3}=\frac{1}{27} \approx 0.037$ .

Incorrect (I) if the response does not meet the criteria for E or P.
Part (b) is scored as follows:
Essentially correct (E) if the response states that the probability from part (a) is small (or insufficiently small), makes an appropriate decision consistent with the probability being small (or insufficiently small), and does so in the context of this situation.

Partially correct (P) if the response shows only one of the following:
Otherwise satisfies the criteria for an E but does so without any context;
OR
States a significance level and makes a decision in context that is appropriate to the given probability in part (a) and the stated significance level, but does not explicitly compare the probability and the significance level;

OR
Otherwise satisfies the criteria for an E but does not explicitly make a decision about whether there is reason to doubt the manager's claim. (For example: "The probability of selecting the three women from among the nine employees is very small so it is unlikely to occur by chance.")

Incorrect (I) if the response does not meet the criteria for E or P.

Notes:

- Each of the following situations is one in which a response that otherwise would be scored as E should be scored as P, and a response that otherwise would be scored as P should be scored as I:

The response includes a statement that the small probability proves that the manager did not make the selection at random (or any equivalent wording).

The response includes a statement that clearly interprets the probability from part (a) to be the probability that the manager selected the three people at random.

- Each of the following situations is one in which the response is scored as I:

The decision is inconsistent with the justification (e.g., "The probability is very small, so there is no reason to doubt the manager's claim").

- The response states or implies that because the selection of three women was not impossible, there is no reason to doubt the manager's claim.

Part (c) is scored as follows:
Essentially correct (E) if the response answers no AND states that the dice outcomes in the proposed simulation are independent AND states that the genders of the selected convention attendees are dependent. The table below shows statements that should be considered equivalent to the required statements of independence and dependence.

Independence of dice outcomes

Dependence of genders

- The probability of rolling a 5 or a 6 is the same on all three dice.
- The dice simulation actually simulates sampling with replacement.

- The genders of the three people are dependent (or not independent).

- The probability of selecting a woman changes after each selection.

- The people are sampled without replacement.

- The genders of the three people are dependent
(or not independent).
- The probability of selecting a woman changes after each selection.

OR
Essentially correct if the response answers no AND computes the correct probability that a trial of the simulation will indicate the selection of three women $\left(\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \approx 0.037\right)$ AND states that the probability is different from the probability found in part (a).

Partially correct (P) if the response correctly answers no and either:
States only that the dice outcomes are independent or states only that the genders of the selected convention attendees are dependent, but not both;

OR
Otherwise meets the criteria for E but has poor communication. An example of poor communication is: "No, because it selects with replacement. It isn't possible for the same person to be selected twice." (There is an apparent shift between the two sentences from describing the simulation to describing the actual selection of people, but that is not made clear.)

Incorrect (I) if the response does not meet the criteria for E or P.
Note: Pointing out that a sample of three people is more than 10\% of the population of nine people should be considered equivalent to stating that the selection of a woman is not independent among the three people selected to attend the convention.

4 Complete Response

All three parts essentially correct

3 Substantial Response

Two parts essentially correct and one part partially correct

2 Developing Response

Two parts essentially correct and one part incorrect

OR

One part essentially correct and one or two parts partially correct

OR

Three parts partially correct

1 Minimal Response

One part essentially correct and two parts incorrect

OR

Two parts partially correct and one part incorrect

[Maximum number: 4]

Ms. Fey is a manager at a restaurant. To improve the dining experience for her customers, she

uses a digital music service to create a playlist of songs that will be played in the restaurant.

The playlist contains 1,000 songs and consists of four different types of music in the following

quantities: 200 country songs, 400 pop songs, 100 rock songs, and 300 jazz songs. The digital

music service will select songs at random from the playlist to be played in the restaurant. Any

song can be replayed at any time.

(a)

A.

(i)

Suppose one song is selected at random to be played. What is the probability that the

song is a rock song? Show your work.

Model Solution
$P(\text{rock song})=\frac{100}{1000}=0.10.$

Scoring components
1. Calculates 0.10.
2. Shows supporting work.

(ii)

Suppose two songs are selected at random to be played. What is the probability that both

songs are rock songs? Show your work.

Model Solution
Because any song can be replayed, the selections are independent:
$P(\text{both rock})=(0.10)(0.10)=0.01.$

Scoring components
3. Calculates the probability consistently with part A(i).
4. Shows supporting work consistent with part A(i).

(b)

In every one-hour period, 20 songs will be played at random and any song can be replayed at

any time. Ms. Fey is interested in how many rock songs will be played in a typical one-hour

period.

(i)

Define the random variable of interest to Ms. Fey, and state how the random variable is

distributed.

(ii)

What is the expected value for the random variable in part B (i)? Show your work.

Model Solution
$E(X)=np=20(0.10)=2\text{ songs}.$

Scoring component
4. Correctly calculates the expected value and shows supporting work. The parameters may also be stated here.

(c)

Recall that in every one-hour period, 20 songs will be played at random and any song can be

replayed at any time.

(i)

Determine the probability that 4 or more rock songs in a particular one-hour period will

be played. Show your work.

Model Solution
$\begin{aligned} P(X\ge4)&=1-P(X\le3)\\ &=1-\sum_{x=0}^{3}\binom{20}{x}(0.10)^x(0.90)^{20-x}\\ &=1-0.867=0.133. \end{aligned}$

Scoring components
1. Gives the correct probability.
2. Shows work supporting the probability.

(ii)

Suppose 4 rock songs are played during a particular one-hour period. Does this provide

strong evidence that the song selection process was not truly random? Justify your

answer without performing an inference procedure.

AP Statistics Unit 2: Probability and Distributions

Covered syllabus points

Covered syllabus points

2 Probability, Random Variables, and Probability Distributions question 1

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)

2 Probability, Random Variables, and Probability Distributions question 2

Question (a)

Question (b)

2 Probability, Random Variables, and Probability Distributions question 3

Question (a)

Question (b)

Question (c)

2 Probability, Random Variables, and Probability Distributions question 4

Question (a)

Question (i)

Question (ii)

Question (b)

Question (i)

Question (ii)

Question (c)

Question (i)

Question (ii)