# data[].attributes.body
Represent the distribution of data with plots on the real number line(stem plots, dot plots, histograms and box plots).

# data[].attributes.body
Decide if a specified model is consistent with the results from a simulation.

# data[].attributes.body
Use data from a sample survey to estimate a population mean or proportion; and explain how bias may be involved in the process.

# data[].attributes.body
Describe events as subsets of a sample space. Use characteristics (or categories) of the outcomes, such as,
● as unions, “A or B”, that are mutually exclusive events and
● as unions, “A or B”, that are non-mutually exclusive events and
● as intersections, “A and B”, and
● as complements of other events, “not A”
to calculate basic probabilities.

# data[].attributes.body
Understand the concept of conditional probability.
a. Understand the conditional probability of A given B as P(A and B)/P(B).
b. 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.
c. Recognize and explain the concept of conditional probability in everyday language and everyday situations.
d. 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.

# data[].attributes.body
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range when comparing medians and standard deviation when comparing means) of different data sets.

# data[].attributes.body
When appropriate, fit a normal distribution to a numerical data set forgiven mean and standard deviation and then estimate population percentages using the Empirical Rule and recognize that there are datasets for which such a procedure is not appropriate.

# data[].attributes.body
Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between estimates or statistics are significant.

# data[].attributes.body
Calculate the expected value of a random variable; interpret it as the mean of the probability distribution and use the value in analyzing decisions.

# data[].attributes.body
Apply the General Multiplication Rule, P(A and B) =P(A)P(B|A) =P(B)P(A|B), in a uniform probability model and interpret the answer in terms of the model.

# data[].attributes.body
Interpret differences in shape, center, and spread in the context of the distributions of the numerical data, accounting for the presence and possible effects of extreme data points (outliers).

# data[].attributes.body
Summarize categorical data for two or more categories in frequency tables. Calculate and interpret joint, marginal, and conditional relative frequencies (probabilities) in the context of the data, recognizing possible associations and trends in the data.

# data[].attributes.body
Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

# data[].attributes.body
Develop a probability distribution for a random variable.
a. Find an expected value based on a sample space in which theoretical probabilities can be calculated.
b. Find an expected value based on a sample space in which empirical probabilities can be calculated.

# data[].attributes.body
Interpret the slope (rate of change) and the intercept (constant term)of a linear model in the context of the data.

# data[].attributes.body
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.
c. Use calculated expected values to make fair decisions and formulate strategies.

# data[].attributes.body
Use permutations and combinations to compute probabilities.
a. Distinguish between situations that can be modeled using counting techniques, including Fundamental Counting Principle, permutations, and combinations.
b. Perform calculations using the appropriate counting technique, including simple probabilities.
c. Use permutations and combinations to compute probabilities of compound events and solve problems.

# data[].attributes.body
Represent data on two quantitative variables on a scatterplot and describe how the explanatory and response variables are related.
a. Calculate an appropriate mathematical model, or use a given mathematical model, for data to solve problems in context.
b. Informally assess the fit of a model (through calculating correlation for linear data, plotting, calculating and/or analyzing residuals).

# data[].attributes.body
Construct and interpret scatter plots for bivariate numerical data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association and nonlinear association.

# data[].attributes.body
Use the equation of a linear model to solve problems in the context of bivariate numerical data, interpreting the slope and intercept.

# data[].attributes.body
Understand the role and purpose of correlation in linear regression.
a. Use technology to compute the correlation coefficient of a linear fit.
b. Interpret the meaning of the correlation within the context of the data.
c. Describe the limitations of correlation when establishing causation.

# data[].attributes.body
Know that lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a line and informally assess the model fit by judging the closeness of the data points to the line.

# data[].attributes.body
Understand the concept of independence.
a. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their individual probabilities, P(A)xP(B)
b. Determine whether two events are independent and provide a justification to support the decision.
c. Recognize and explain the concept of independence in everyday language and everyday situations.

# data[].attributes.body
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 appropriate graphical displays as for data distributions.

# data[].attributes.body
Understand statistics as a process for making inferences and justifying conclusions about population parameters based on a random sample from that population.

# data[].attributes.body
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 whether events are independent and to approximate conditional probabilities.