A large fish tank is filled with table tennis balls with numbers written on them. Jada chooses 10 table tennis balls from the tank and writes down their numbers.
1
3
5
1
3
2
4
1
5
3
A second tank is filled with golf balls with numbers written on them. Jada chooses 10 golf balls from the tank and writes down their numbers.
1
4
5
2
6
2
2
1
4
8
To win a prize, Jada must get a ball with an even number. Should she try to win the prize using the tank of table tennis balls or the tank of golf balls? Explain your reasoning.
Show Solution
Jada should use the tank of golf balls. Sample reasoning: From the tank of table tennis balls, Jada only gets 2 even numbers out of the 10 she chooses. From the tank of golf balls, she gets 7 even numbers out of the 10 she chooses. There seems to be a better chance of her getting even-numbered balls from the tank that has golf balls.
According to market research, a business has a 75% chance of making money in the first three years.
According to lab testing, 65 of a certain kind of experimental light bulb will work after three years.
According to experts, the likelihood of a car needing major repairs in the first three years is 0.7.
Write these situations in order of likelihood from least to greatest after three years: The business makes money, the light bulb still works, and the car needs major repairs.
Select one of the situations. Write a chance experiment and event that has the same likelihood as the situation you selected.
Show Solution
The car needs major repairs, the business makes money, the light bulb still works.
Sample responses:
flipping two coins and at least one not landing on heads
rolling a standard number cube and it landing with any number other than 1 face up
selecting a number greater than 3 when selecting a number between 1 and 10 randomly
A mother decides to teach her son about a letter each day of the week. She will choose a letter from the name of the day. For example, on Saturday she might teach about the letter S or the letter U, but not the letter M.
What letters are possible to teach using this method? (There are 15.)
What are 4 letters that can't be taught using this method?
On TUESDAY, the mother writes the word on a piece of paper and cuts it up so that each letter is on a separate piece of paper. She mixes up the papers and picks one. What is the probability that she will choose the piece of paper with the letter Y? Explain your reasoning.
Show Solution
A, D, E, F, H, I, M, N, O, R, S, T, U, W, Y
Any 4 of B, C, G, J, K, L, P, Q, V, X, Z
71. Sample reasoning: There are 7 outcomes in the sample space, all outcomes are equally likely, and there is only 1 outcome that corresponds to the letter Y.
A librarian is curious about the habits of the library's patrons. He records the type of item that the first 10 patrons check out from the library.
patron
item type
1
fiction book
2
nonfiction book
3
fiction book
4
fiction book
5
audiobook
6
nonfiction book
7
DVD
8
nonfiction book
9
fiction book
10
DVD
Based on the information from these patrons . . .
Estimate the probability that the next patron will check out a fiction book. Explain your reasoning.
Estimate the number of DVDs that will be checked out for every 100 patrons. Explain your reasoning.
Show Solution
104 or equivalent. Sample reasoning: 4 of the 10 patrons in the list checked out fiction books.
20 DVDs. Sample reasoning: Since 2 of the 10 patrons in the list checked out DVDs, which is 51 of the patrons, we can expect 51 of every 100 patrons to check out DVDs. 51 of 100 is 20.
Jada, Diego, and Elena each use the same spinner that has four (not necessarily equal sized) sections marked A, B, C, and D.
Jada says, “The probability of spinning B is 0.3 because I spun 10 times, and it landed on B 3 times.”
Diego says, “The probability of spinning B is 20% because I spun 5 times, and it landed on B once.”
Elena says, “The probability of spinning B is 72 because I spun 7 times, and it landed on B twice.”
Based on their methods, which probability estimate do you think is the most accurate? Explain your reasoning.
Andre measures the spinner and finds that the B section takes up 41 of the circle. Explain why none of the methods match this probability exactly.
Show Solution
Sample response:
Jada's method is probably the most accurate since she had the most attempts.
Since Jada spun it 10 times, she could only get estimates in increments of 0.1. Since Diego spun it 5 times, he could only get estimates in increments of 20%. Since Elena spun it 7 times, she could only get estimates in increments of 71. If they had spun the spinner more times, their results would probably get closer to 41.
In a video game, the chance of rain each day is always 30%. At the beginning of each day in the video game, the computer generates a random integer between 1 and 50. Explain how you could use this number to simulate the weather in the video game.
Show Solution
Sample response: If the number is between 1 and 15, the video game should create a rainy day. If the number is between 16 and 50, the video game should not create a rainy day.
Section A Check
Section A Checkpoint
Problem 1
A kindergarten teacher has a set of 20 cards to help students learn to count. Each card has a number of animals on it, from 1 to 20.
One child can count to 15 well, but has problems with 16 and after. When this child picks a card at random, what is the probability that it will be a number they have problems with?
Show Solution
205,41,25% (or equivalent)
Problem 2
An environmental scientist can determine the health of a river based on whether a particular insect is present when they collect a sample of water. The scientist and students collect 25 samples from different parts of a river and notice that the insect is present in 12 of them.
The river is considered healthy if the probability of finding the insect in a sample is 21. Do you think this river could be considered healthy? Explain your reasoning.
What could the scientist do to better estimate the probability of finding the insect in a sample of river water?
Show Solution
Sample response:
I think it might be considered healthy. Sample reasoning: With a probability of 21, we might expect that 12.5 of the 25 samples have the insect. Because 12 samples had the insect, and that is pretty close to the expected 12.5, it might be close to a probability of 21.
The probability of a certain brand of battery going dead within 15 hours is 31. Noah has a toy that requires 4 of these batteries. He wants to estimate the probability that at least one battery will die before 15 hours are up.
Noah will simulate the situation by putting marbles in a bag. Drawing one marble from the bag will represent the outcome of one of the batteries in the toy after 15 hours. Red marbles represent a battery that dies before 15 hours are up, and green marbles represent a battery that lasts longer.
How many marbles of each color should he put in the bag? Explain your reasoning.
After doing the simulation 5 times, Noah has these results. What should he use as an estimate of the probability that at least one battery will die within 15 hours?
trial
result
1
GGRG
2
GRGR
3
GGGG
4
RGGG
5
GGGR
Show Solution
1 red marble and 2 green marbles (or some multiple of these). Sample reasoning:Based on the probability of each battery dying, 31 of the marbles should be red.
Lin plays a game that involves a standard number cube and a deck of ten cards numbered 1 through 10. If both the cube and card have the same number, Lin gets another turn. Otherwise, play continues with the next player.
What is the probability that Lin gets another turn?
Show Solution
606 (or equivalent), since there are 6 outcomes for which the numbers match and 60 equally likely outcomes in the sample space (6⋅10=60)
Elena is programming a video game. She needs to simulate the power-up that the player gets when they reach a certain level. The computer can run a program to return a random integer between 1 and 100. Elena wants the best power-up to be given 15% of the time.
Explain how Elena could use the computer to simulate the player getting the best power-up at least 2 out of 3 times.
Show Solution
Sample response: Elena could have the computer generate 3 random integers between 1 and 100. If at least 2 of the numbers are between 1 and 15, then the player gets the best power-up at least twice. She could repeat this process many times and estimate the probability as the proportion of trials for which at least 2 of the numbers are between 1 and 15.
Section B Check
Section B Checkpoint
Problem 1
A cube is labeled so that when it is rolled, it shows only 1, 2, or 3 with equal chances of each one coming up. This cube is rolled, then 2 different coins are flipped.
How many outcomes are in the sample space? Explain your reasoning.
What is the probability that the cube shows 2, and the coins land showing the same face? Explain your reasoning.
Show Solution
12. Sample reasoning: The outcomes are 1HH, 1HT, 1TH, 1TT, 2HH, 2HT, 2TH, 2TT, 3HH, 3HT, 3TH, 3TT.
122 or equivalent. Sample reasoning: There are 2 outcomes that match the situation: 2HH and 2TT. There are 12 outcomes in the sample space, so the probability is 122.
Problem 2
A computer game shows that opening a prize box has a 25% chance that it contains your favorite character to play in the game.
How could you use coins to simulate opening 3 prize boxes to find the probability of getting your favorite character?
Show Solution
Sample response: Flip 2 coins three times to simulate opening 3 boxes. If both coins show heads any of the times, then it represents getting my favorite character. To find the probability, repeat this process many times and find the fraction of times this happens.
Noah's parents are interested in moving to another part of town. They look up all the prices of the homes for sale and record them in thousands of dollars.
neighborhood 1 (mean: 75, MAD: 15)
80
55
80
120
60
90
60
80
55
70
neighborhood 2 (mean: 124, MAD: 15.6)
110
120
160
110
100
110
140
150
120
120
neighborhood 1
A dot plot, neighborhood 1, Home prices in Thousand Dollars, with the numbers 40 through 180, in intervals of 20, are indicated. The data are as follows: 55 thousand dollars, 2 dots. 60 thousand dollars, 2 dots. 70 thousand dollars, 1 dot. 80 thousand dollars, 3 dots. 90 thousand dollars, 1 dot. 120 thousand dollars, 1 dot.
neighborhood 2
A dot plot, neighborhood 2, Home prices in Thousand Dollars, with the numbers 40 through 180, in intervals of 20, are indicated. The data are as follows: 100 thousand dollars, 1 dot. 110 thousand dollars, 3 dots. 120 thousand dollars, 3 dots. 140 thousand dollars, 1 dot. 150 thousand dollars, 1 dot. 160 thousand dollars, 1 dot.
Decide whether the two groups are very different or not.
Show Solution
They are very different since the difference in means is 49, which is more than 3 times the larger MAD.
Andre is designing a website that will display reviews of school lunches. Each item on the menu is rated from 0 to 5 stars. The main display can only show 6 reviews, so Andre needs to decide how to choose which reviews to show at the top.
This is a dot plot of all 40 reviews for the lasagna.
This is a dot plot of the stars shown on the first page of results.
If each rating also has a sentence or two explaining the rating, what are some good reasons to keep this sample displayed first? What are some good reasons to change the sample that is displayed first?
Is the sample representative of the population?
Show Solution
Sample response: It might be good to keep it so that students can see the wide range of reviews possible for the lasagna. It might be good to change it because there are a lot more 0 and 5 star ratings than ones in the middle, so maybe there should be more of those ratings shown.
It is not representative since the shape of the distributions are not similar.
A public health expert is worried that a recent outbreak of a disease may be related to a batch of spinach from a certain farm. She wants to test the plants at the farm, but it will ruin the crop if she tests all of them.
If the farm has 5,000 spinach plants, describe a method that would produce a random sample of 10 plants.
Why would a random sample be useful in this situation?
Show Solution
Sample responses:
She could number the plants from 1 to 5,000 and have a computer select 10 random numbers between 1 and 5,000, then test the plants that correspond to the numbers the computer generated.
Since it is not known where the disease may have originated, a random sample would hopefully produce a wide selection of plants that would be representative of the entire crop.
Section C Check
Section C Checkpoint
Problem 1
You want to know what percentage of schools in the United States have a math teacher as the football coach.
What is the population for this situation?
Why might it be important to use a sample for this situation?
What is a possible sample you could use for this situation?
Show Solution
All of the schools in the United States
Sample response: It would be difficult to get data from every school in the United States, so we could use a sample to estimate the percentage.
Sample response: We could pick a few schools from each state.
Problem 2
Kiran wants to know what students at his school think about the theme for the homecoming dance. He is sitting with a group of friends at lunch, so he decides to ask them and use their responses as a sample.
Why might this method for getting a sample not reflect the data for the population?
Why might using more randomness in selecting people for the sample be better?
Show Solution
Sample response:
Because all of the people in the sample are friends, they might have similar interests and be biased one way or another and not match what the people in the school think.
It will help avoid bias and be more likely to collect information from a group more representative of the population.
A chemical engineer is trying to increase the amount of the useful product in a reaction. She performs the reaction with her new equipment 10 times and gets the following amounts of the useful product in grams:
47.1
48.2
48.3
47.5
48.5
48.1
47.2
48.2
48.4
48.3
What proportion of the reactions are above the 48 grams threshold?
Other chemists typically get 65% of their reactions to produce more than 48 grams. Should the engineer say that she is able to increase the useful product when compared to the other chemists?
Show Solution
0.7, since 7 of the 10 reactions have more than 48 grams of the useful product
Sample response: She could be optimistic, but her proportion does not seem far from what others have done. She should run more reactions to be more sure of the improvement. With only 10 values in her data set, 0.7 (and 0.6) is as close to 0.65 as she could get.
Jada collects data about the number of letters people get in the mail each week. The population distribution is shown in the dot plot.
A dot plot with the numbers 14 through 36, in increments of 2, indicated. The data are as follows: 14, 0 dots. 15, 1 dot. 16, 1 dot. 17, 1 dot. 18, 0 dots. 19, 2 dots. 20, 3 dots. 21, 3 dots. 22, 10 dots. 23, 16 dots. 24, 11 dots. 25, 11 dots. 26, 14 dots. 27, 4 dots. 28, 11 dots. 29, 2 dots. 30, 3 dots. 31, 3 dots. 32, 1 dot. 33, 1 dot. 34, 2 dots. 35, 0 dots. 36, 0 dots.
Which of these dot plots are likely to represent the means from samples of size 10 from this population? Explain your reasoning.
Dot Plot 1
A dot plot labeled "Dot plot 1" with the numbers 14 through 36, in increments of 2, indicated. The data are as follows: 22, 1 dot. 23, 2 dots. 24, 4 dots. 25, 6 dots. 26, 6 dots. 27, 8 dots. 28, 7 dots. 29, 3 dots. 30, 6 dots. 31, 3 dots. 32, 3 dots. 33, 1 dot.
Dot Plot 2
Show Solution
Dot Plot 2. Sample reasoning: It has the same center, but less variability than the population data. Dot Plot 1 also has less variability, but has a different center than the population data, so it is probably not generated by sample means from the original population.
Noah is interested in comparing the number of movies watched by students and teachers over the winter break. He takes a random sample of 10 students and 10 teachers and makes a dot plot of their responses.
students
teachers
<p>A dot plot for “movies watched over break” titled "Teacher." The numbers 0 through 8 are indicated. The data are as follows: 1 movie, 1 dot. 2 movies, 4 dots. 3 movies, 3 dots. 4 movies, 1 dot. 5 movies, 1 dot.</p>
Noah then computes the measures of center and variability for each group:
Students: mean: 5.7 movies, MAD: 0.76 movies
Teachers: mean: 2.7 movies, MAD: 0.9 movies
Should Noah conclude that there is a meaningful difference in the mean number of movies watched over winter break between the two groups? Explain your reasoning.
Show Solution
Yes. Sample reasoning: Because the difference in the means is greater than 2 MADs, there is a meaningful difference in the mean number of movies watched. (5.7−2.7)÷0.9≈3.33
What measure of center is shown in the box plot? What measure of variability? What are the values for each of these characteristics?
Draw another box plot with the same measure of variability that is meaningfully different from the one shown.
Show Solution
Median: 48, IQR: 6
Correct responses show a box plot with an IQR of 6 and median greater than or equal to 60 (or less than or equal to 36).
Section D Check
Section D Checkpoint
Problem 1
A company selects 20 people at random to say whether they use social media to get news. The people can select from the following options: never, rarely, sometimes, or often. Here are the results.
often
sometimes
never
sometimes
never
rarely
sometimes
never
sometimes
never
never
never
never
rarely
rarely
never
sometimes
sometimes
often
often
Based on this sample, what proportion of the population gets their news from social media sometimes or often?
A headline for a newspaper uses this study to say, “Half of Americans Get Their News from Social Media.” Is the headline misleading or does it fit with what the sample shows? Explain your reasoning.
Show Solution
209
Sample response: I think it fits with what the sample shows. 209 is very close to 21, and it is reasonable to use that benchmark fraction for this case.
Problem 2
A traveler is visiting Buffalo, New York, and wants to check the difference in price between getting a hotel room and renting a house for a short-term stay. The mean price of each option for one night is listed.
hotel: $144
house: $212
What is a value for the MAD that would indicate that there is a meaningful difference between the price of the two options? Explain your reasoning.
Show Solution
Sample response: Any value less than $34. Sample reasoning: (212−144)÷2=34, so the difference of the means would be greater than twice the MAD.
