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### S Statistics and Probability

- S.ID Interpreting Categorical and Quantitative Data
- S.ID.A Summarize, represent and interpret data on a single count or measurement variable.
- S.ID.A.1 Represent data with plots on the real number line (dot plots, histograms and box plots).
- S.ID.A.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.A.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.A.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.

- S.ID.B Summarize, represent, and interpret data on two categorical and quantitative variables.
- S.ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
- S.ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
- S.ID.B.6.a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
- S.ID.B.6.b Informally assess the fit of a function by plotting and analyzing residuals.
- S.ID.B.6.c Fit a linear function for a scatter plot that suggests a linear association.

- S.ID.C Interpret linear models.
- S.ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
- S.ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
- S.ID.C.9 Distinguish between correlation and causation.

- S.ID.A Summarize, represent and interpret data on a single count or measurement variable.
- S.IC Making Inferences and Justifying Conclusions
- S.IC.A Understand and evaluate random processes underlying statistical experiments.
- S.IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
- S.IC.A.1.a Introduce sampling distributions as a means to explore variability in sample statistics and ultimately evaluate a claim about a population.

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

- S.IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
- S.IC.B Make inferences and justify conclusions from sample surveys, experiments and observational studies.
- S.IC.B.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
- S.IC.B.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.B.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
- S.IC.B.6 Evaluate reports based on data.
- S.IC.B.7 Conduct statistical investigations.
- S.IC.B.7.a Conduct observational studies.
- S.IC.B.7.b Conduct statistical experiments.

- S.IC.A Understand and evaluate random processes underlying statistical experiments.
- S.CP Conditional Probability and the Rules of Probability
- S.CP.A Understand independence and conditional probability and use them to interpret data.
- S.CP.A.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.A.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.A.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.A.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.A.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

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

- S.CP.B.6 Find the conditional probability of

- S.CP.A Understand independence and conditional probability and use them to interpret data.
- S.MD Using Probability to Make Decisions
- S.MD.A Calculate expected values and use them to solve problems.
- S.MD.A.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 for data distributions.
- S.MD.A.2 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
- S.MD.A.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.A.4 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

- S.MD.B Use probability to evaluate outcomes of decisions.
- S.MD.B.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
- S.MD.B.5.a Find the expected payoff for a game of chance.
- S.MD.B.5.b Evaluate and compare strategies on the basis of expected values.

- S.MD.B.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
- S.MD.B.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

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

- S.MD.A Calculate expected values and use them to solve problems.