Unit 6

Probability and Statistics

 

Grade 7

Math 

Unit Length and Description:

 

30 days

 

The initial work in Unit 6 extends students’ prior knowledge of statistics to include population statistics. Students learn to draw inferences about populations based on random samples.  Data used in this unit will be from a real-world context with an emphasis on representative samples and their corresponding populations. Students will utilize both measures of center and measures of variability to compare and make inferences for two populations.  After working with population statistics, students start to explore chance events and are introduced to the concept of probability. This is the students’ first experience with probability, and it begins with the conceptual understanding that the probability of a chance event is a rational number between 0 and 1. Students will explore simple probability through collecting data on a chance process as well as through developing probability models. Probability is designed to enhance the students’ statistical thinking and reasoning.  The latter part of Unit 6 extends the students’ work with probability to include finding probabilities of compound events. Students are not expected to calculate the probability of a compound event arithmetically; rather, they will be expected to find the probability of a compound event using organized lists, tables, tree diagrams, and simulations. Students should understand that the probability of a compound event is analogous to the probability of a simple event in that both are ratios comparing the number of favorable outcomes within a sample space to the entire sample space. Students will design and use simulations to generate frequencies for compound events. This unit should provide students with an appreciation of how statistics and probability can be applied in a real-world context.

 

Standards:

 

7.SP.A.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.

 

7.SP.A.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.

 

7.SP.B.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variability, 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.

 

7.SP.B.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.

 

7.SP.C.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.

 

o   Standards Clarification:  Students use proportional reasoning and percentages when working with probability. Students will need a strong conceptual understanding of probability to be successful when working with compound probabilities.

 

7.SP.C.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.

 

7.SP.C.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

a.   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.

b.   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?

 

7.SP.C.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

a.   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.

b.   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.

c.    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?

 

  • Standards Clarification:  This standard does not include finding probability of compound events arithmetically. Neither the Addition nor the Multiplication Rule is taught in this course.

 

Focus Standards of Mathematical Practice:

 

MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

 

Instructional Outcomes:

Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:

 

7.SP.A.1

·        I can apply statistics terms such as population, sample, sample size, random sampling, generalizations, valid, biased and unbiased.

·        I can recognize sampling techniques such as convenience, random, systematic and voluntary.

·        I can recognize that generalizations about a population from a sample are valid only if the sample is representative of that population.

·        I can apply statistics to gain information about a population from a sample of the population.

·        I can generalize that random sampling tends to produce representative samples and support valid inferences.

 

7.SP.A.2

·        I can define random sample.

·        I can identify an appropriate sample size.

·        I can analyze and interpret data from a random sample to draw inferences about a population with an unknown characteristic of interest.

·        I can generate multiple samples (or simulated samples) of the same size to determine the variation in estimates or predictions by comparing and contrasting the samples.

 

7.SP.B.3

·        I can identify measures of central tendency (mean, median, and mode) in a data distribution.

·        I can identify measures of variation including upper quartile, lower quartile, upper extreme-maximum, lower extreme minimum, range, interquartile range, and mean absolute deviation (i.e. box-and-whisker plots, line plot, dot plots, etc.)

·        I can compare two numerical data distributions on a graph by visually comparing data displays, and assessing the degree of visual overlap.

·        I can compare the differences in the measure of central tendency in two numerical data distributions by measuring the difference between the centers and expressing it as a multiple of a measure of variability.

 

7.SP.B.4

·        I can find measures of central tendency (mean, median, and mode) and measures of variability (range, quartile, etc.).

·        I can analyze and interpret data using measures of central tendency and variability.

·        I can draw informal comparative inferences about two populations from random sample.

 

7.SP.C.5

·        I can understand that probability is expressed as a number between 0 and 1.

·        I can understand that a random event with a probability of ½ is equally likely to happen.

·        I can understand that as probability moves closer to 1 it is increasingly likely to happen.

·        I can understand that as probability moves closer to 0 it is decreasingly likely to happen.

·        I can draw conclusions to determine that a greater likelihood occurs as the number of favorable outcomes approaches the total number of outcomes.

 

7.SP.C.6

·        I can determine relative frequency (experimental probability) is the number of times an outcome occurs divided by the total number of times the experiment is completed.

·        I can determine the relationship between experimental and theoretical probabilities by using the law of large numbers.

·        I can predict the relative frequency (experimental probability) of an event based on the (theoretical) probability.

 

7.SP.C.7a

·        I can use models to determine the probability of events.

·        I can recognize uniform (equally likely) probability.

·        I can develop a uniform probability model and use it to determine the probability of each outcome/event.

 

7.SP.C.7b

·        I can use models to determine the probability of events.

·        I can develop a probability model (which may not be uniform) by observing frequencies in data generated from a change process.

·        I can analyze a probability model and justify why it is uniform or explain the discrepancy if it is not.

 

7.SP.C.8a

·        I can determine that the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

·        I can identify the outcomes in the sample space for an everyday event.

 

7.SP.C.8b

·        I can define and describe a compound event.

·        I can find probabilities of compound events using organized lists, tree diagrams, etc. and analyze the outcomes.

·        I can choose the appropriate method such as organized lists, tables and tree diagrams to represent sample spaces for compound events.

 

7.SP.C.8c

·        I can define simulation.

·        I can design and use a simulation to generate frequencies for compound events.

 

Enduring Understandings:

 

     The way that data is collected, organized and displayed influences interpretation.

     Measures of center and measures of variability can be compared and used to make inferences for two populations.

     The probability of a chance event is a rational number between 0 and 1.

     The probability of an event’s occurrence can be predicted with varying degrees of confidence.

     The probability of a compound event can sometimes be found using organized lists, tables, tree diagrams, and simulations.

     The probability of a compound event is similar to the probability of a simple event in that both are ratios comparing the number of favorable outcomes within a sample space to the entire sample space.

 

Essential Questions:

 

·        How can you predict the outcome of future events?

·        Why is data collected and analyzed?

·        How do you know which type of graph to use when displaying data?

·        How do people use data to influence others?

·        How can predictions be made based on data?

·        How can the probability of an event be determined?

·        What is the reliability of the determination of the probability of an event?