Inference for Regression (Chapter 14) A.P. Stats Review Topic #3

Inference for Slope: Topics Compute confidence intervals and perform hypothesis tests for the slope of a least squares regression line. The AP Statistics syllabus does not include inference for the y-intercept of a LSRL, so don’t worry about that. * The LSRL is given by

The y-intercept “a” and the slope “b” of a regression line vary with different samples, so these are statistics. The true regression line is given by the equation . It has slope and intercept . It is the regression line for the entire population of ordered pairs, so it is a parameter. Inference procedures for the true slope are based on knowledge of the sampling distribution of sample slopes. When conditions are satisfied, sample slopes will be normally distributed.

Conditions for Slope Inference LINS Linear model is correct: check the scatter plot for linearity and the residual plot for no pattern. Independent observations. This is a design issue that should be addressed in information about the data. Random sample of data points will suffice.

Conditions, page 2 Normality – the y-values vary normally about the true regression line. Check that residuals are approximately normally distributed using a histogram, boxplot, dotplot, normal probability plot, or stem and leaf plot. Standard deviation of y-values is the Same for every value of x. Check the residual plot to be sure that the spread of the residuals about the horizontal axis is approximately uniform.

Steps for Confidence Intervals Be sure to include the usual steps for a confidence interval: C = state and check conditions, C = perform computations, and C = interpret the confidence interval in context. Formula for Confidence interval is: “b” is the slope from your sample regression line t* is based on the confidence level; note n-2 degrees of freedom sb is the standard deviation of the sample slope. It is provided on computer output. Note: You can usually get this interval straight from your calculator.

The 4 Steps for a Significance Test Steps for a hypothesis test are the same as for all tests: H = hypotheses, C = conditions, C = calculate test statistic and P-value, and C = state conclusion with connection in context. The statement of the null hypothesis usually assumed by computers and calculators is . However, the null hypothesis may specify any value for , for example, .

Model Utility Test = Hypothesis Test When the null hypothesis is β=0, the null hypothesis can be thought of as a statement that there is no useful relationship between the variables. Rejecting the null hypothesis leads to the conclusion that the linear regression model with x as the explanatory variable is useful for predicting y.

Here’s the thinking….. If two variables X and Y were completely unrelated, then their ordered pairs would be scattered around the coordinate plane, and the best-fitting line would be a horizontal one. Horizontal lines have their slope = 0. Thus, if two variables X and Y are related somehow, then the best-fitting line would be one with a non-zero slope. Thus the hypotheses: (usually)

Hypothesis Tests for Beta The test statistic is: