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Statistics & Probability

 

Interpreting Categorical and Quantitative Data

Summarize, Represent, and Interpret Data on a Single Count or Measurement Variable

S-ID.1. Represent data with plots on the real number line (dot plots, histograms, and box plots).

 

S-ID.2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

 

S-ID.3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

 

S-ID.4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

 

Summarize, Represent, and Interpret Data on Two Categorical and Quantitative Variables

S-ID.5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including join, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

 

S-ID.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a.     Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

b.     Informally assess the fit of a function by plotting and analyzing residuals.

c.      Fit a linear function for a scatter plot that suggests a linear association.

 

Interpret Linear Models

S-ID.7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

 

S-ID.8. Compute (using technology) and interpret the correlation coefficient of a linear fit.

 

S-ID.9. Distinguish between correlation and causation.

 

Making Inferences and Justifying Conclusions

Understand and Evaluate Random Processes Underlying Statistical Experiments

S-IC.1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

 

S-IC.2. Decide if a specified model is consistent with results from a given data-generating process, e.g. using simulation.

 

Make Inferences and Justify Conclusions form Sample Surveys, Experiments, and Observational Studies

S-IC.3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

 

S-IC.4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

 

S-IC.5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

 

S-IC.6. Evaluate reports based on data.

 

Conditional Probability and the Rules of Probability

Understand Independence and Conditional Probability and use them to Interpret Data

S-CP.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or”, “and”, “not”)

 

S-CP.2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

 

S-CP.3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

 

S-CP.4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

 

S-CP.5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

 

Use the Rules of Probability to Compute Probabilities of Compound Events in a Uniform Probability Model

S-CP.6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

 

S-CP.7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

 

S-CP.8. apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

 

S-CP.9. Use permutations and combinations to compute probabilities of compound events and solve problems.

 

Using Probability to Make Decisions

            Calculate Expected Values and Use Them to Solve Problems

S-MD.1. Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as far data distributions.

 

S-MD.2. Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

 

S-MD.3. Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

 

S-MD.4. Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

 

            Use Probability to Evaluate Outcomes of Decisions

S-MD.5. Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

a.     Find the expected payoff for a game of chance.

b.     Evaluate and compare strategies on the basis of expected values.

 

S-MD.6. Use probabilities to make fair decisions e.g. drawing by lots, using random number generator

 

S-MD.7. Analyze decisions and strategies using probability concepts e.g. product testing, medical testing, pulling a hockey goalie at the end of a game.

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