רווח סמך לשיפוע
הוצגThe 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.
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 $x$).
השיפוע המותאם וחוסר הוודאות שלו
האינטרוול b ± t*·SE_b מספק ערכים סבירים לשיפוע האמיתי.
להסקת שיפוע עם n = 20, מהי דרגות החופש (n − 2)?
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df = n − 2 = 18 (הוערכו שיפוע ויחידת חיתוך).
b = 4.2, SE_b = 1.0, t* = 2.10. מצאו את שגיאת הסטייה t*·SE_b.
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2.10 × 1.0 = 2.10.
בתפוקת מחשב של רגרסיה, השיפוע b מופיע בעמודה המסומנת...
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ההערכה למשתנה הסבר היא השיפוע.
הסקת שיפוע משתמשת ב-n − 1 דרגות חופש.
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זהו n − 2 — שני פרמטרים (שיפוע ויחידת חיתוך) מתוערכים.
הזיכרון הקצר לתנאי ההסקה לחמישה שיפועים הוא ___ (חמש אותיות).
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ליניארי, עצמאי, נורמלי, SD שווה, אקראי.