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AP Statistics Syllabus and Objectives

John McPeak

Northgate High School

California Department of Education Math Standards

Advanced Placement Probability and Statistics

This discipline is a technical and in-depth extension of probability and statistics. In particular, mastery of academic content for advanced placement gives students the background to succeed in the Advanced Placement examination in the subject.

There are 19 specific standards that students should be able to meet as a result of taking this course. They are:

1.0 Students solve probability problems with finite sample spaces by using the rules for addition, multiplication, and complementation for probability distributions and understand the simplifications that arise with independent events.

2.0 Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.

3.0 Students demonstrate an understanding of the notion of discrete random variables by using this concept to solve for the probabilities of outcomes, such as the probability of the occurrence of five or fewer heads in 14 coin tosses.

4.0 Students understand the notion of a continuous random variable and can interpret the probability of an outcome as the area of a region under the graph of the probability density function associated with the random variable.

5.0 Students know the definition of the mean of a discrete random variable and can determine the mean for a particular discrete random variable.

6.0 Students know the definition of the variance of a discrete random variable and can determine the variance for a particular discrete random variable.

7.0 Students demonstrate an understanding of the standard distributions (normal, binomial, and exponential) and can use the distributions to solve for events in problems in which the distribution belongs to those families.

8.0 Students determine the mean and the standard deviation of a normally distributed random variable.

9.0 Students know the central limit theorem and can use it to obtain approximations for probabilities in problems of finite sample spaces in which the probabilities are distributed binomially.

10.0 Students know the definitions of the mean, median, and mode of distribution of data and can compute each of them in particular situations.

11.0 Students compute the variance and the standard deviation of a distribution of data.

12.0 Students find the line of best fit to a given distribution of data by using least squares regression.

13.0 Students know what the correlation coefficient of two variables means and are familiar with the coefficient's properties.

14.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.

15.0 Students are familiar with the notions of a statistic of a distribution of values, of the sampling distribution of a statistic, and of the variability of a statistic.

16.0 Students know basic facts concerning the relation between the mean and the standard deviation of a sampling distribution and the mean and the standard deviation of the population distribution.

17.0 Students determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error.

18.0 Students determine the P- value for a statistic for a simple random sample from a normal distribution.

19.0 Students are familiar with the chi- square distribution and chi- square test and understand

The specific syllabus of the Northgate course begins below.

We will use as our text The Practice of Statistics by Yates, Moore & McCabe. For each chapter and subject below, here are the things you should be able to do.

Chapter 1: Exploring Data

Section 1.1: Displaying distributions with graphs

Define and contrast Quantitative Data and Categorical Data; give an example of each.
Describe a distribution in terms of center, shape and spread
Given a 1-variable data set, create a manual dot plot.
Given a 1-variable data set, create a manual histogram.
Given a visual display of a distribution:

Ø Describe the overall pattern by center and spread;

Ø Identify any outliers;

Ø State whether the distribution is symmetric, skewed left, skewed right, or “other”.

Given a data set, construct a simple Stemplot or a back-to-back Stemplot.
Given a time-dependent data set, construct a time plot.
Use calculator LINK fundtion to send and receive lists

Sec 1.2: Describing distributions with numbers

Given a dataset, find the mean and median.
Describe a data set for which the mean and median agree or disagree.
Given a data set, find the range, quartiles, interquartile range (IQR) and outlier boundaries.
Represent a given data set by a 5 number summary or a modified boxplot.
Given a data set, use a formula to find the Variance and Standard Deviation of the data.

Chapter 2: The Normal Distributions

Sec 2.1: Density Curves and the Normal Distributions

Draw a density curve based on a definition
Locate the mean, median & quartiles of a density curve
Draw a normal curve and axis labeled in terms of mean and std. dev.
Apply the Empirical Rule (68-95) to answer questions about a distribution
Find the percentile of an observation

Sec 2.2: Standard Normal Calculations

Calculate the z-score of an observation
Use Table A or the TI to find the area within a specified region under a standard normal curve
Use Table A or the TI to find a z-score associated with a given area
Calculate the x value associated with a z-score
Assess whether a distribution is approximately normal based on shape, rule & normal probability plot.

Chapter 3: Relationships in Two-Variable Data

Sec 3.1: Scatterplots

· Given a 2 variable data set, construct and interpret a scatterplot.

Identify which of two variables is explanatory and which is responsive.
Describe a scatterplot in terms of direction, shape and strength.
Construct and interpret a scatterplot with 3 variables, one of which is categorical.

Sec 3.2: Correlation

· Calculate the Correlation Constant r from linear models by using a formula and on calculator.

Use the Correlation Constant r to describe strength and direction of a relation.

Sec 3.3: Least Squares Regression

Find the Least Squares Regression Line (LSLR) by using formulas and calculator.
Create a Residual Plot on calculator and draw conclusions about goodness of fit of LSLR.
Use LSLR to predict y for a given x.
Describe how outliers and influential observations affect LSLR.
Use r-squared to describe how much of the y-variation comes from the linear relation with x

Chapter 4: Nonlinear Two-Variable Data

Sec 4.1: Modeling nonlinear data

Given a data set that grows exponentially, use transformations to find its equation.
Given a data set that behaves like a polynomial, use transformations to find its equation.
Determine from problem context and residual plots whether the exponential or polynomial model gives the best fit of the data.

Sec 4.2: Interpreting Correlation and Regression

Interpret regression in light of extrapolation limitations, “lurking” variables, and the issue of association vs. causation.

Sec 4.3: Relations in Categorical Data

From a two-way table of counts, find the marginal distributions of both variables.
Express conditional distributions as percents.
Describe the relationship between two categorical variables by comparing percents.

Chapter 5: Producing Data

Sec 5.1: Designing Samples

Define the terms population and sample.
Define each type of sample: Probability Sample, Simple Random Sample (SRS), Stratified Random Sample, Systematic Random Sample, and Multistage Sample.
Use a Table of Random Numbers or a calculator Random Number Generator to select an SRS from a population.
Define each type of problems in samples: Undercover age, Non-response, Response Bias, Wording Effects.

Sec 5.2: Designing Experiments

· Define and give examples of each term: Observation vs. Experiment, Subjects & Treatment, Factors & Levels.

Design experiments that require randomization and a control group. Express the design as a schematic drawing and in a written paragraph.

Sec 5.3: Simulating Experiments

· State and carry out the five steps of a simulation.

Use a calculator to perform a simulation.
Find the empirical probability of an event based on simulations.

Chapter 6: Probability

Sec 6.0: Counting Theory

· Use tree diagrams and organized lists to find all possible outcomes of a trial.

Use permutation and combination techniques to find the number of possible outcomes of a trial.

Sec 6.1: Random Outcomes

· Describe what is meant by random outcomes of a trial

· Describe what is meant by the probability of a given outcome

Sec 6.2: Probability Models

· Define and give examples of each term: Disjoint, Independent, Sample Space, Replacement, Equally Likely Outcomes.

Use the rules of probability to determine whether a probability model is legitimate.
Given a description of an event, state its complement.
Given the probability of an event, find the probability of its complement.

Sec 6.3: Conditional Probability

· Define and give examples of each term: Union, Intersection, Conditional Probability.

· Given two events A and B, use the appropriate addition or multiplication rule for conditional probability to find P(A or B) and/or P(A and B).

· Given a two-way table of frequencies, find probabilities of specified events.

· Given sufficient information about two events A and B, use a formula to evaluate the conditional probability P(B|A).

Use counting techniques to find theoretical probability in problems involving coins, dice, cards and marbles.
Use experiment and simulation to find empirical probabilities involving coins, dice, cards and marbles.

Chapter 7: Random Variables

Sec 7.1: Discrete and Continuous Random Variables

· Given the definition of a discrete Random Variable, write a Probability Distribution Function (PDF) as either a table or a histogram.

· Display the PDF of a continuous random variable as a density curve.

· Find the probability that a given continuous random variable is within a specified interval by finding the area under a density curve.

Sec 7.2: Means and Variances of Random Variables

· Find the mean of a discreet random variable from formula and on calculator

· Calculate the variance and standard deviation of a discrete random variable from formula and calculator

· Find the mean and variance of a random variable formed from a linear transformation of a given random variable with known mean and variance.

· Find the mean and variance of a random variable formed from a linear combination of two given random variables with known mean and variance.

Chapter 8: Binomial and Geometric Distributions

Sec 8.1: The Binomial Distribution

· Identify whether a random variable is in a binomial setting.

· Use TI-83 or the binomial probability formula to find binomial probabilities and construct probability distribution tables and histograms.

· Calculate cumulative distribution functions for binomial random variables and construct cumulative distribution tables and histograms.

· Calculate the mean (expected value) and standard deviation (and variance) of a given binomial random variable from formulas and on calculator.

Sec 8.2: The Geometric Distribution

· Identify whether a random variable is in a geometric setting.

· Use formulas or a TI-83 to determine geometric probabilities and construct probability distribution tables and histograms.

· Calculate cumulative distribution functions for geometric random variables and construct cumulative distribution tables and histograms.

· Calculate the mean (expected value) of a given geometric random variable.

Chapter 9: Sampling Distributions (10 days)

Sec 9.1: Sampling Distributions

· Describe the difference between parameters and statistics.

· Define what is meant by the terms variability, sample means and sample proportions.

· Describe the bias and variability of a statistic in terms of the mean and spread of its sampling distribution.

· Describe the relationship between variability of a statistic and sample size.

Sec 9.2: Sample Proportions

· Find the mean and standard deviation of the distribution of sample proportions taken from an SRS of size n from a population having population proportion p.

· Describe and use the relationship between sample proportion standard deviation and sample size.

· Apply appropriate tests (Rules of Thumb) to determine whether it is appropriate to use the normal approximation to the sampling distribution.

· Use the normal approximation to calculate probabilities concerning sample proportions.

Sec 9.3: Sample Means

· Find the mean and standard deviation of the distribution of sample means taken from an SRS of size n from a population whose mean and standard deviation are known.

· Describe and use the relationship between sample proportion standard deviation and sample size.

· State the Central Limit Theorem.

· Use the normal approximation to calculate probabilities concerning sample means.

Chapter 10: Introduction to Inference

Sec 10.1: Confidence Intervals

· Describe what is meant by a Confidence Level, Confidence Interval, and Margin of Error

· Calculate the margin of error and the confidence interval for a given sample from a population

· Calculate the sample size needed to give a desired margin of error

Sec 10.2: Hypotheses & Tests of Significance

· Given an experimental situation, state the Null and Alternative Hypotheses (H_o & H_a)

· Calculate the z test statistic and the P-value for a given sample mean

· Determine whether the sample mean is statistically significant.

· Describe the logic of hypothesis testing in words and through a sketch.

Sec. 10.3: Using Significance Tests

· Apply critical judgment to the use of significance tests, particularly as to the nature of the sample taken and the subjectivity of confidence levels

Sec 10.4: Making Decisions from inference

· Define a Type I Error and a Type II Error

· Find the probability of Type I and Type II Errors

· Define and evaluate the Power of a significance test

· Describe the relationship between power and sample size in a significance test

Chapter 11: Inference for Population Means

Sec 11.1: One sample means

· State the assumptions needed for inference about means

· Find a t confidence interval for a population mean

· Perform a t-test for hypotheses about population means

· Find the power of a t-test

Sec 11.2: Two sample means

· State the assumptions needed for inference about two-sample means

· Find a confidence interval for the difference of two population means

· Perform a t-test for hypotheses about the difference of two population means

Chapter 12: Inference for Population Proportions

Sec 12.1: One-sample proportions

· State the assumptions needed for inference about proportions

· Find a confidence interval for a population proportion

· Perform a test for hypotheses about population proportions

Sec 12.2: Two sample proportions

· State the assumptions needed for inference about two-sample proportions

· Find a confidence interval for the difference of two population proportions

· Perform a test for hypotheses about the difference of two population proportions

Chapter 13: Inference for Tables

Sec 13.1: Test for Goodness of Fit

· Calculate expected counts for a distribution

· Calculate the chi-square statistic based on the difference between expected and observed counts

· Perform a test for hypotheses about a distribution

Sec 13.2: Inference for Two-Way Tables

· Create a two-way table of counts for two categorical variables

· Calculate expected counts for a table

· Calculate the chi-square statistic based on the difference between expected and observed counts.

· Perform a test for hypotheses about a table.

Chapter 14: Inference for Regression

Sec 14.1: Inference about the Model

· Given a two-variable data set, determine explanatory and response variables and create a scatterplot with appropriately labeled axes

· Use a calculator or computer software to find the least squares regression line. Comment on the goodness of fit based on outliers, influential points and the values of r and r2.

· Estimate the three parameters alpha, beta, and sigma of the regression model

· Find a confidence intervbal for the regression slope

· Test the hypothesis of no linear relationship

Sec 14.2: Inference about prediction

· Find a confidence interval for the predicted mean value of all responses to a given input

· Find a prediction interval for the predicted value of one response to a given input

Sec 14.3: Checking assumptions

· State and check the underlying assumptions used in regression analysis

Chapter 15: Analysis of Variance (ANOVA)

Sec 15.1: Inference for population spread

· State the assumptions controlling the F-test

· Perform an F-test to compare two population standard deviations

Sec 15.2: One-way analysis of variance

· State the Null and Alternative Hypotheses in comparing three or more population means

· Perform the ANOVA on calculator and state the appropriate results

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