Stat 201 - Chapter 11

When making predictions based on regression​ lines,

-Use the regression equation for predictions only if the graph of the regression line on the scatterplot confirms that the regression line fits the points reasonably well. -If the regression equation does not appear to be useful for making​ predictions, the best predicted value of a variable is its point estimate. -Use the regression equation for predictions only if the linear correlation coefficient r indicates that there is a linear correlation between the two variables.

Correlation

A correlation exists between two variables when the values of one variable are somehow associated with the values of the other variable. If the​ x-values increase as the corresponding​ y-values increase, we say that there is a positive correlation between x and y. If the​ x-values decrease as the corresponding​ y-values increase, we say that there is a negative correlation between x and y.

residual plot

A residual plot is a scatterplot of the​ (x,y) values after each of the​ y-coordinate values has been replaced by the residual value y − y(hat) ​(where y(hat) denotes the predicted value of​ y). That​ is, a residual plot is a graph of the points ​(x,y− y(hat) ​).

Influential points

Points that strongly affect the graph of the regression line are known as influential points. Paired sample data may include one or more such points.

Response variable

Response variable measures the outcome of a study (dependent)

Linear correlation coefficient r

The linear correlation coefficient r measures the strength of the linear correlation between the paired quantitative​ x- and​ y-values in a sample. Because the linear correlation coefficient r is calculated using sample​ data, it is a sample statistic used to measure the strength of the linear correlation between x and y.

presence of linear correlation

The presence of a linear correlation between two variables does not imply that one of the variables is the cause of the other variable.

Scatterplot

When determining whether there is a correlation between two​ variables, one should use a scatterplot to explore the data visually.

Use the regression line for predictions only if

data do not go far beyond the scope of the available sample data. In​ fact, predicting too far beyond the scope of the available data is called​ extrapolation, and it could result in bad predictions

Explanatory

Explanatory helps explain or influences the changes in the response variable (independent)

Residual

For a pair of sample​ x- and​ y-values, the residual is the difference between the observed sample value of y and the​ y-value that is predicted by using the regression equation.

Determine whether there is sufficient evidence to support the claim of a linear correlation between the two variables.

If the computed linear correlation coefficient r lies in the left tail beyond the leftmost critical value or if it lies in the right tail beyond the rightmost critical​ value, reject H0 and conclude that there is sufficient evidence to support the claim of a linear correlation. If the computed linear correlation coefficient lies between the two critical​ values, fail to reject H0 and conclude that there is no sufficient evidence to support the claim of a linear correlation.