A Confidence Interval for the Slope
Đã giới thiệuThe LINER conditions
- Inference about a slope needs five conditions — remember LINER:
- Linear (the true relationship is linear) · Independent observations.
- Normal (responses vary normally around the line) · Equal SD (constant scatter across $x$).
- Random (data from a random sample or experiment). Residual plots help check L, N, E.
Read the computer output
- Slope inference almost always starts from computer output.
- Find the row for the explanatory variable: its estimate is the slope $b$.
- The next column is the standard error $SE_b$.
- Output also reports a $t$ statistic and p-value — but first we build an interval.
A confidence interval gives the plausible values of the population slope
Build the t-interval
- The slope interval is estimate $\pm$ margin of error:
-
$$b \pm t^{*}\, SE_b$$
- Use the $t$-distribution with $df = n - 2$ (two estimated: slope and intercept).
- $t^{*}$ comes from that $df$ at your confidence level.
Interpret in context
- "We're $95\%$ confident the true slope $\beta$ is between the endpoints."
- Say what the slope means: the predicted change in $y$ per one-unit increase in $x$.
- Name the real variables and units.
- The interval is a range of plausible values for $\beta$.
Slope inference uses $df = n - 2$, not $n - 1$. Two parameters are estimated (slope and intercept), so you lose two degrees of freedom. Read $b$ and $SE_b$ straight from the output — don't recompute them — and remember the interval is $b \pm t^{*}SE_b$, the same estimate-$\pm$-margin form as every other $t$-interval.
Output: slope $b = 4.2$, $SE_b = 1.0$, $n = 20$. Build a $95\%$ interval. ($df = 18$, $t^{*} \approx 2.10$.)
- Margin of error: $2.10 \times 1.0 = 2.10$.
- Interval: $4.2 \pm 2.10 = (2.1,\ 6.3)$.
- Interpret: we're $95\%$ confident each extra hour predicts between $2.1$ and $6.3$ more points.
After the LINER conditions, read $b$ and $SE_b$ from the output and build a slope $t$-interval $b \pm t^{*}\,SE_b$ using $df = n - 2$. Interpret it in context — a range of plausible values for the true slope $\beta$ (the predicted change in $y$ per unit $x$).
Write from a slope estimate of 0.8
- Computer output gives a slope of 0.8 with standard error 0.1.
- There are 16 observations, so the degrees of freedom are 14. The lesson slope interval uses 18 and 2.1.
What is the point estimate of the slope?
The fitted slope is the point estimate.
What is the point estimate of the slope?
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The output slope is the estimate. Other options — 0.1 (the standard error); 2.1 (the other output).
The fitted slope and its uncertainty
The interval b ± t*·SE_b gives plausible values for the true slope.
For slope inference with n = 20, what are the degrees of freedom (n − 2)?
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df = n − 2 = 18 (slope and intercept estimated).
b = 4.2, SE_b = 1.0, t* = 2.10. Find the margin of error t*·SE_b.
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2.10 × 1.0 = 2.10.
In computer regression output, the slope b is found in the column labeled...
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The explanatory variable's estimate is the slope.
Slope inference uses n − 1 degrees of freedom.
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It's n − 2 — two parameters (slope and intercept) are estimated.
The mnemonic for the five slope-inference conditions is ___ (five letters).
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Linear, Independent, Normal, Equal SD, Random.