High School Practice Problems: Data, Statistics & Probability

Evaluate the use, or justify the choice, of summary statistics used to describe a set of data—including mean, median, mode, range, interquartile range, and standard deviation— and identify misleading uses of these statistics. Example:

• Which is more likely to be affected by an outlier: the interquartile range or the standard deviation?

Evaluate the effect of linear transformations of a data set (univariate) on summary statistics. Examples:

• A small company decides to give every one of its employees a $5000 raise. What happens to the mean and standard deviation of the salaries as a result?
• A small company decides to double each of its employee’s salaries. What happens to the mean and standard deviation of the salaries as a result?

Provide evidence about whether a linear pattern is apparent in a scatterplot of bivariate data, approximate a trend line or determine a line of best fit, interpret the slope and y-intercept of the trend line in the context of the data, interpolate and extrapolate to make predictions, and defend the resulting predictions. Statistical functions on a calculator or computer may be useful tools for determining lines of best fit.

Provide evidence about whether a recognizable nonlinear pattern is apparent in a scatterplot of bivariate data, and, if so, approximate the exponential or quadratic curve of best fit, interpolate and extrapolate to make predictions, and defend the resulting predictions.

Determining curves of best fit at this level is best done using a calculator or technological tool.

State whether the correlation of data represented in a given scatterplot is strong or weak and whether it is positive or negative, and relate the strength of the correlation to the value of the correlation coefficient.

Compare and draw conclusions about two or more univariate data sets using graphic displays of their distributions to answer questions and solve problems.

Use the defining characteristics of a normal distribution to identify and justify common examples that are reasonably modeled with normal distribution and those that are not.

Apply the empirical rule to approximate the percentage of the population meeting certain criteria in a normal distribution. The empirical rule is sometimes called the 68-95-99.7 rule.

Identify and justify possible causes and effects of skewed and clustered distributions, including outliers.

Determine whether arguments based on data confuse association with causation and, when they do, formulate valid inferences and conclusions based on the data.

Evaluate the reasonableness of and make judgments about claims, reports, studies, and conclusions.

Use examples to illustrate how a given set of data can be used to support different points of view.

Determine possible sources of bias in questions, data collection methods, and sampling techniques, and describe how these can impact the accuracy of the results of an experiment or study. Example:

• Identify a possible source of bias in this question and suggest an unbiased alternative: Since smoking is so bad for a person’s health, shouldn’t there be a tax on the sale of cigarettes to discourage people from buying them?

Describe the nature and purpose of sample surveys, statistical experiments, and observational studies, relating each to the types of research questions they are best suited to address. Example:

• Explain why observational studies generally do not lead to good cause-and-effect statements.

Select and justify an appropriate data collection method (e.g., survey, statistical experiment, or observation) for a given research question, and, conversely, identify specific research questions that could be addressed by a given data collection method. Example: A study questions whether teenagers who take physical education do better in school. Which of the following data collection methods would best serve to answer this question?

• A comparison of cumulative grade points of students enrolled in physical education and students not enrolled in physical education.
• An experiment in which one group of teenagers eats only fast food and another eats only salads.
• Phone calls are made to homes during the school day asking about their exercise routines and their grades.

Formulate a question that can be answered by analyzing data, identify relevant data sources, create an appropriate data display, select appropriate statistical techniques to answer the question, report results, and draw and defend conclusions. The focus is on identifying relevant and reliable existing data sources, rather than collecting original data.

Critique various methods of experimental design, data collection, and data presentation used to investigate real-world problems, including those reported in public studies.

Describe and evaluate methods to select a simple random sample, and explain the rationale for using randomness in research designs.

Distinguish between random sampling in surveys and random assignment to treatments in experiments.

Simulate the collection of data to carry out a statistical experiment or use reliable sample data to make conjectures, draw conclusions, and answer a research question.

Apply the Fundamental Counting Principle and the ideas of order and replacement to calculate probabilities in situations including those arising from two-stage experiments (compound events). Examples:

• What is the probability of getting exactly two heads when tossing a fair coin three times?
• What is the probability of getting at least two heads in three tosses of a fair coin?

Given a finite sample space consisting of equally likely outcomes and containing events A and B, determine whether A and B are independent or dependent, and find the conditional probability of A given B. Example:

• What is the probability of drawing a heart from a standard deck of cards on a second draw, given that a heart was drawn on the first draw and discarded.

Compute permutations and combinations, and use the results to calculate probabilities.

Describe the relationship between theoretical probability and empirical frequency of events in mathematical and applied settings; and use simulations to estimate the probabilities of events where theoretical values are difficult or impossible to compute.

Use experimental and theoretical probability to investigate, represent, make, and defend decisions about practical situations involving uncertainty.

Recognize, interpret, and correct common misconceptions about odds and risk in terms of their relationship to probability. Example:

• Sixteen-year-old Christopher just got his driver’s license. His older brother, Tony, is 25 years old, has a full-time job, and is married. Their mother, Renee, is 48 years old and works for a large corporation. They are quoted the following rates for car insurance on the car they all share equally: Christopher: $200 per month;   Tony: $125 per month;   Renee: $75 per month.  For any two of the individuals listed, give possible reasons for the differences in their insurance quotes.

Solve probability problems involving area models, including nonrectangular models. Example:

• A dartboard measures 30 cm in diameter. It consists of an inner region, painted white, that is 10 cm in diameter, and an outer region, painted blue, that makes up the rest of the dartboard. If a darts hits the board at random point, and each location on the board is equally likely, what is the probability of hitting the inside portion?

Describe whether the outcome of a first event affects the probability of a later event.

Describe the difference between dependent and independent events. 

Describe the relationship between theoretical probability and empirical frequency of dependent events using simulations with and without technology.

Determine the sample space for independent or dependent events.

Determine probabilities of dependent and independent events. 

Determine the outcomes and probability of multiple independent or dependent events.

Modify or revise a simple game based on independent probabilities so that all players have an equal probability of winning. 

Create a simple game based on conditional probabilities.

Determine whether claims made about results are based on biased data due to sampling.

Collect data using appropriate questions, samples, and/or methods to control for bias.

Examine sources of bias in data collection questions, samples, and/or methods and describe how such bias can be controlled.

Examine methods and technology used to investigate a research question. 

Determine how data collection methods impact the accuracy of the results. 

Determine whether the underlying model for a set of data is linear.  

Determine whether an equation for a line is appropriate for a given set of data and supports the judgment with data. 

Match an equation with a set of data or a graphic display.

Identify trends in a set of data in order to make a prediction based on the information.

Determine whether a prediction is reasonable based on the given data or graph.

Explain how the same set of data can support different points of view.

Explain, using data, how statistics have been used or misused to support a point of view or argument.

Use statistics to support different points of view.  

Use a set of statistics to develop a logical point of view.