Vocabulary* Chance Process: The repeated observations of random outcomes of a given event.

• Compound Event: Any event which consists of more than one outcome.

• Empirical: A probability model based upon observed data generated by the process. Also, referred to as the experimental probability.

• Event: Any possible outcome of an experiment in probability. Any collection of outcomes of an experiment. Formally, an event is any subset of the sample space.

• Experimental Probability: The ratio of the number of times an outcome occurs to the total amount of trials performed. The number of times an event occurs

Experimental probability = the number of times an event occurs/the total number of trials

• Independent events: Two events are independent if the occurrence of one of the events gives us no information about whether or not the other event will occur; that is, the events have no influence on each other.

• Probability: A measure of the likelihood of an event. It is the ratio of the number of ways a certain event can occur to the number of possible outcomes.

• Probability Model: It provides a probability for each possible non-overlapping outcome for a change process so that the total probability over all such outcomes is unity. This can be either theoretical or experimental.

• Relative Frequency of Outcomes: Also, Experimental Probability

• Sample space: All possible outcomes of a given experiment.

• Simple Event: Any event which consists of a single outcome in the sample space. A simple event can be represented by a single branch of a tree diagram.

• Simulation: A technique used for answering real-world questions or making decisions in complex situations where an element of chance is involved.

• Theoretical Probability: The mathematical calculation that an event will happen in theory. It is based on the structure of the processes and its outcomes.

• Tree diagram: A tree-shaped diagram that illustrates sequentially the possible outcomes of a given event.

Enduring Understandings• Probabilities are fractions derived from modeling real world experiments and simulations of chance.

• Modeling real world experiments through trials and simulations are used to predict the probability of a given event.

• Chance has no memory. For repeated trials of a simple experiment, the outcome of prior trials has no impact on the next.

• The probability of a given event can be represented as a fraction between 0 and 1. • Probabilities are similar to percents. They are all between 0 and 1, where a probability of

0 means an outcome has 0% chance of happening and a probability of 1 means that the outcome will happen 100% of the time. A probability of 50% means an even chance of the outcome occurring.

• If we add the probabilities of every outcome in a sample space, the sum should always equal 1.

• The experimental probability or relative frequency of outcomes of an event can be used to estimate the exact probability of an event.

• Experimental probability approaches theoretical probability when the number of trials is large.

• Sometimes the outcome of one event does not affect the outcome of another event. (This is when the outcomes are called independent.)

• Tree diagrams are useful for describing relatively small sample spaces and computing probabilities, as well as for visualizing why the number of outcomes can be extremely large.

• Simulations can be used to collect data and estimate probabilities for real situations that are sufficiently complex that the theoretical probabilities are not obvious.

OverviewIn this unit students will: • use real-life situations to show the purpose for using random sampling to make inferences about a population. • understand that random sampling guarantees that each element of the population has an equal opportunity to be selected in the sample. • compare the random sample to the population, asking questions like, “Are all the elements of the entire population represented in the sample?” and “Are the elements represented proportionally?” • make inferences given random samples from a population along with the statistical measures. • learn to draw inferences about one population from a random sampling of that population. • draw informal comparative inferences about two populations. • deal with small populations and determine measures of center and variability for a population. • compare measures of center and variability and make inferences. • use graphical representations of data to compare measures of center and variability. • begin to develop understanding of the benefits of the measures of center and variability by analyzing data with both methods. • understand that when they study large populations, random sampling is used as a basis for the population inference. • understand that measures of center and variability are used to make inferences on each of the general populations. • make comparisons for two populations based on inferences made from the measures of center and variability.

Enduring Understandings• 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.

• 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.

• 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.

• Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

Vocabulary• Box and Whisker Plot: A diagram that summarizes data using the median, the upper and lowers quartiles, and the extreme values (minimum and maximum). Box and whisker plots are also known as box plots. It is constructed from the five-number summary of the data: Minimum, Q1 (lower quartile), Q2 (median), Q3 (upper quartile), Maximum.

• Frequency: the number of times an item, number, or event occurs in a set of data

• Grouped Frequency Table: The organization of raw data in table form with classes and
frequencies.

• Histogram: a way of displaying numeric data using horizontal or vertical bars so that the height or length of the bars indicates frequency

• Inter-Quartile Range (IQR): The difference between the first and third quartiles. (Note that the first quartile and third quartiles are sometimes called upper and lower quartiles.)

• Maximum value: The largest value in a set of data.

• Mean Absolute Deviation: the average distance of each data value from the mean. The MAD is a gauge of “on average” how different the data values are form the mean value.

• Mean: The “average” or “fair share” value for the data. The mean is also the balance

• Measures of Center: The mean and the median are both ways to measure the center for a set of data.

• Measures of Spread: The range and the mean absolute deviation are both common ways

to measure the spread for a set of data.

• Median: The value for which half the numbers are larger and half are smaller. If there are two middle numbers, the median is the arithmetic mean of the two middle numbers. Note: The median is a good choice to represent the center of a distribution when the distribution is skewed or outliers are present.

• Minimum value: The smallest value in a set of data.

• Mode: The number that occurs the most often in a list. There can more than one mode, or no mode.

• Mutually Exclusive: two events are mutually exclusive if they cannot occur at the same time (i.e., they have not outcomes in common).

• Outlier: A value that is very far away from most of the values in a data set.

• Range: A measure of spread for a set of data. To find the range, subtract the smallest value from the largest value in a set of data.

• Sample: A part of the population that we actually examine in order to gather information.

• Simple Random Sampling: Consists of individuals from the population chosen in such a way that every set of individuals has an equal chance to be a part of the sample actually selected. Poor sampling methods, that are not random and do not represent the population well, can lead to misleading conclusions.

• Stem and Leaf Plot: A graphical method used to represent ordered numerical data. Once the data are ordered, the stem and leaves are determined. Typically the stem is all but the last digit of each data point and the leaf is that last digit.

Picture

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. Video: Calculate the probability of an event by creating a ratio Video: Describe the probability of an event using a number line Video: Calculate the probability of an event by making a sum of 1

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. Video: Find the experimental probability by creating a ratio Video: Compare experimental and theoretical probability to interpret data Video: Predict the frequency of an event using results from experiments Video: Predict the frequency of an event using the theoretical probability

7.SP.7a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. Video: Analyze the probability of an event by assigning equal probability to all outcomes Video: Find the probability of events with multiple possibilities

7.SP.7b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. Video: Understand the law of large numbers by comparing experimental results to the theoretical probability Video: Explain discrepancies in results from a probability model by comparing the experimental and theoretical probabilities

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. Video: Analyze independent and dependent events

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. Video: Find the probability of a compound event by creating an organized list Video: Find the probability of a compound event by creating a tree diagram Video: Find the probability of a compound event by creating a table7.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. Video: Identify a random sample Video: Identify a representative sample Video: Generate a representative sample Video: Understanding biased samples

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. Video: Make inferences about a population by analyzing random samples Video: Use proportional reasoning to make estimates about a population Video: Assess whether an inference is valid by analyzing data Video: Make estimates about a population using the mean of multiple 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. Video: Compare populations using mean Video: Compare populations using median Video: Make inferences about range Video: Compare IQR using box plots Video: Compare two populations using mean absolute deviation

Chance Process:The repeated observations of random outcomes of a given event.•

Compound Event:Any event which consists of more than one outcome.•

Empirical:A probability model based upon observed data generated by the process. Also, referred to as the experimental probability.•

Event:Any possible outcome of an experiment in probability. Any collection of outcomes of an experiment. Formally, an event is any subset of the sample space.•

Experimental Probability:The ratio of the number of times an outcome occurs to the total amount of trials performed. The number of times an event occursExperimental probability = the number of times an event occurs/the total number of trials

•

Independent events:Two events are independent if the occurrence of one of the events gives us no information about whether or not the other event will occur; that is, the events have no influence on each other.•

Probability:A measure of the likelihood of an event. It is the ratio of the number of ways a certain event can occur to the number of possible outcomes.•

Probability Model:It provides a probability for each possible non-overlapping outcome for a change process so that the total probability over all such outcomes is unity. This can be either theoretical or experimental.•

Relative Frequency of Outcomes:Also, Experimental Probability•

Sample space:All possible outcomes of a given experiment.•

Simple Event:Any event which consists of a single outcome in the sample space. A simple event can be represented by a single branch of a tree diagram.•

Simulation:A technique used for answering real-world questions or making decisions in complex situations where an element of chance is involved.•

Theoretical Probability: The mathematical calculation that an event will happen in theory. It is based on the structure of the processes and its outcomes.•

Tree diagram:A tree-shaped diagram that illustrates sequentially the possible outcomes of a given event.• Modeling real world experiments through trials and simulations are used to predict the probability of a given event.

• Chance has no memory. For repeated trials of a simple experiment, the outcome of prior trials has no impact on the next.

• The probability of a given event can be represented as a fraction between 0 and 1. • Probabilities are similar to percents. They are all between 0 and 1, where a probability of

0 means an outcome has 0% chance of happening and a probability of 1 means that the outcome will happen 100% of the time. A probability of 50% means an even chance of the outcome occurring.

• If we add the probabilities of every outcome in a sample space, the sum should always equal 1.

• The experimental probability or relative frequency of outcomes of an event can be used to estimate the exact probability of an event.

• Experimental probability approaches theoretical probability when the number of trials is large.

• Sometimes the outcome of one event does not affect the outcome of another event. (This is when the outcomes are called independent.)

• Tree diagrams are useful for describing relatively small sample spaces and computing probabilities, as well as for visualizing why the number of outcomes can be extremely large.

• Simulations can be used to collect data and estimate probabilities for real situations that are sufficiently complex that the theoretical probabilities are not obvious.

• use real-life situations to show the purpose for using random sampling to make inferences about a population.

• understand that random sampling guarantees that each element of the population has an equal opportunity to be selected in the sample. • compare the random sample to the population, asking questions like, “Are all the elements of the entire population represented in the sample?” and “Are the elements represented proportionally?”

• make inferences given random samples from a population along with the statistical measures.

• learn to draw inferences about one population from a random sampling of that population.

• draw informal comparative inferences about two populations.

• deal with small populations and determine measures of center and variability for a population.

• compare measures of center and variability and make inferences.

• use graphical representations of data to compare measures of center and variability.

• begin to develop understanding of the benefits of the measures of center and variability by analyzing data with both methods.

• understand that when they study large populations, random sampling is used as a basis for the population inference.

• understand that measures of center and variability are used to make inferences on each of the general populations.

• make comparisons for two populations based on inferences made from the measures of center and variability.

• Understand that random sampling tends to produce representative samples and support valid inferences.

• 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.

• 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.

• Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations

.Box and Whisker Plot: A diagram that summarizes data using the median, the upper and lowers quartiles, and the extreme values (minimum and maximum). Box and whisker plots are also known as box plots. It is constructed from the five-number summary of the data: Minimum, Q1 (lower quartile), Q2 (median), Q3 (upper quartile), Maximum.•

Frequency: the number of times an item, number, or event occurs in a set of data•

Grouped Frequency Table: The organization of raw data in table form with classes andfrequencies.

•

Histogram: a way of displaying numeric data using horizontal or vertical bars so that the height or length of the bars indicates frequency•

Inter-Quartile Range (IQR): The difference between the first and third quartiles. (Note that the first quartile and third quartiles are sometimes called upper and lower quartiles.)•

Maximum value: The largest value in a set of data.•

Mean Absolute Deviation:the average distance of each data value from the mean. The MAD is a gauge of “on average” how different the data values are form the mean value.•

Mean: The “average” or “fair share” value for the data. The mean is also the balance•

Measures of Center: The mean and the median are both ways to measure the center for a set of data.•

Measures of Spread: The range and the mean absolute deviation are both common waysto measure the spread for a set of data.

•

Median: The value for which half the numbers are larger and half are smaller. If there are two middle numbers, the median is the arithmetic mean of the two middle numbers. Note: The median is a good choice to represent the center of a distribution when the distribution is skewed or outliers are present.•

Minimum value: The smallest value in a set of data.•

Mode: The number that occurs the most often in a list. There can more than one mode, or no mode.•

Mutually Exclusive: two events are mutually exclusive if they cannot occur at the same time (i.e., they have not outcomes in common).•

Outlier: A value that is very far away from most of the values in a data set.•

Range: A measure of spread for a set of data. To find the range, subtract the smallest value from the largest value in a set of data.•

Sample:A part of the population that we actually examine in order to gather information.•

Simple Random Sampling: Consists of individuals from the population chosen in such a way that every set of individuals has an equal chance to be a part of the sample actually selected. Poor sampling methods, that are not random and do not represent the population well, can lead to misleading conclusions.•

Stem and Leaf Plot: A graphical method used to represent ordered numerical data. Once the data are ordered, the stem and leaves are determined. Typically the stem is all but the last digit of each data point and the leaf is that last digit.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.Video: Calculate the probability of an event by creating a ratio

Video: Describe the probability of an event using a number line

Video: Calculate the probability of an event by making a sum of 1

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.Video: Find the experimental probability by creating a ratio

Video: Compare experimental and theoretical probability to interpret data

Video: Predict the frequency of an event using results from experiments

Video: Predict the frequency of an event using the theoretical probability

7.SP.7a.Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.Video: Analyze the probability of an event by assigning equal probability to all outcomes

Video: Find the probability of events with multiple possibilities

7.SP.7b.Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.Video: Understand the law of large numbers by comparing experimental results to the theoretical probability

Video: Explain discrepancies in results from a probability model by comparing the experimental and theoretical probabilities

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.Video: Analyze independent and dependent events

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.Video: Find the probability of a compound event by creating an organized list

Video: Find the probability of a compound event by creating a tree diagram

Video: Find the probability of a compound event by creating a table

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. Video: Identify a random sampleVideo: Identify a representative sample

Video: Generate a representative sample

Video: Understanding biased samples

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.Video: Make inferences about a population by analyzing random samples

Video: Use proportional reasoning to make estimates about a population

Video: Assess whether an inference is valid by analyzing data

Video: Make estimates about a population using the mean of multiple 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.Video: Compare populations using mean

Video: Compare populations using median

Video: Make inferences about range

Video: Compare IQR using box plots

Video: Compare two populations using mean absolute deviation