Inference for Regression Math Example 2

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Example 2

hard
Construct a 95% confidence interval for the slope β\beta given: b=1.8b=1.8, SEb=0.5SE_b=0.5, n=25n=25, and t0.025,23∗=2.069t^*_{0.025,23}=2.069.

Solution

  1. 1
    95% CI formula: b±t∗×SEbb \pm t^* \times SE_b
  2. 2
    1.8±2.069×0.5=1.8±1.0351.8 \pm 2.069 \times 0.5 = 1.8 \pm 1.035
  3. 3
    95% CI: (1.8−1.035, 1.8+1.035)=(0.765,2.835)(1.8 - 1.035,\ 1.8 + 1.035) = (0.765, 2.835)
  4. 4
    Interpretation: we are 95% confident the true slope β\beta is between 0.765 and 2.835

Answer

95% CI for β\beta: (0.765,2.835)(0.765, 2.835). Since 0 is not in the interval, the slope is significant.
A confidence interval for the slope contains the true population slope with 95% confidence. If 0 is NOT in the interval, we reject H0:β=0H_0: \beta=0 (consistent with the t-test). The CI also gives practical information: we're 95% confident the true effect size is between 0.765 and 2.835.

About Inference for Regression

Using hypothesis tests and confidence intervals to draw conclusions about the true population slope β1\beta_1 of the linear relationship y=β0+β1x+εy = \beta_0 + \beta_1 x + \varepsilon, based on sample data.

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