EXAMPLE 10.6

Confidence interval for the slope. A confidence interval for β₁ requires a critical value t* from the t(n − 2) = t(98) distribution. In Table D, there are entries for 80 and 100 degrees of freedom. The values for these rows are very similar. To be conservative, we will use the larger critical value, for 80 degrees of freedom. Find the confidence level values at the bottom of the table. In the 95% confidence column, the entry for 80 degrees of freedom is t* = 1.990.

To compute the 95% confidence interval for β₁, we combine the estimate of the slope with the margin of error:

= −0.655 ± 0.314

The interval is (−0.969, −0.341). As expected, this is slightly wider than the interval given by software (see Excel output in Figure 10.4). We estimate that, on average, an increase of 1000 steps per day is associated with a decrease in BMI of between 0.341 and 0.969 kg/m².