Summarize categorical data for two categories in two-way frequency tables and represent using segmented bar graphs.
Interpret relative frequencies in the context of categorical data (including joint, marginal, and conditional relative frequencies).
Identify possible associations and trends in categorical data. [Algebra I with Probability, 35]
Generate a two-way categorical table in order to find and evaluate solutions to real-world problems.
Aggregate data from several groups to find an overall association between two categorical variables.
Recognize and explore situations where the association between two categorical variables is reversed when a third variable is considered (Simpson’s Paradox). [Algebra I with Probability, 36]
Example: In a certain city, Hospital 1 has a higher fatality rate than Hospital 2. But when considering mildly-injured patients and severely-injured patients as separate groups, Hospital 1 has a lower fatality rate among both groups than Hospital 2, since Hospital 1 is a Level 1 Trauma Center. Thus, Hospital 1 receives most of the severely-injured patients who are less likely to survive overall but have a better chance of surviving in Hospital 1 than they would in Hospital 2.
Use mathematical and statistical reasoning with bivariate categorical data in order to draw conclusions and assess risk. [Algebra I with Probability, 32]
Example: In a clinical trial comparing the effectiveness of flu shots A and B, 21 subjects in treatment group A avoided getting the flu while 29 contracted it. In group B, 12 avoided the flu while 13 contracted it. Discuss which flu shot appears to be more effective in reducing the chances of contracting the flu. Possible answer: Even though more people in group A avoided the flu than in group B, the proportion of people avoiding the flu in group B is greater than the proportion in group A, which suggests that treatment B may be more effective in lowering the risk of getting the flu.
Design and carry out an investigation to determine whether there appears to be an association between two categorical variables, and write a persuasive argument based on the results of the investigation. [Algebra I with Probability, 33]
Example: Investigate whether there appears to be an association between successfully completing a task in a given length of time and listening to music while attempting to complete the task. Randomly assign some students to listen to music while attempting to complete the task and others to complete the task without listening to music. Discuss whether students should listen to music while studying, based on that analysis.
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“). [Algebra I with Probability 37]“
Explain whether two events, A and B, are independent, using two-way tables or tree diagrams. [Algebra I with Probability, 38]
Compute the conditional probability of event A given event B, using two-way tables or tree diagrams. [Algebra I with Probability, 39]
Recognize and describe the concepts of conditional probability and independence in everyday situations and explain them using everyday language. [Algebra I with Probability, 40]
Example: Contrast the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
Explain why the conditional probability of A given B is the fraction of B’s outcomes that also belong to A, and interpret the answer in context. [Algebra I with Probability, 41]
Example: the probability of drawing a king from a deck of cards, given that it is a face card, is $\frac{4/52}{12/52}$, which is $\frac{1}{3}$.
Informally justify the Pythagorean Theorem and its converse. [Grade 8, 26]
Apply the Pythagorean Theorem to find the distance between two points in a coordinate plane. [Grade 8, 27]
Apply the Pythagorean Theorem to determine unknown side lengths of right triangles, including real-world applications. [Grade 8, 28]
Extend understanding of irrational and rational numbers by rewriting expressions involving radicals, including addition, subtraction, multiplication, and division, in order to recognize geometric patterns.
Use units as a way to understand problems and to guide the solution of multi-step problems.
Choose and interpret units consistently in formulas.
Choose and interpret the scale and the origin in graphs and data displays.
Define appropriate quantities for the purpose of descriptive modeling.
Choose a level of accuracy appropriate to limitations of measurements when reporting quantities.
Find the coordinates of the vertices of a polygon determined by a set of lines, given their equations, by setting their function rules equal and solving, or by using their graphs.
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
Example: Rearrange the formula for the area of a trapezoid to highlight one of the bases.
Verify that the graph of a linear equation in two variables is the set of all its solutions plotted in the coordinate plane, which forms a line.
Derive the equation of a circle of given center and radius using the Pythagorean Theorem.
Given the endpoints of the diameter of a circle, use the midpoint formula to find its center and then use the Pythagorean Theorem to find its equation.
Derive the distance formula from the Pythagorean Theorem.
Use mathematical and statistical reasoning with quantitative data, both univariate data (set of values) and bivariate data (set of pairs of values) that suggest a linear association, in order to draw conclusions and assess risk.
Example: Estimate the typical age at which a lung cancer patient is diagnosed, and estimate how the typical age differs depending on the number of cigarettes smoked per day.
Use technology to organize data, including very large data sets, into a useful and manageable structure.
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Represent the distribution of univariate quantitative data with plots on the real number line, choosing a format (dot plot, histogram, or box plot) most appropriate to the data set, and represent the distribution of bivariate quantitative data with a scatter plot. Extend from simple cases by hand to more complex cases involving large data sets using technology.
Use statistics appropriate to the shape of the data distribution to compare and contrast two or more data sets, utilizing the mean and median for center and the interquartile range and standard deviation for variability.
Explain how standard deviation develops from mean absolute deviation.
Calculate the standard deviation for a data set, using technology where appropriate.
Interpret differences in shape, center, and spread in the context of data sets, accounting for possible effects of extreme data points (outliers) on mean and standard deviation.
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