EXAMPLE 7 Using confidence intervals to perform the t test for the slope

  1. Construct and interpret a 99% confidence interval for the slope for the text messaging data in Table 1.
  2. Use the confidence interval in (a) to test whether a linear relationship exists between age and text messages, using level of significance .

textms

Solution

  1. The regression assumptions were verified in Example 2. Also,
    • In Example 1, we found .
    • In Example 3, we calculated , and .

    From the table, we find that, for 99% confidence, for degrees of freedom is . So, our margin of error si

    The 99% confidence interval for is then given by

    We are 99% confident that the interval (−1.9448, −1.0552) captures the slope of the population regression line. That is, we are 99% confident that, for each additional year of age, the decrease in the number of text messages lies between 1.9448 and 1.0552.

  2. The hypotheses are

    No linear relationship exists between age and text messages.

    A linear relationship exists between age and text messages.

The confidence interval from (a) does not contain zero, so we may conclude that a linear relationship exists between age and text messages, at level of significance .

NOW YOU CAN DO

Exercises 31–38.