# Correlations

# Statistics: The Art and Science of Learning from Data ©2007

Alan Agresti and Christine Franklin

### Correlated to: Advanced Placement® (AP®) Statistics Standards (Grades 9–12)

## I. Exploring Data: Describing patterns and departures from patterns

A. Constructing and interpreting graphical displays of distributions of univariate data (dotplot, stemplot, histogram, cumulative frequency plot) | |

1. Center and spread | 2.3, 2.4 |

2. Clusters and gaps | 2.2 |

3. Outliers and other unusual features | 2.3, 2.5 |

4. Shape | 2.2 |

B. Summarizing distributions of univariate data | |

1. Measuring center: median, mean | 2.3 |

2. Measuring spread: range, interquartile range, standard deviation | 2.4,2.5 |

3. Measuring position: quartiles, percentiles, standardized scores (z-scores) | 2.5 |

4. Using boxplots | 2.5 |

5. The effect of changing units on summary measures | |

C. Comparing distributions of univariate data (dotplots, back-to back stemplots, parallel boxplots) | |

1. Comparing center and spread: within group, between group variation | Ch. 2 |

2. Comparing clusters and gaps | Ch. 2 |

3. Comparing outliers and other unusual features | Ch. 2 |

4. Comparing shapes | Ch. 2 |

D. Exploring bivariate data | |

1. Analyzing patterns in scatterplots | 3.2 |

2. Correlation and linearity | 3.2 |

3. Least-squares regression line | 3.3 |

4. Residual plots, outliers, and influential points | 3.3, 3.4 |

5. Transformations to achieve linearity: logarithmic and power transformations | 11.5 |

E. Exploring categorical data | |

1. Frequency tables and bar charts | 3.1 |

2. Marginal and joint frequencies for two-way tables | 3.1 |

3. Conditional relative frequencies and association | 3.1 |

4. Comparing distributions using bar charts | 3.1 |

## II. Sampling and Experimentation: Planning and conducting a study

A. Overview of methods of data collection | |

1. Census | 4.1 |

2. Sample survey | 4.1 |

3. Experiment | 4.1 |

4. Observational study | 4.1 |

B. Planning and conducting surveys | |

1. Characteristics of a well-designed and well-conducted survey | 4.2 |

2. Populations, samples, and random selection | 4.2 |

3. Sources of bias in surveys | 4.2 |

4. Sampling methods, including simple random sampling, stratified random sampling, and cluster sampling | 4.4 |

C. Planning and conducting experiments | |

1. Characteristics of a well-designed and well-conducted experiment | 4.3 |

2. Treatments, control groups, experimental units, random assignments, and replication | 4.3 |

3. Sources of bias and confounding, including placebo effect and blinding | 4.2, 4.3 |

4. Completely randomized design | 4.4 |

5. Randomized block design, including matched pairs design | 4.4 |

D. Generalizability of results and types of conclusions that can be drawn from observational studies, experiments, and surveys | Ch.4 |

## III. Anticipating Patterns: Exploring random phenomena using probability and simulation

A. Probability | |

1. Interpreting probability, including long-run relative frequency interpretation | 5.1 |

2. "Law of large numbers" concept | 5.1 |

3. Addition rule, multiplication rule, conditional probability, and independence | 5.2, 5.3 |

4. Discrete random variables and their probability distributions, including binomial and geometric | 6.1, 6.3 |

5. Simulation of random behavior and probability distributions | 5.4 |

6. Mean (expected value) and standard deviation of a random variable, and linear transformation of a random variable | 6.1 |

B. Combining independent random variables | |

1. Notion of independence versus dependence | 9.1, 5.2 |

2. Mean and standard deviation for sums and differences of independent random variables | |

C. The normal distribution | |

1. Properties of the normal distribution | 6.2 |

2. Using tables of the normal distribution | 6.2 |

3. The normal distribution as a model for measurements | 6.2 |

D. Sampling distributions | |

1. Sampling distribution of a sample proportion | 6.4 |

2. Sampling distribution of a sample mean | 6.5 |

3. Central Limit Theorem | 6.5 |

4. Sampling distribution of a difference between two independent sample proportions | 9.1 |

5. Sampling distribution of a difference between two independent sample means | 9.2 |

6. Simulation of sampling distributions | 5.4 |

7. t-distribution | 7.3 |

8. Chi-square distribution | 10.2 |

## IV. Statistical Inference: Estimating population parameters and testing hypotheses

A. Estimation (point estimators and confidence intervals) | |

1. Estimating population parameters and margins of error | 7.1 |

2. Properties of point estimators, including unbiasedness and variability | 7.1 |

3. Logic of confidence intervals, meaning of confidence level and confidence intervals, and properties of confidence intervals | 7.1 |

4. Large sample confidence interval for a proportion | 7.2 |

5. Large sample confidence interval for a difference between two proportions | 9.1 |

6. Confidence interval for a mean | 7.3 |

7. Confidence interval for a difference between two means (unpaired and paired) | 9.2 |

8. Confidence interval for the slope of a least-squares regression line | 11.4 |

B. Tests of significance | |

1. Logic of significance testing, null and alternative hypotheses; p-values; one- and two-sided tests; concepts of Type I and Type II errors; concept of power | 8.1, 8.4, 8.6 |

2. Large sample test for a proportion | 8.2 |

3. Large sample test for a difference between two proportions | 9.1 |

4. Test for a mean | 8.3 |

5. Test for a difference between two means (unpaired and paired) | 9.2 |

6. Chi-square test for goodness of fit, homogeneity of proportions, and independence (one- and two-way tables) | 10.2 |

7. Test for the slope of a least-squares regression line | 11.3 |

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