IT/Mathematics: Statistical Science

Abstract

This chapter is devoted to the mathematics component of the FI³T project, which focused on statistical science with applications in three increasingly important areas of scientific inquiry: public health and medical data, environmental issues, and manufacturing reliability and safety issues. We describe the workshops of the capacity building first year as well as the project-related work during the second year for both cohorts of students in the project. The descriptions of workshop activities, of the work of visiting scientists, and of field trips to local businesses exhibit the applications and importance of mathematics and statistics in STEM related fields. The chapter also provides information about career opportunities and how the workshop themes align with the high school Common Core Standards.

Appendix

6.1.1 Appendix: A—Common Core Statistics Standards

Content Standards for Statistics

Middle School Standards

Grade 7 Standards (7.SP: Statistics and Probability)

Cluster: Use random sampling to draw inferences about a population.

• 1. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

• 2. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.

Cluster: Draw informal comparative inferences about two populations.

• 3. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.

• 4. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

Cluster: Investigate chance processes and develop, use, and evaluate probability models.

• 5. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

• 6. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.

• 7. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

  • a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.

  • b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

• 8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

  • a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

  • b. Represent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.

  • c. Design and use a simulation to generate frequencies for compound events.

Grade 8 Standards (8.SP : Statistics and Probability)

Cluster: Investigate patterns of association in bivariate data.

• 1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

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

• 3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

• 4. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

High School Standards (Statistics and Probability)

Interpreting Categorical and Quantitative Data (S-ID)

Cluster: Summarize, represent, and interpret data on a single count or measurement variable.

• 1. Represent data with plots on the real number line (dot plots, histograms, and boxplots).

• 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.

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

• 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.

Cluster: Summarize, represent, and interpret data on two categorical and quantitative variables.

• 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.

• 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. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

  • 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.

Cluster: Interpret linear models.

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

• 8. Compute (using technology) and interpret the correlation coefficient of a linear fit. [Note: Students used Minitab, Fathom, and Excel for computer-based analysis of data.]

• 9. Distinguish between correlation and causation.

Making Inferences and Justifying Conclusions (S-IC)

Cluster: Understand and evaluate random processes underlying statistical experiments.

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

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

Cluster: Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

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

• 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.

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

• 6. Evaluate reports based on data.

Conditional Probability and the rules of Probability (S-CP)

Cluster: Understand independence and conditional probability and use them to interpret data.

• 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”).

• 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.

• 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.

• 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. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

• 5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model

• 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.

• 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.

• 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.

• 9. Use permutations and combinations to compute probabilities of compound events and solve problems.

Using Probability to make decisions (S-MD)

Cluster: Calculate expected values and use them to solve problems

• 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.

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

• 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.

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

Cluster: Use probability to evaluate outcomes of decisions.

• 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.

• 6. Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

• 7. Analyze decisions and strategies using probability concepts (e.g., product testing).