Inference for Regression

With regression, we are looking at whether there is a significant linear relationship between two QUANTITATIVE variables. We specifically make inferences with regard to the SLOPE of the regression line.

Recall

where b0 is the sample y-intercept and b1 is the sample slope. If this model represented the whole population, it would be

Confidence Interval for Slope

Assumptions:

random sample

residuals must be independent and normally distributed

linear relationship between x and y

constant variance in y for all values of x

Notice df = n - 2 with regression (2 variables). SEb1 is called the standard error of the slope and refers to the variation in the different sample slopes of possible sample regression lines.

Interpretation: I am 95% confident that, on average, for a 1 unit change in x, there will be a change in y between __ and ___.

If 0 is contained in this interval, it gives you evidence that there is not a significant linear relationship between the variables. If (+, +) or (-, -), then there is evidence of either a significant positive or significant negative linear relationship between the variables.

T-Test for Slope

same parameter and assumptions as above with the confidence interval

The Ho implies there is no significant linear relationship whereas the Ha implies there is a significant linear relationship between the variables.

Test Statistic:

If you have the raw data, using the TI-83, run STAT, TEST, E:LinRegTTest. This will give you your test statistic and p-value. Otherwise expect a computer printout with information.

The calculator will also give you s. This is called standard error of the residuals and is NOT the same as standard error of the slope. S measures the average spread of the observed y-values from the predicted y-values.

The calculator does not give you SEb1, so if you need it you can use the calculator to get t and b1 and then use the formula:

Example

Number of chirps / 22 / 27 / 35 / 15 / 28 / 30 / 39 / 23 / 25 / 18 / 35 / 29
Temperature (F) / 64 / 68 / 78 / 60 / 72 / 76 / 82 / 66 / 70 / 62 / 80 / 74

Consider the following data on the number of cricket chirps in 15 seconds from various temperatures. We want to determine if there is a linear relationship between temperature and number of cricket chirps (if temperature can be predicted by the number of cricket chirps).

A computer printout of the data is as follows: