Review AP Statistics Unit 2 through probability rules, conditional events, random variables, binomial and normal models, and sampling distributions.
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.
Complete the segmented bar graphs representing the relative frequencies of the number of bottles sold for each size by students at each school.
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.
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.
Distribution of the Number of Bottles Sold by High School
Which of the two high schools sold a greater proportion of large bottles? Justify your answer.
Which of the two high schools sold a greater number of large bottles? Justify your answer.
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.
Majority Type of Baseball Cards and Months of Collecting Baseball Cards
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.
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.
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.
Calculate the probability that randomly selecting 3 people from a group of 6 men and 3 women will result in selecting 3 women.
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.
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.
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.
Suppose one song is selected at random to be played. What is the probability that the song is a rock song? Show your work.
Suppose two songs are selected at random to be played. What is the probability that both songs are rock songs? Show your work.
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.
Define the random variable of interest to Ms. Fey, and state how the random variable is distributed.
What is the expected value for the random variable in part B (i)? Show your work.
Recall that in every one-hour period, 20 songs will be played at random and any song can be replayed at any time.
Determine the probability that 4 or more rock songs in a particular one-hour period will be played. Show your work.
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.