Student Learning Outcomes and Reporting Categories

Probability and Statistics

Student Learning Outcomes and Reporting Categories

A. / Variablesand Statistics

Identify observational units (cases or individuals) and variables
Distinguish types of variables as categorical or quantitative

Distinguish between populations and samples
Distinguish between parameters and statistics
Distinguish between explanatory and response variables

B. / Sampling

Carry out a simple random sample
Randomly assign subjects to treatments to form experimental groups

Identify sources of bias in poor sampling methods as in voluntary response and convenience samples
Identify non-sampling sources of bias such as the wording of a survey question
Understand that random sampling is an unbiased method where precisionis relatedto sample size alone

C. / Drawing Conclusions from Studies

Distinguish between observational studies and experimental studies
Understand the different types of conclusions that can be drawn from each

Understand principles of control including comparison, replication, randomization, blindness, and blocking
Carry out a well-controlled experiment
Critique studies by suggesting possible confounding or lurking variables not accounted for in a study and making a plausible argument as to how this variable is related to the explanatory and response variable

Understand the difference between an association and a cause and effect relationship.

D. / Comparing Distributions: Categorical versus Categorical

Compute and interpret conditional and marginal distributions
Create and interpret segmented bar graphs
Describe the relationship between two variables (categorical versus categorical)

Determine whether two categorical variables are independent
Understand Simpson's Paradox

E. / Graphs of a Single Variable

Describe the overall pattern of variability (distribution) for a variable by discussing the center (tendency), spread (consistency), and shape of a distribution verbally and in context
Identify departures from the overall pattern of a distribution (outliers)
Interpret various displays of a distribution or distributions (pie chart, bar graph, dotplot, stemplot, histogram)
Create by hand and using technology various displays of the distribution of a variable (pie chart, bar graph, dotplot, stemplot, histogram)
Determine which displays are appropriatefor different situations

F. / Measures of Center and Spread

Compute and interpret numerical measures of the center of a distribution including mean, median, and mode.
Apply the concept of resistance to measures of center
Determine which measures of center and spread apply for different situations
Compute and interpret measures of spread for a distribution including IQR, standard deviation, and range
Apply the concept of resistance to measures of spread
Understand the relationship between the shape of a distribution and the relative position of the mean and median
Empirical Rule
Apply the empirical rule for interpreting the standard deviation for mound-shaped distributions

G. / Comparing Distributions: Quantitative versus Categorical

Use the Five Number Summary to create and interpret boxplots
Understand and apply the concepts of statistical tendency and consistency
Be able to create displays for comparing distributions of two or more variables (including back-to-back stemplots, stacked dotplots, and modified boxplots)
Compare and contrast two or more distributions with regard to center, spread, and shape
Use the 1.5*IQR rule for identifying possible outliers and constructing modified boxplots

H. / Comparing Distributions: Quantitative versus Quantitative

Create and interpret scatter plots
Describe the direction, strength, and form of the relationship between two variables (measurement versus measurement) and identify outliers

Calculate and interpret the correlation coefficient

I. / Elementary Probability

Understand that a probability is a number between 0 and 1 that specifies the likelihood of an event
Understand that in a valid probability model the sum of the probabilities must be 1
Compute probabilities using an appropriate sample space, an area diagram, or a tree diagram
Understand the relative frequency interpretation of probability and the Law of Large Numbers
Conduct simulations with physical models and random number generators to make empirical estimates of probability
Distinguish between correct and incorrect conceptions of probability including the Gambler's Fallacy and the fallacious Law of Small Numbers

J. / Normal Distributions

Calculate z-scores and use them to compare measurements made on different scales
Sketch the graph of a normal distribution
Understand that a continuous probability distribution represents the probabilities in terms of area under the curve
Calculate probabilities/proportions pertaining to a normal distribution
Calculate values of a variable corresponding to given probabilities or proportions

K. / Linear Regression

Calculate the regression equation

Graph and make predictions based on the LSR line
Interpret the slope and intercept of the regression equation

Understand the Least Squares criterion for line of best fit

Use R2 to determine the proportion of variability explained by the model
Plot residuals to evaluate the appropriateness of the model
Apply a transformation of the data to create a more appropriate model

L. / Basic Counting Rules

Use the multiplication and addition rules of counting
Construct and apply tree diagrams and tables for systematically listing possibilities

M. / Advanced Counting Rules

Compute and apply factorials
Use combinations and permutations to solve counting problems
Distinguish between situations where combinations, permutations, or other counting rules apply
Compute probabilities using counting rules including permutations and combinations

N. / Probability with Compound Events

Solve probability problems involving unions, intersections, and complementary events
Apply the addition and multiplication rules
Use tree diagrams and area diagrams to compute probabilities
Compute and interpret conditional probabilities
Determine if two events are independent
Explain the concepts of conditional probability and independence in everyday situations

O. / Expected Value

Define a random variable by assigning probabilities to events
Graph the probability distribution of a random variable
Compute expected value and interpret it as the mean of the probability distribution

Make and defend decisions based on expected value calculations
Determine if a game of chance is fair

P. / Bayesian Probability

Compute the inverses of conditional probabilities using either a table, a tree diagram, or Bayes’ Theorem

Q. / The Binomial Probability Distribution

Compute the probability of k successes in n trials with a fixed probability of success p using a binomial distribution
Graph a binomial distribution
Carry out informal tests of statistical significance for counts or proportions using binomial distributions

R. / Sampling Distributions

Understand the concept of sampling variability and how sampling variability relates to sample size
Understand bias and variability (accuracy and precision) with respect to sampling distributions
Generate the sampling distribution of a statistic
Compute probabilities based on sampling distributions
Use data from a survey to estimate a population proportion and determine the margin or error of the estimate using a simulation model of random sampling
+Apply the Central Limit Theorem to describe the center, spread, and shape for the distribution of sample means or sample proportions and compute relevant probabilities

S. / Understanding Tests of Significance

Recognize that an apparent effect may be best explained by random sampling variability orpre-existing differences in the groups created by random assignment of subjects to treatments.
State null and alternative hypotheses in words

Understand the meaning of “statistical significance” and its relation to the significance level, alpha
Distinguish between practical significance and statistical significance

Explain what is meant by a p-value as a conditional probability
Explain Type I and Type II errors
+Understand the concept of statistical power
+Explain the relationships between power, Type I error rates, Type II error rates, the size of the effect, and sample size

T. / Carrying Out Randomization Tests

Understand the reasoning of a test of significance
Carry out and interpret the results of a randomization test by (1) specifyinghypotheses, (2) collecting and summarizingthe data, (3) constructing a sampling distribution by simulating data, (4) computing ap-value, (5) making decision, and (6) interpreting the results in context)

U. / Inference for Proportions

Carry out and interpret tests of significance for a single population proportion and for the comparison of two population proportions
Construct and interpret confidence interval estimatesfor a single population proportion and for the comparison of two population proportions
Determine the sample size needed for a desired level of confidence and margin of error
Understand the complementary uses of significance tests and confidence intervals
Verify that the technical conditions for z-tests for proportions are met

V. / Inference for Means

Compute the t statistic and compute probabilities based on the t distribution with the appropriate number of degrees of freedom
Carry out and interpret the results of t-tests for a single population mean, the difference between population means, and for the population mean difference in paired data
Construct and interpret the results of t-intervals for estimating a single population mean, the difference between population means, and for the population mean difference in paired data
Verify that the technical conditions for a t-test are met

W. / Inference for Categorical Variables

Apply a chi-square goodness of fit test and interpret the results
Apply a chi-square test for a two-way table and interpret the results
Distinguish among the three scenarios in which a chi-square test for a two-way table applies: independent samples from different populations, random assignment to groups, and one random sample with two categorical variables
Verify that the technical conditions for applying a chi-square test are met

X. / Inference for Correlation and Regression

Apply a t test for a correlation coefficient and interpret the results
Apply a t-test for a slope coefficient and interpret the results
Construct and interpret a t-interval for a slope coefficient
Use a residual plot to check technical conditions for applying t-procedures

Y. / Analysis of Variance (ANOVA)

Apply and interpret an ANOVA F-test for comparing population means for three or more groups
Verify that the technical conditions for applying the ANOVA F-test are met