Confidence Intervals for the Slope

You’ll be able to

Construct a t-interval for the population slope β
Read the slope and standard error from regression output
Interpret a slope interval in context

The slope t-interval

A confidence interval for the true slope has the same shape as any t-interval: *b ± t\ · SE_b, with df = n − 2**. The sample slope b and its standard error SE_b are read directly from computer regression output — SE_b appears in the row for the explanatory variable, next to the slope coefficient.

Interpreting the interval

Interpret a slope interval in context: "We are C% confident that the true slope — the average change in y per one-unit increase in x — is between (low) and (high)." A key check: if the interval contains 0, then a slope of zero (no linear relationship) is plausible, so there is no convincing evidence of a linear association. If the interval is entirely above or below 0, there is evidence of a relationship.

Confidence interval for the slope

b ± t\* · SE_b with df = n − 2

t\* comes from the t-distribution with n − 2 degrees of freedom at the chosen confidence level. An interval that excludes 0 signals a statistically significant linear relationship.

Worked example

Regression output (n = 12) gives slope b = 3.2, SE_b = 0.8. Construct a 95% confidence interval for β (t\* = 2.228 for df = 10) and interpret it.

1.df = n − 2 = 12 − 2 = 10, so t\* = 2.228 for 95% confidence.
2.Margin of error: t\*·SE_b = 2.228 × 0.8 ≈ 1.78.
3.Interval: 3.2 ± 1.78 = (1.42, 4.98).
4.The interval lies entirely above 0, so a positive linear relationship is supported.

Answer: The 95% interval is 3.2 ± 1.78 = (1.42, 4.98). We are 95% confident the true slope — the average change in y per one-unit increase in x — is between 1.42 and 4.98. Because 0 is not in the interval, there is evidence of a positive linear relationship.

Checkpoint

A 95% confidence interval for the slope of a regression line is (−1.5, 2.3). What does this suggest?

On the exam

A slope interval interpretation must name the context and units: "average change in [y-variable] per one-unit increase in [x-variable]." And always check whether 0 is inside — that single fact tells you whether a linear relationship is plausible.

Checkpoint

Regression output shows a slope of 4.0 with a standard error of 1.0, based on n = 22 observations. The margin of error for a slope interval using t\* = 2.086 is closest to:

Answer the 2 checkpoints as you read.