# Quantitative Skills

Which of these quantitative skills are needed for students in your major?

Absolute Value Functions

1. Definition of absolute value and absolute value functions.
2. Graph of absolute value functions.
3. Solve absolute value equations.
4. Solve absolute value inequalities.

Algorithms

1. Redefine a complex problem into a sequential set of parts that can be translated into the language of programming logic.

Coding

1. Design, write, test, and debug computer programs.

Conic sections

1. Graph parabolas, hyperbolas, and ellipses.
2. Find vertices and foci for hyperbolas and ellipses and vertices and directrixes for parabolas.
3. Use parabolas, hyperbolas, and ellipses to model tunnels, satellite dishes, and other objects.

Data – Bivariate

1. Represent bivariate quantitative data using a scatter plot and describe how the variables might be related.
2. Compute and interpret a correlation coefficient given bivariate numerical data.
3. Distinguish between correlation and causation and between conspiracy and coincidence.
4. Critique graphical displays in the media.

Data – Categorical

1. Summarize categorical data by constructing frequency tables and relative frequency tables.
2. Display categorical data with bar graphs.
3. Exploring two categorical variables by analyzing contingency tables.
4. Critique graphical displays in the media.

Data – Collection

1. Distinguish between an observational study and a statistical experiment.
2. Describe the purpose of random selection in an observational study and the purpose of random assignment in a statistical experiment.
3. Understand the types of conclusions that can be drawn from an observational study and from an experiment.
4. Describe a method for selecting a random sample from a population.

Data Structures

1. Incorporate fundamental data management concepts such as data structures within computer programs.

Data – Univariate

1. Summarize univariate data using an appropriate graphical display.
2. Describe the shape of a distribution of numerical data and identify any outliers in the data set.
3. Summarize univariate data using appropriate numerical summary measures.
4. Interpret differences in shape, center and spread in the context of the data sets, accounting for possible extreme data points.
5. When appropriate, use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages.
6. Critique graphical displays in the media.

Derivatives

1. Calculate derivatives using product, quotient, and chain rules.
2. Solve related rates problems.
3. Use derivatives to find maximum and minimum values.
4. Use the tangent line to approximate functions.

Exponential Functions

1. Definition of exponential functions.
2. Use exponential functions to model growth and decay.
3. Graph of exponential functions.
4. Solving exponential equations.

Finance

1. Explore essentials of creating a family/personal budget.
2. Understand the difference between simple and compound interest and their effects on savings and expenditures.
3. Explore saving and investment accounts.
4. Explore loan payments, credit card accounts and mortgages.
5. Explain debt and deficit
6. Explain tax systems

Integrals

1. Calculate integrals using the idea of an antiderivative.
2. Calculate integrals using u-substitution.
3. Use integrals to represent area, net change, and other quantities approximated by a sum.

Linear Functions

1. Definition of linear functions.
2. Slope-intercept form and point-slope form.
3. Parallel and perpendicular lines.
4. Piecewise functions.
5. Solve linear equations.
6. Solve linear inequalities.
7. Interpret slope and intercept.
8. Use linear functions to model relationships.
9. Fit a line to data points that are perfectly linear.

Linear Regression

1. Construct a scatterplot of bivariate numerical data.
2. Calculate a correlation coefficient.
3. Calculate the least squares regression line of best fit.
4. Interpret the slope and y-intercept (if appropriate) of the least squares regression line in context.

Logarithmic Functions

1. Definition of logarithmic function.
2. Properties of logarithm.
3. Graph of logarithm functions.
4. Solving logarithm equations.
5. Using logarithm to solve exponential equations.

Logic:

1. Translate arguments into formal language and symbols
2. Recognize and apply standard techniques of logical deduction.
3. Recognize and avoid common logical fallacies.

Mathematical Modeling

1. Use function notation, understand functions as processes, and interpret statements that use function notation in terms of a context.
2. Construct graphs and tables that model changing quantities and interpret key features in terms of the quantities.
3. Interpret the slope and the intercept of a linear model in the context of the data.
4. Graph linear and exponential functions and identify critical points.
5. Compute and interpret the correlation coefficient of a linear fit.
6. Distinguish between situations that can be modeled with linear functions and those modeled with exponential functions.
7. Use linear and exponential functions to model contextual situations such as costs and growth of savings accounts.

Normal Distributions

1. Describe characteristics of a normal distribution and calculate and interpret a z-score using the mean and standard deviation of the normal distribution.
2. Calculate areas under a normal curve and interpret these areas as probabilities in context.
3. Approximate population percentages using a normal distribution.

Parametric Equations

1. Convert between parametric and rectangular (x, y) equations.
2. Graph parametric equations.

Polar Coordinates

1. Convert between polar and rectangular (x, y) equations.
2. Graph polar equations.

Polynomial Functions

1. Definition of polynomial functions.
2. Characteristics of polynomial functions: degree, zeros, multiplicity, turning points.
3. Long division and/or synthetic division.
4. Solve polynomial equations.
5. Solve polynomial inequalities.

Probability – Theory and Computation

1. Define sample space and events.
2. Distinguish between discrete and continuous random variables.
3. Explain that a probability distribution describes the long-run behavior of a random variable.
4. Calculate probabilities of unions, intersections and complements of events.
5. Use permutations and combinations to compute probabilities of compound events and solve problems.
6. Discuss the law of large numbers vs. law of averages myth.
7. Estimate probabilities empirically and interpret probabilities and long-run relative frequencies.
8. Distinguish between independent events and dependent events.
9. Use data in two-way tables to calculate probabilities, including conditional probabilities.
10. Calculate expected value.
11. Calculate standard deviation of a discrete variable

Probability – Reasoning and Interpretation

1. Explain probability as a measure of the likelihood that an event will occur.
2. Describe events as subsets of a sample space using characteristics of the outcomes, or as unions, intersections, or complements of other events.
3. Interpret probabilities of the union and intersection of independent and dependent events.
4. Interpret probabilities in context.
5. Interpret expected value in context.
6. Interpret conditional probabilities in cases such as the false positive paradox.
7. Analyze risk in health situations and understand the difference between absolute changes in risk and relative changes in risk.
8. Interpret expected payoff for a game of chance.

1. Definition of quadratic functions
2. Graph quadratic functions: vertex, intercepts, maximum or minimum
3. Solve quadratic equations: factoring, quadratic formula, graphically.
4. Solve quadratic inequalities.

Measurement

1. Understand the use of units, thinking of numbers as adjectives.
2. Study multiple ways of comparing quantities including the use of indices, e. g. the consumer price index and its relationship to the changing value of the dollar.
3. Investigate ways of finding exact and approximate areas and volumes of geometric and irregular shapes.
4. Estimate quantities.
5. Evaluate plausibility of numerical answers.

Rates of Change

1. Display time series data using a time series plot.
2. Compute and interpret average rate of change from data, graphs, or function equations.
3. Estimate and interpret instantaneous rate of change from data, graphs, or function equations.

Rational Functions

1. Definition of rational functions.
2. Graph of rational functions (including asymptotes).
3. Solve rational equations.
4. Solving rational inequalities.

Recursive Functions:

1. Write and use functions that call themselves to perform calculations, solve problems, or process data.

Statistical Inference – Theory and Computation

1. Describe characteristics of the sampling distribution of a sample mean and of a sample proportion.
2. Define and apply the central limit theorem for random samples.
3. Calculate a confidence interval for a population mean given a random sample.
4. Calculate a confidence interval for a population proportion.
5. Calculate a confidence interval for the difference in two population means or two population proportions.
6. Carry out a test of hypotheses about a population mean given a random sample.
7. Carry out a test of hypotheses about a population proportion.
8. Carry out a test of hypotheses about the difference in two population means or two population proportions.

Statistical Inference – Interpretation

1. Describe statistics as a process for making inferences about population parameters based on a random sample from that population.
2. Explain the concept of sample-to-sample variability and describe how this understanding relates to statistical inference.
3. Explain the meaning of margin of error and interpret margin of error in context.
4. Explain the meaning of a confidence interval and interpret a confidence interval in context.
5. Interpret a P-value in context and use a P-value to reach a conclusion in a hypothesis testing context.
6. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
7. Evaluate reports or print media articles based on statistical data.

Systems of Equations

1. Solving system of equations.
2. Set up systems of equations to solve problems involving mixtures, travel, and work rates, and other contexts.

Trigonometric Functions

1. Graph trigonometric functions (sine, cosine, tangent).
2. Use sine and cosine functions to model periodic phenomena.
3. Solve triangles based on partial information about angles and sides, in applied settings.
4. Solve trigonometric equations.