AP Stats Chapter 3-4 Flashcards

Questions and Answers

What is the difference between a response variable and an explanatory variable?

A response variable measures an outcome of a study. An explanatory variable attempts to explain the observed outcomes.

How are response and explanatory variables related to dependent and independent variables?

The explanatory variable is usually called independent and the response variable is called dependent.

When is it appropriate to use a scatterplot to display data?

A scatterplot displays the relation between two quantitative variables.

Which variable always appears on the horizontal axis of a scatterplot?

The explanatory variable is always on the horizontal (x-) axis. Signup and view all the answers

Explain the difference between a positive association and a negative association.

A positive association is when above-average values of one variable tend to accompany above-average values of the other. Below-average values also tend to occur together. Two variables are negatively associated when above-average values of one tend to accompany below-average values of the other, and vice versa. Signup and view all the answers

What does correlation measure?

The correlation measures the direction and strength of the linear relationship between two quantitative variables. Signup and view all the answers

Explain why two variables must both be quantitative in order to find the correlation between them.

If the variables are not quantitative we cannot do the arithmetic required in the formulas for r. Signup and view all the answers

What is true about the relationship between two variables if the r-value is: 1.Near 0?; 2.Near 1?; 3.Near -1?; 4.Exactly 1?; 5.Exactly -1?

Very weak linear relationship Signup and view all the answers

Is correlation resistant to extreme observations?

False Signup and view all the answers

What does it mean if two variables have high correlation?

They have a strong linear relation. Signup and view all the answers

What does it mean if two variables have weak correlation?

They have a weak linear relation. Signup and view all the answers

What does it mean if two variables have no correlation?

They have no linear relation. Signup and view all the answers

How is correlation affected when you change the units of measurement for one or both variables?

It is not affected if you change the units of measurement. Signup and view all the answers

How useful is correlation in describing the strength of curved relationships between variables?

Not at all. Signup and view all the answers

In what way is a regression line a mathematical model?

It describes how a response variable y changes as an explanatory variable x changes. Signup and view all the answers

What is a least-squares regression line?

A least-squares regression line is a regression line that minimizes the sum of the squares of the residuals. Signup and view all the answers

What is the formula for the equation of the least-squares regression line?

(ŷ) = a + bx ; b = r(sy/sx) ; a = ymean - b(xmean) Signup and view all the answers

How is correlation related to least-squares regression?

There is a close connection between correlation and the slope of the least-squares line. Signup and view all the answers

The least-squares regression line always passes through what point?

The grand mean (xmean, ymean). Signup and view all the answers

How is the least-squares regression line affected if we interchange the explanatory and response variables?

False Signup and view all the answers

What is the formula for calculating the coefficient of determination?

r² = (E(y−ymean)² - E(y−ŷ)²) / E(y−ymean)² Signup and view all the answers

If r² = 0.95, what can be concluded about the relationship between x and y?

95% of the variation in y can be explained by the least-squares regression of y on s. Signup and view all the answers

Define residual.

A residual is the difference between an observed value of the response variable and the value predicted by the model. Signup and view all the answers

Study Notes

Variables

A response variable measures the outcome of a study, while an explanatory variable attempts to explain observed outcomes.
The explanatory variable is also known as the independent variable; the response variable is referred to as the dependent variable, though this terminology is less commonly used in statistics.

Data Visualization

A scatterplot is appropriate for displaying the relationship between two quantitative variables.
The explanatory variable is always plotted on the horizontal (x-) axis in scatterplots.

Associations

A positive association indicates that above-average values of one variable tend to occur with above-average values of another, whereas below-average values occur together.
A negative association indicates that above-average values of one variable correspond with below-average values of the other.

Correlation

Correlation measures both the direction and strength of a linear relationship between two quantitative variables.
In order to calculate correlation, both variables must be quantitative as non-quantitative variables cannot support the necessary arithmetic operations.

Understanding r-values

An r-value near 0 indicates a very weak linear relationship.
An r-value near 1 indicates a strong positive linear relationship.
An r-value near -1 indicates a strong negative linear relationship.
An r-value of exactly 1 indicates a perfect positive linear relationship.
An r-value of exactly -1 indicates a perfect negative linear relationship.
Outliers can significantly impact the correlation value (r), thus correlation is not resistant to extreme observations.

Correlation Characteristics

High correlation indicates a strong linear relation between two variables.
Weak correlation suggests a weak linear relation between two variables.
No correlation implies no linear relation exists between the variables.
Changing the units of measurement for either variable does not affect the correlation value.
Correlation is ineffective in describing the strength of curved relationships between variables.

Regression Analysis

A regression line is a mathematical model that represents how a response variable changes as an explanatory variable changes, and it predicts the value of the response variable for given values of the explanatory variable.
The least-squares regression line minimizes the sum of the squares of the residuals (vertical distances from the data points to the line).
The formula for the least-squares regression line is:
( \hat{y} = a + bx )
Where ( b = r \left(\frac{s_y}{s_x}\right) ) and ( a = \bar{y} - b \bar{x} ).
Correlation is closely linked to the slope of the least-squares regression line.

Key Points on Regression Line

The least-squares regression line passes through the grand mean ((\bar{x}, \bar{y})).
Interchanging the explanatory and response variables does not alter the outcome of the least-squares regression line.

Coefficient of Determination

The formula for the coefficient of determination (( r^2 )) is:
( r^2 = \frac{E(y - \bar{y})^2 - E(y - \hat{y})^2}{E(y - \bar{y})^2} )
An ( r^2 ) value of 0.95 implies that 95% of the variation in ( y ) can be explained by its linear relationship with ( x ).

Residuals

A residual is the difference between the observed value of the response variable and the predicted value based on the regression line.

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Description

Test your knowledge of key concepts from Chapters 3 and 4 of AP Statistics. This set of flashcards focuses on the definitions and relationships between response variables, explanatory variables, and their roles as dependent and independent variables. Perfect for exam preparation and understanding statistical relationships.