Grade 8 Practice Problems: Data, Statistics & Probability
Examples from Standards Revision and GLEs
8D-28) Answer: see spinner below.
Draw a spinner that fits all of the following clues
a) There are five numbers on the spinner. Five is the greatest number.
b) One of the numbers on the spinner comes up about ½ of the time.
c) The number three comes up 1/8 of the time.
d) The number two comes up most often.
e) The probabilities of 1, 4, and 5 coming up are equal.
Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-27) Answer: 2/5; this is not a fair game. Reasons may vary.
Jack and Jill are playing a game. There are six tiles in a box: 3 red and 3 blue. A player picks two tiles without looking. Jill gets a point if the tiles do not match; Jack gets a point if they do match. What are the odds that both tiles will have the same color? Is this a fair game? Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-26) Answer: 49
Monroe uses license plates for its motorcycles. Each license plate consists of one vowel followed by one digit. There are 5 possible vowels and 10 possible digits. Since the zero and the capital O look the same, they cannot be used together. How many different license plates are possible? Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-25) Answer: 92
Kyle has earned scores of 85, 88, 73, and 80 on the first four math tests. If the final counts as two tests, what score does Kyle need to earn to earn an 85% average for the quarter? Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-24) Answer: 82
Emma has an average of 74 on the first three tests in her algebra class. On the final four tests, she scored 92, 90, 94, and 76. What was her average for the entire year? Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-23) Answer: 14
In a collection of 80 books, 34 have leather covers and 18 were published before 1850. If 42 of these books were published after 1850 and do not have leather covers, how many books were published before 1850 and have leather covers? Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-22) Answer: 3/25
Harriet has 50 dollar bills stuffed in a jar in her room. There are half as many five dollar bills as there are one dollar bills. There are one-third as many tens as there are ones. There are half as many twenties as there are fives. If Harriet reaches into the jar and randomly pulls out a bill, what is the probability that the bill will be a twenty? State your answer as a fraction in lowest terms. Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-21) Answer: 1/6
The height of the shaded rectangle is one-half the height of the square, and the width of the rectangle is one-third the width of the square. If a point is chosen at random inside the square, what is the probability that it is also inside the square? Express your answer as a common fraction. Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-20) Answer: 5/8
A bag contains 3 red and 5 blue marbles. If one marble is randomly drawn from the bag, what is the probability that it is blue? Express your answer as a common fraction. Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-19) Answer: 5/14
A pocket full of change consists of 5 nickels, 2 dimes and 1 quarter. Two coins are randomly chosen from the bag without replacement. What is the probability that their combined value is 15 cents? Express your answer as a common fraction. Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-18) Answer: 1/4
Two fair coins are flipped. What is the probability that both show heads? Express your answer as a common fraction. Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-17) Answer: 1/5
Two of the digits 1-5 are selected at random without replacement. What is the probability that the positive difference between the two numbers is 3? Express your answer as a common fraction in lowest terms. Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-16) Answer: 0
Mary selected a positive multiple of 7 less than 70. Roger selected a positive multiple of 11 less than 70. What is the probability that they selected the same integer? Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-15) Answer: 9/100
Alex has a 3/10 chance of making a free throw. What is the probability that she will make both of her next two free throws? Express your answer as a common fraction. Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-14) Answer: 10/81
Tiles numbered 1-6 are each placed randomly into one of three different boxes. What is the probability that each box contains 2 tiles? Express your answer as a common fraction. Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-13) Answer: 1/28
Michael is given a jar with eight marbles, six of which are red and two of which are green. Michael then randomly draws marbles without replacement. If he selects a red marble, he puts it in the first box; similarly, he continues to place all red marbles he selects into the first box until a green marble is chosen. When a green marble is chosen, he puts the first box aside. He then selects and places red marbles in the second box until the second green marble is chosen. When the second green marble is selected, he will put the second box aside and place all remaining red marbles in a third box. After all marbles are drawn, what is the probability that each box contains two red marbles? Express your answer as a common fraction. Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-12) Answer: 1/2
One letter is randomly selected from the word MATH and one letter is randomly selected from the word SOLVER. What is the probability that both letters selected are consonants? Express your answer as a common fraction. Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-11) Answer: 250,000
In the town of Ornery, a survey was conducted. Of all people surveyed, 241,750 were identified as employed, and 3.3% of were identified as unemployed. About how many people were surveyed? Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-10) Answer: 12
A set of three numbers has an average of 8. What number should be added to the set to make the average of all four numbers 9? Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-9) Answer: Hank's
A baseball player's batting average is computed as the ratio of hits to at-bats. Hammerin' Hank had 160 hits in 452 at-bats. Joltin' Joe had 148 hits in 421 at-bats. Whose average was higher? Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-8) Answers will vary.
When Ben first started to play the electric guitar, his skill increased quite rapidly. Over time, as Ben continued to practice, he seemed to improve more slowly. Sketch a graph to show how Ben's guitar-playing skill progressed over time since he began to play. Your graph shows the relationship between two variables. What are they? What other variables might affect the rate at which Ben's playing improves?
Make a table and a graph of (time, temperature) data that fit the following information about a day on the road: ü We started riding at 8 A.M. The day was quite warm, with dark clouds in the sky. ü About midmorning the temperature dropped quickly to 630F, and there was a thunderstorm for about an hour. ü After the storm, the sky cleared and there was a warm breeze. ü As the day went on, the sun steadily warmed the air. When we reached our campground at 4 PM it was 89° F.
Here are the box office earnings (in millions of dollars) for a popular movie after each of the first eight weeks following its release.
Weeks in theaters |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
Weekly earnings |
16 |
22 |
18 |
12 |
7 |
4 |
3 |
1 |
Make a coordinate graph showing the weekly earnings after each week. Since a film's weekly earnings depend on the number of weeks it is in theaters, put the weeks in theaters on the x-axis and the weekly earnings on they-axis. Write a short description of the pattern of change in the data table and in your graph. Explain how the movie's weekly earnings changed as time passed, how this change is shown in the table and the graph, and why this change might have occurred. What were the total earnings of the movie in the eight weeks? Make a coordinate graph showing the total earnings after each of the first eight weeks. Write a short description of the pattern of change in your graph of total earnings. Explain how the movie's total earnings changed over time, how this change is shown in the table and the graph, and why this change might have occurred. Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-7) Answer: 12
Customers at a particular yogurt shop may select one of three flavors of yogurt. They may choose one of four toppings. How many one-flavor, one-topping combinations are possible? Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-6) Answer: 20
What is the positive difference between the mean of set A {12,13,23,34,143} and the mean of set B {12,13,23,34,43}? Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-5) Answer: 1
What is the positive difference between the range and the median of the set {1,3,3,5,5,5,7,7,7,7}? Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-4) Answer: no more than 28 canoes
Carla painted identification numbers on the canoes in her new franchise. Each canoe has three numbers. The first number must be a 1 (her franchise number), and the next two digits must be in ascending order. No zeros were used, and no digits were repeated on any canoe. What is the maximum number of canoes Carla might have had? Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-3) Answer: 15 games
There are 16 entries in the seventh-grade ping-pong elimination tournament. In this tournament, two players meet, with the loser being eliminated. How many matches must be played to get a single winner? Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-2) Answer: 107/1087 that Angela will pull out a key; there is a statistical advantage of 1% to going first rather than 25th only if the 24 people ahead of the 25th each pulled out a key.
Riverside Mall is holding its annual Midnight Mall Mayfest Contest. Twenty-five people, including Angela, have won the chance to reach into the Pirate's Chest and pull out the key to the mall. There are 214 keys in the chest along with 860 key-shaped pretzels, 500 plastic key rings, and 600 five-dollar gift certificates folded into small metal tubes. What are the odds that Angela will reach in and pull out a key? Given the number of prizes in the pirate's chest, is there a statistical advantage in going first rather than 25th? Explain in detail how you found your answer using words, numbers, and/or pictures.
8D-1) Answer: 30 combinations
In Surreytown, they have decided to put a license plate on every bicycle. Each license plate will consist of a letter followed by a numeral. The possible letters are the vowels: A, E, I, 0, and U. The possible numerals are 1, 2, 3, 4, 5, and 6. How many different license plates are possible? Explain in detail how you found your answer using words, numbers, and/or pictures
Use measures of center (mean, median, and mode) and spread (range and interquartile range) to summarize and compare data sets and explain the influence of outliers on each measure. Select, construct, and analyze data displays—including Venn diagrams, stem-and-leaf plots, box-and-whisker plots, histograms, circle graphs, and line plots—to compare two sets of data. Create a scatterplot for a two-variable data set, and, when appropriate, sketch and use a trend line to analyze the data. Identify different methods of selecting samples and analyze the strengths and weaknesses of each method. Evaluate conclusions of statistical studies reported in the media in terms of the processes used and possible sources of bias. Given data that address a question, select, construct, and analyze a data display, draw conclusions, and use mathematics to verify the conclusion. Use the concepts of complementary, mutually exclusive, dependent, and independent events to determine probabilities. Solve problems using counting techniques, including the fundamental counting principle, lists, tables, and tree diagrams. Example:
Apply proportionality concepts to make predictions and test conjectures about the results of probability experiments. |
Examples of Probability and Statistics from the 2006 GLEs – Grade 8
Determine and explain when events are compound. Describe the difference between compound events involving “and” or “or”. Describe or represent compound events. Determine the sample space for simple experiments involving independent or compound events. Calculate the probability of two independent events occurring simultaneously using various methods including organized lists, tree diagrams, counting procedures, and area models. Explain the relationship between theoretical and empirical probability of compound events. Predict the probability of outcomes of experiments and relates the predictions to empirical results. Design a situation that would produce a given probability. Design a game using compound probabilities with equal chances of winning for all players. Describe bias in population samples and explains a procedure for selecting an unbiased representative sample. Examine the results of a survey given to two different sample groups to determine if differences in survey results were caused by differences in samples. Determine whether claims made about results are based on biased data due to sampling. Select an appropriate population for a given survey question. Determine whether a sampling method will result in a representative sample. Identify clusters and outliers and determine how they may affect measures of central tendency. Modify a set of data so that the median is a more reasonable measure of central tendency than the mean. Examine variations in data, including clusters and outliers, to select the most appropriate measure of central tendency to describe a given set of data. Determine and/or use the mean, median, mode, and/or range for a set of data. Describe trends or patterns in data presented in a table of ordered pairs or a scatter plot. Read and interpret the data in Venn Diagrams, tables of ordered pairs, and/or scatter plots. Select a line of best fit for a set of data to predict a future value of a variable to interpolate between existing data values. Draw trend lines with or without technology and makes predictions about real world situations. Explain whether stem and leaf plot, box and whisker plot, or scatter plot is more appropriate for a given set of data, a particular situation, or purpose, or answers a question most effectively. Determine whether claims made about results are based on biased representations of data. Predict an outcome given a linear relationship involving non negative rational numbers. Explain how the same set of data can support different points of view. Explain how data have been used or misused to support a point of view. |