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Common Core Learning StandardsGRADE 7 Mathematics

STATISTICS & PROBABILITY

Common Core Learning Standards Concepts Embedded Skills Vocabulary

Use random sampling to draw inferences about a population.

random sampling

Explain how statistics is used to gain information about a population.

Population Sample Representative

sample Biased sample Random

sampling Inferences Validity

Evaluate the validity of a statistical sample from a population.

7.SP.1. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

Explain why random sampling produces a sample representative of a population.

SAMPLE TASKS

Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Alignment Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.

I. Some music students want to start their own band. They need a name for the band that would be appealing to teenagers. The music students plan to survey a group of teenagers to find out which name is best.

Group 1: 300 teens at a middle school

Group 2: 25 teens at a football game

Group 3: 600 teens at a variety of concerts

a. Which of the sample space groups listed above will provide the best results?

_____________________________________

b. Explain how you determined the best sample group.



Use random sampling to draw inferences about a population.

drawing inferences

Draw inferences about a population with a certain characteristic from data gathered from a random sample.

Inference Random

sampling Population Characteristic

Gather data from multiple random samples of the same size in reference to a certain characteristic.

7.SP.2. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.


SAMPLE TASKSI. The first 10 monetary donations from boys and the first 10 monetary donations from girls were recorded.

Donation Amounts from Boys ($)15, 10, 10, 5, 12, 20, 15, 5, 10, 10

Donation Amounts from Girls ($)10, 20, 5, 1, 20, 25, 15, 15, 10, 5

a. What inferences can you make about boys and girls donations?

b. If 25 boys donated, predict how much money was raised by the boys?

II. The chart below represents the number of words on select pages of a 4 th grade and 7th grade Social Studies textbook. 4th Grade 7th Grade

35, 32, 40, 20, and 38 54, 68, 72, 59, and 63

Compare the number of words in the 4th grade book to the number of words in the 7th grade book. To support your answer, describe how you used one of the measures of center (mean or median) or variability (range, inter-quartile range, or mean absolute deviation),.



Draw informal comparative inferences about two populations.

variability of data

distributions

Describe the variability of two numerical data sets. variability

(how far away from the mean)

mean absolute deviation

range outlier interquartile

range

Compute the mean absolute deviation, range, and interquartile range.

7.SP.3. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

Describe how many times larger/smaller the variability of one data set is to another.Read and interpret data from statistical representations (box-and-whisker plot, line/dot plot).

SAMPLE TASKSI. Find the mean absolute deviation for the following quiz scores: 6, 9, 6, 9, 8, and 10.


II. The following box and whisker plot shows the weights of children between the ages of 8 and 10. a. What is the inter-quartile range?

b. What is the range of the data?

III. The following box-and-whisker plot shows the number of sales that Angela and Carl made in a week. a. Complete the chart below to compare the measures of

variation of both salespeople.

b. Using the data from the chart, what can you conclude about the number of sales of the salespeople?


Angela CarlMedianRangeLower QuartileUpper QuartileInterquartile Range


Draw informal comparative inferences about two populations.

measures of central

tendency and variability to

make inferences

Compare/contrast measures of central tendency to draw conclusions about two random samples.

measures of central tendency (mean, median, mode)

variability range outlier interquartile

range

Compare/contrast variability of two data sets to draw conclusions about two random samples.

7.SP.4. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh grade science book are generally longer than the words in a chapter of a fourth-grade science book.

Read and interpret data from statistical representations (box-and-whisker plot, line/dot plot).

SAMPLE TASKSI. Box-and-whisker plots can sometimes compare two sets of data. What conclusion can be drawn from the box-and-

whisker plot below?


II. The following table shows the annual salaries for two different companies.

Company A$12,000 $15,000 $27,500 $28,000 $80,000

Company B $27,000 $27,500 $27,500 $40,000 $45,000a. Find the mean, median, and mode of the two companies. Show your work in the space below.

Company A:

Mean: ____________________ Median: _____________________ Mode: ____________________

Company B:

Mean: ____________________ Median: _____________________ Mode: ____________________

b. Which company would you rather work for? Justify your answer.



Investigate chance processes and develop, use, and evaluate probability models.

Probability Define probability as number between 0 and 1. Probability Event Likely event Unlikely event

Describe a situation in which the event is unlikely.

Identify the probability of an unlikely event as a number near 0.

7.SP.5. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

Describe a situation in which the event is likely.

Identify the probability of a likely event as a number near 1. Describe a situation in which the event is neither likely nor unlikely.

Identify the probability of an event that is neither likely nor unlikely as a number near ½.


SAMPLE TASKSI. Sara has box that holds 7 blue marbles, 5 purple marbles, 3 white marbles, and 15 red marbles. She pulls one marble out of the

box without looking.

a. Is it more likely, less likely, or equally likely that Sara will pick a blue marble than a purple marble from the box? Explain why you chose your answer.

b. Describe an event involving Sara’s box of marbles in which the outcome would have a probability of 0.

c. Describe an event involving Sara’s box of marbles in which the outcome would have a probability of 1.

d. Describe an event involving Sara’s box of marbles in which the outcome would have a probability of 1/2.

II. The 7th grade math class has a number cube and a fair coin. Order the following probabilities from least likely to most likely: flipping a tail, flipping a head or a tail, rolling a number that is not 1, and rolling a 1.




Approximating probability

Predict the number of times an event occurs by multiplying the theoretical probability by the number of trials.

Probability Event Outcomes Possible

outcomes Favorable

outcomes Theoretical

probability Experimental

probability Trials

Compute the experimental probability of an event occurring through repeated trials.

7.SP.6. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

Compare the theoretical probability of an event occurring and the experimental probability.

Predict future probabilities based on data collected.


SAMPLE TASKSI. a. What is the theoretical probability of rolling a 2 on a number cube?

b. Predict the number of time you will roll a 2 if you roll the die 50 times.

c. Roll a number cube 50 times and record your results in the table below.

Number Rolled Tally Total1

2

3

4

5

6

d. Calculate the experimental probability of rolling a 2.

e. Explain any similarities or differences between your experimental probability and theoretical probability.


II. Keith and his grandfather are counting the types of birds that arrive at a feeder in their backyard. The results of their observations are in the table below.

Birds at the FeederGoldfinch Robins

21 7

Use the experimental probability to predict how many of the next 20 birds will be goldfinches.




Probability of equally likely

eventsCreate a uniform probability model (a situation in which all outcomes are equally likely).

Outcomes Events simple

probability Equally likely Uniform

probability model

Calculate simple probabilities of events.

7.SP.7a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.

SAMPLE TASKSI. Define uniform probability.

II. Timothy is going to randomly choose a marble from the bag of marbles shown below. a) What is the probability of Timothy choosing a striped marble?

P(striped) = ______________

b) Explain whether this situation shows a uniform probability model.


III. Create a situation in which all probabilities would be uniform.




experiments Design an experiment to investigate the likelihood of an outcome.

probability model (not uniform)

probability model (uniform)

frequencies data

Compare the results of a series of trials and draw conclusions. 7.SP.7b.

Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?


SAMPLE TASKSI. Design and describe a probability model in which an experiment will be conducted.

Experiment description: _________________________________________________________________________________________________

__________________________________________________________________________________________________

Conduct your experiment 25 times, then record your results in the table below.

Now, conduct your experiment another 25 times, and record your results in the table below.

Compare the results of your trials and draw conclusions.




Compound probability

Calculate compound probabilities. Probability tree diagrams simulation sample space compound

events simple events outcomes Fundamental

Counting Principle

Determine the total number of possible outcomes (sample space or Counting Principle).

7.SP.8a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

Define compound probabilities as fractions of the sample space taken from.

SAMPLE TASKSI. Thomas has a number cube and a coin. What is the probability of him rolling a number less than three and then flipping a heads

on a coin?

II. Find the probability of flipping a heads, then a tails, then a heads again on a coin.

III. Samantha has a hat with 20 pieces of paper labeled 1-20. Samantha’s work for finding the probability of picking out an even number, not returning it to the hat, and then picking out another even number, is shown below.


1020∙ 1020

=100400

=14

Is she correct? Why or why not?



compound sample space

Construct a tree diagram, list, or table to illustrate all possible outcomes of a compound event.

Tree diagrams Lists Tables Compound

events

Calculate the probability of a compound event based on a table, list, or tree diagram.7.SP.8b.

Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.

SAMPLE TASKSI. Joshua is buying a new bicycle. For the style he likes best, the bicycle shop sells 26-inch and 29-inch frames in blue, red, and

black. a. Create a table to represent the sample space.

b. What is the probability that he selects a black 29-inch bicycle?


II. Which of the following is an organized list to help you find the probability for flipping a coin twice?a. HH, HT, TH, HHb. TT, TH, HT, HHc. TH, HT, TH, HTd. TT, HH, TT, HH

III. A new car company allows you to design your own car online. The car model options are shown in the table below.Car Model Options

Color: Red Black SilverDoors: Two FourStyle: Sunroof Convertible

a. Use a tree diagram to show all the different ways you can design the car.


b. Does it matter what order you list the different ways to design the car? Explain why or why not.



simulation Design a simulation to generate data for compound events.

Simulation Compound

events Data

Calculate the probability of a compound event from data generated in a simulation.

7.SP.8c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

SAMPLE TASKSI. A deli has 2 breads, 6 meats, and 3 cheeses to choose from to make a sandwich. Describe, using pictures and/or words, a

simulation you could use to model the sandwiches ordered by the next three customers.


II. You have 5 shirts, 2 hats, and 4 pairs of shoes to choose from for your weekend outfits. Describe, using pictures and/or words, a simulation you could use to model your outfits for the next 2 weekend days.


III. Suppose each box of a popular brand of cereal contains a pen as a prize. The pens come in four colors, blue, red, green and yellow. Each color of pen is equally likely to appear in any box of cereal. Design and carry out a simulation to help you answer each of the following questions.

1. What is the probability of having to buy at least five boxes of cereal to get a blue pen?

2. What is the probability of having to buy at least ten boxes of cereal to get a full set of pens (all four colors)?


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