In the equation y = a + bx, the number b is the slope, which gives the change in y for each one-unit increase in x.
Missing Word
slope
The slope describes how steep a line is and in which direction it tilts. In regression it is the predicted change in y for each one-unit increase in x.
Category: Linear Equations
Question 2
The difference y - y-hat for a data point is often loosely called an error, but its standard name is the residual.
Missing Word
residual
A residual is the observed value minus the value predicted by the regression line. Positive residuals belong to points above the line and negative residuals to points below it.
Category: Least-Squares Line
Question 3
A graph that shows how two quantitative variables relate by plotting each (x, y) pair as a point is called a scatter plot.
Missing Word
scatter
A scatter plot places one point for each pair of values. It is the first step in regression, because it shows whether a straight line is a sensible model.
Category: Scatter Plots
Question 4
A correlation coefficient near 0 can hide a strong curved pattern, because r measures only linear association.
Missing Word
linear
The correlation coefficient describes how closely points cluster around a straight line. A U-shaped or other curved pattern can be strong yet give r near 0, which is why the scatter plot should always be checked.
Category: Correlation
Question 5
Using a regression line to predict y at an x value outside the range of the sample data is called extrapolation.
Missing Word
extrapolation
Extrapolation goes beyond the observed x values, where there is no evidence that the linear pattern continues. Such predictions can be badly wrong and should be avoided or treated with caution.
Category: Prediction
Question 6
If a model was fit to heights recorded for ages 2 through 10, estimating the height at age 7 from the line is an example of interpolation.
Missing Word
interpolation
Interpolation is predicting at an x value inside the range of the data. Age 7 lies between 2 and 10, so the prediction rests on the observed pattern.
Category: Prediction
Question 7
A data point lying unusually far above or below the least-squares line, compared with the other points, is called an outlier.
Missing Word
outlier
An outlier sits far from the regression line in the vertical direction, so its residual is unusually large. It should be checked for errors before any decision to remove it.
Category: Outliers
Question 8
When larger x values tend to go with smaller y values, the correlation coefficient is negative.
Missing Word
negative
A negative correlation means y tends to decrease as x increases, so the points trend downward from left to right. The regression line then has a negative slope.
Category: Correlation
Question 9
A point lying above the least-squares line has a y - y-hat value that is positive.
Missing Word
positive
A point above the line has an observed y larger than the predicted y-hat, so y - y-hat is positive. The line underestimates y at that point.
Category: Least-Squares Line
Question 10
The correlation coefficient of a whole population is written with the Greek letter rho, while r is computed from a sample.
Missing Word
rho
Rho is the population parameter and r is the sample statistic that estimates it. Significance tests state their hypotheses about rho, such as H0: rho = 0.
Category: Significance Testing
Question 11
The correlation coefficient of an entire population is a parameter, while the correlation r computed from a sample is a statistic.
Missing Word
parameter
A parameter is a fixed number that describes a whole population, and it is usually unknown. The population correlation is a parameter, and the sample value r is the statistic used to estimate it and to test whether it equals 0.
Category: Significance Testing
Question 12
The coefficient of determination is found by squaring the correlation coefficient r.
Missing Word
squaring
The coefficient of determination is r^2, the correlation coefficient multiplied by itself. Expressed as a percent, it gives the share of the variation in y that the regression line on x accounts for, and it can never be negative.
Category: Correlation
Question 13
The quantity y - y-hat measures the vertical distance between a data point and the regression line.
Missing Word
vertical
Residuals are measured straight up or down, parallel to the y-axis, because the line predicts y from x. The least-squares line minimizes the sum of these squared vertical distances.
Category: Least-Squares Line
Question 14
The t test for whether a sample correlation r is significant uses n - 2 degrees of freedom.
Missing Word
2
The test of H0: rho = 0 uses a t distribution with df = n - 2, where n is the number of (x, y) pairs. Two degrees of freedom are used up because the line estimates both a slope and an intercept.
Category: Significance Testing
Question 15
The significance test for r assumes that, for each x, the y values are normally distributed about the regression line.
Missing Word
normally
The test assumes the y values at each x follow a normal distribution centered on the line, with the same spread for every x. This assumption allows the t distribution with n - 2 degrees of freedom to be used.
Category: Significance Testing