Inference for a Regression Slope: Legacy AP Statistics Guide
Inference for regression slope is preserved here for students using older AP Statistics materials or college-introductory statistics courses. It is not current-core content for the revised AP Statistics course effective 2026–27, because the former slope-inference unit was removed.
Current AP Statistics status
The revised AP Statistics course took effect for the 2026–27 school year. College Board removed the former Unit 9 content on inference for quantitative-data slopes. Therefore this page is intentionally labeled legacy/enrichment. It can still help students reading older textbooks, reviewing pre-2027 released materials, or taking a college statistics course that includes regression-slope inference.
Current AP Statistics still uses regression analysis as a way to describe and predict quantitative relationships. Students should understand scatterplots, correlation, least-squares lines, residuals, and data-based prediction. What changed is the required inferential procedure for a population slope, not the usefulness of regression itself.
Keeping the status explicit prevents an older syllabus from silently becoming a 2027 exam checklist. If your immediate goal is the revised AP exam, prioritize current regression-analysis skills. If your goal is historical review or broader statistics mastery, the legacy inference procedure below remains mathematically valuable.
Population slope, sample slope, and the hypotheses
The parameter is the population regression slope β, which describes the expected change in the mean response for a one-unit increase in the explanatory variable under the population linear model. The sample least-squares slope b estimates β. A common legacy test uses H₀:β=0 against a two-sided or directional alternative.
Do not write the hypothesis as H₀:b=0. The statistic b comes from the sample and is already observed; inference concerns the unknown population parameter β. This distinction is the regression analogue of using μ rather than x̄ in a one-sample mean hypothesis.
A nonzero slope describes a linear association in the model; it does not automatically imply causation. Causal interpretation depends on the study design. In observational data, a convincing slope can coexist with confounding, lurking variables, reverse direction, or selection effects.
| Quantity | Role |
|---|---|
| β | Population regression slope parameter |
| b | Sample least-squares slope estimate |
| SE(b) | Estimated standard error of b |
| t | (b−β₀)/SE(b) |
| df | n−2 in the simple-linear-regression legacy procedure |
Legacy t test for a regression slope
For the common null value β₀=0, the legacy test statistic is t = b / SE(b). More generally, t=(b−β₀)/SE(b). Under the simple linear regression conditions, the reference distribution uses n−2 degrees of freedom because two line parameters—the intercept and slope—are estimated.
The sign of t follows the sign of b when β₀=0. In a two-sided test, a large magnitude in either direction contributes evidence against zero. For a directional alternative, the direction must be specified from the research question rather than chosen after seeing the sample slope.
A small p-value supplies evidence against the null slope under the model conditions; it is not a measure of practical importance. With a very large sample, a tiny slope can be statistically detectable while explaining little variation. Report the slope’s units and consider the substantive size of the change.
Legacy confidence interval for the population slope
A legacy confidence interval has the form b ± t* × SE(b), using the appropriate t critical value with n−2 degrees of freedom. The interval estimates plausible values of β under the regression model. If a two-sided 95% interval excludes 0, the corresponding two-sided test at α=.05 rejects H₀:β=0.
Interpret the interval in slope units: plausible change in the mean response per one-unit increase in x. Avoid saying that 95% of individual observations fall within the slope interval; the interval is about the population slope parameter, not individual y-values or prediction errors.
An interval can be statistically separated from zero yet too wide for a precise practical decision. Conversely, an interval tightly clustered around a small nonzero slope may show high precision but modest substantive impact. Inference and practical significance answer different questions.
Interpretation rules that still matter beyond the legacy procedure
Slope language should include units and direction. “The slope is 2.4” is incomplete; “the model predicts an average increase of about 2.4 minutes in response time for each additional kilometer” communicates the actual relationship. If extrapolation would be required, state that the relationship is supported only over the observed x-range.
Residual diagnostics matter because a t statistic cannot rescue a badly chosen linear model. Curvature, changing residual spread, influential observations, or dependence can make the standard error and reference distribution inappropriate. The conditions page linked below focuses on those diagnostics rather than repeating the calculation workflow.
Correlation and slope are related but not interchangeable. Correlation is unitless and standardized; slope has response-units per explanatory-unit. Rescaling x changes b but leaves correlation unchanged. A hypothesis about β therefore belongs to the chosen measurement scale and model.
Twenty legacy regression-slope worked cases
Legacy worked case 1: Study hours and quiz score
A historical simple-regression analysis reports b=1.8, SE(b)=0.55, and n=42. Testing H₀:β=0 gives t=3.273 with df=40. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 1.8 points per hour in the increasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 1: Study hours and quiz score rather than substituting a result from another scenario.
Legacy worked case 2: Temperature and electricity use
A historical simple-regression analysis reports b=-2.4, SE(b)=0.73, and n=36. Testing H₀:β=0 gives t=-3.288 with df=34. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 2.4 kWh per degree in the decreasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 2: Temperature and electricity use rather than substituting a result from another scenario.
Legacy worked case 3: Distance and delivery time
A historical simple-regression analysis reports b=3.1, SE(b)=0.62, and n=50. Testing H₀:β=0 gives t=5.000 with df=48. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 3.1 minutes per kilometer in the increasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 3: Distance and delivery time rather than substituting a result from another scenario.
Legacy worked case 4: Training sessions and completion time
A historical simple-regression analysis reports b=-1.25, SE(b)=0.41, and n=31. Testing H₀:β=0 gives t=-3.049 with df=29. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 1.25 minutes per session in the decreasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 4: Training sessions and completion time rather than substituting a result from another scenario.
Legacy worked case 5: Rainfall and reservoir level
A historical simple-regression analysis reports b=0.84, SE(b)=0.29, and n=44. Testing H₀:β=0 gives t=2.897 with df=42. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 0.84 meters per centimeter in the increasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 5: Rainfall and reservoir level rather than substituting a result from another scenario.
Legacy worked case 6: Advertising and weekly sales
A historical simple-regression analysis reports b=2.75, SE(b)=0.88, and n=39. Testing H₀:β=0 gives t=3.125 with df=37. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 2.75 sales units per thousand dollars in the increasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 6: Advertising and weekly sales rather than substituting a result from another scenario.
Legacy worked case 7: Sleep and reaction time
A historical simple-regression analysis reports b=-7.2, SE(b)=2.1, and n=28. Testing H₀:β=0 gives t=-3.429 with df=26. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 7.2 milliseconds per hour in the decreasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 7: Sleep and reaction time rather than substituting a result from another scenario.
Legacy worked case 8: Class size and feedback delay
A historical simple-regression analysis reports b=0.63, SE(b)=0.21, and n=46. Testing H₀:β=0 gives t=3.000 with df=44. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 0.63 days per student in the increasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 8: Class size and feedback delay rather than substituting a result from another scenario.
Legacy worked case 9: Elevation and temperature
A historical simple-regression analysis reports b=-0.006, SE(b)=0.0018, and n=52. Testing H₀:β=0 gives t=-3.333 with df=50. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 0.006 degrees per meter in the decreasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 9: Elevation and temperature rather than substituting a result from another scenario.
Legacy worked case 10: Practice sets and error rate
A historical simple-regression analysis reports b=-0.45, SE(b)=0.14, and n=34. Testing H₀:β=0 gives t=-3.214 with df=32. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 0.45 percentage points per set in the decreasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 10: Practice sets and error rate rather than substituting a result from another scenario.
Legacy worked case 11: Age and systolic pressure
A historical simple-regression analysis reports b=0.52, SE(b)=0.19, and n=60. Testing H₀:β=0 gives t=2.737 with df=58. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 0.52 mmHg per year in the increasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 11: Age and systolic pressure rather than substituting a result from another scenario.
Legacy worked case 12: Speed and stopping distance
A historical simple-regression analysis reports b=1.47, SE(b)=0.31, and n=45. Testing H₀:β=0 gives t=4.742 with df=43. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 1.47 meters per km/h in the increasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 12: Speed and stopping distance rather than substituting a result from another scenario.
Legacy worked case 13: Screen time and sleep duration
A historical simple-regression analysis reports b=-0.28, SE(b)=0.09, and n=41. Testing H₀:β=0 gives t=-3.111 with df=39. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 0.28 hours per screen-hour in the decreasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 13: Screen time and sleep duration rather than substituting a result from another scenario.
Legacy worked case 14: Machine cycles and wear
A historical simple-regression analysis reports b=0.035, SE(b)=0.011, and n=48. Testing H₀:β=0 gives t=3.182 with df=46. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 0.035 millimeters per 100 cycles in the increasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 14: Machine cycles and wear rather than substituting a result from another scenario.
Legacy worked case 15: Fertilizer and plant height
A historical simple-regression analysis reports b=0.91, SE(b)=0.27, and n=32. Testing H₀:β=0 gives t=3.370 with df=30. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 0.91 centimeters per gram in the increasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 15: Fertilizer and plant height rather than substituting a result from another scenario.
Legacy worked case 16: Attendance and final score
A historical simple-regression analysis reports b=0.67, SE(b)=0.22, and n=55. Testing H₀:β=0 gives t=3.045 with df=53. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 0.67 points per attendance point in the increasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 16: Attendance and final score rather than substituting a result from another scenario.
Legacy worked case 17: Humidity and drying time
A historical simple-regression analysis reports b=0.38, SE(b)=0.12, and n=37. Testing H₀:β=0 gives t=3.167 with df=35. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 0.38 minutes per humidity point in the increasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 17: Humidity and drying time rather than substituting a result from another scenario.
Legacy worked case 18: Income and expenditure
A historical simple-regression analysis reports b=0.62, SE(b)=0.15, and n=49. Testing H₀:β=0 gives t=4.133 with df=47. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 0.62 thousand dollars per thousand dollars in the increasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 18: Income and expenditure rather than substituting a result from another scenario.
Legacy worked case 19: Water depth and light intensity
A historical simple-regression analysis reports b=-4.6, SE(b)=1.3, and n=30. Testing H₀:β=0 gives t=-3.538 with df=28. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 4.6 lux units per meter in the decreasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 19: Water depth and light intensity rather than substituting a result from another scenario.
Legacy worked case 20: Run length and recovery pulse
A historical simple-regression analysis reports b=1.15, SE(b)=0.36, and n=43. Testing H₀:β=0 gives t=3.194 with df=41. The numerical evidence is based on how many estimated standard errors the sample slope lies from zero.
The slope should be interpreted as about 1.15 beats per minute per kilometer in the increasing direction for a one-unit increase in the explanatory variable, within the range and model represented by the data. Statistical evidence for a nonzero β would not, by itself, justify a causal statement.
For current AP Statistics preparation, treat the calculation as enrichment and redirect exam-focused study toward interpreting regression output, residuals, predictions, and limitations. For an older AP or college statistics syllabus, continue by obtaining the t tail probability and, if requested, constructing b±t*SE(b). Case-specific checkpoint: use the evidence from Legacy worked case 20: Run length and recovery pulse rather than substituting a result from another scenario.
What remains current and what is legacy
Current-core regression work includes describing direction, form, strength, correlation, least-squares prediction, residual interpretation, and recognizing limitations of linear models. These skills continue to matter because regression remains a central way to analyze relationships between quantitative variables.
The formal one-sample t inference procedure for the population regression slope—the test statistic, slope confidence interval, and its dedicated model-inference conditions—belongs to older AP materials and broader introductory statistics rather than the revised required AP content. This page keeps those tools clearly labeled so older resources remain usable without confusing the current exam scope.
When a legacy textbook assigns a slope-inference exercise, you can still solve it correctly. When building a 2027 AP review schedule, however, do not let this legacy topic displace current five-unit content or the revised exam practices.
Legacy regression-slope multiple-choice practice
Question 1. Legacy slope inference
Which symbol represents the population regression slope?
Answer: B
β is the population slope; b is its sample estimate.
Question 2. Legacy slope inference
For H₀:β=0, b=1.8 and SE(b)=0.6. What is t?
Answer: C
t=b/SE(b)=1.8/.6=3.
Question 3. Legacy slope inference
A simple regression uses n=27 observations. What df is used in the legacy slope t procedure?
Answer: A
Simple-linear-regression slope inference uses n−2=25 df.
Question 4. Legacy slope inference
Which interpretation is appropriate for b=-2.1?
Answer: B
Slope is a model-based change in predicted mean response per x unit.
Question 5. Legacy slope inference
Why is H₀:b=0 incorrectly parameterized?
Answer: A
Inference is about β, the population parameter.
Question 6. Legacy slope inference
A 95% slope interval excludes 0. What is the corresponding two-sided α=.05 test relationship under the same model?
Answer: B
The confidence-interval and two-sided test decisions agree at matching confidence/significance levels.
Question 7. Legacy slope inference
Which design feature is required for a causal slope claim?
Answer: C
Statistical association does not create causation.
Question 8. Legacy slope inference
What is the unit of a regression slope?
Answer: B
Slope carries y-units divided by x-units.
Question 9. Legacy slope inference
Which issue most directly threatens a linear slope model?
Answer: A
Curvature indicates that a straight-line mean function is inadequate.
Question 10. Legacy slope inference
What current-course status should a 2027 AP student assign to formal slope inference?
Answer: B
The revised 2026–27 AP Statistics framework removed Unit 9 slope inference.
Question 11. Legacy slope inference
If b=.12 with a very small SE, what distinction is still important?
Answer: A
A precisely estimated tiny slope can be statistically significant but practically small.
Question 12. Legacy slope inference
What does extrapolation mean?
Answer: A
Regression support is strongest within the data range; extrapolation relies on unverified continuation of the pattern.
Question 13. Legacy slope inference
Correlation r=.8 proves which statement?
Answer: C
Correlation describes linear association and does not prove causation.
Question 14. Legacy slope inference
Which expression is the legacy slope confidence interval?
Answer: A
The sample slope plus/minus a t critical multiplier times its standard error is the standard form.
Question 15. Legacy slope inference
A slope estimate changes when x is measured in meters instead of centimeters. What need not change?
Answer: C
Correlation is invariant to positive linear rescaling, while slope rescales.
Question 16. Legacy slope inference
What does a residual represent?
Answer: A
Residuals measure vertical prediction errors and are central to checking the linear model.
Legacy slope-inference free-response practice
FRQ 1. Test statistic
An older textbook reports b=2.4, SE(b)=0.75, n=32. Calculate the legacy test statistic and df for H₀:β=0.
Model response
t=2.4/.75=3.20.
df=n−2=30. A two-sided p-value would come from a t distribution with 30 df; the result concerns the population slope β, not the observed b.
FRQ 2. Interval interpretation
A 95% legacy interval for β is (0.8, 2.6) points per hour. Interpret it and state the matching two-sided test decision at α=.05.
Model response
The interval gives plausible values for the population mean-response change per additional hour, under the regression model and study design.
Because 0 is not in the interval, the corresponding two-sided test rejects H₀:β=0 at α=.05.
FRQ 3. Causation warning
An observational regression gives a tiny p-value for a positive slope between tutoring hours and score. Explain why “tutoring causes higher scores” is not established.
Model response
Students were not necessarily randomly assigned to tutoring exposure, so motivation, prior achievement, course difficulty, or other variables may confound the association.
The slope can describe/predict the sample relationship and may support population association under a suitable random sample, but causal attribution requires an appropriate causal design.
FRQ 4. Legacy status
A student sees a pre-2027 AP review chapter on regression-slope inference. Explain how it should be used for the revised exam.
Model response
The mathematical material is legitimate historical/enrichment statistics, but the former AP Unit 9 slope-inference requirement was removed beginning with the 2026–27 revised course.
For current AP preparation, prioritize current regression analysis and the revised five-unit framework; use the old chapter only for enrichment or when a teacher/college course specifically assigns it.
FRQ 5. Slope versus correlation
A dataset is rescaled from centimeters to meters for x. Explain how slope and correlation respond.
Model response
The numerical slope changes because its denominator unit changes; it is expressed in response units per meter rather than per centimeter.
Correlation remains the same under a positive linear rescaling because it is unitless and standardized.
FRQ 6. Model diagnostics
A legacy slope test produces p=.004, but the residual plot shows strong U-shaped curvature. Explain the problem.
Model response
The standard linear regression inference procedure assumes the linear mean structure is an appropriate model. Strong curvature indicates systematic model misspecification.
The small p-value should not be treated as validating the straight-line model. Reconsider the functional form or use a method suited to the relationship before interpreting slope inference.
Extended topic-specific mastery cases
Population Slope Parameter mastery extension 1
Legacy mastery case 1 centers on population slope parameter. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating population slope parameter as a required 2027 inferential procedure.
Legacy T Statistic mastery extension 2
Legacy mastery case 2 centers on legacy t statistic. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating legacy t statistic as a required 2027 inferential procedure.
Slope Units mastery extension 3
Legacy mastery case 3 centers on slope units. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating slope units as a required 2027 inferential procedure.
Confidence Interval mastery extension 4
Legacy mastery case 4 centers on confidence interval. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating confidence interval as a required 2027 inferential procedure.
Causation Limits mastery extension 5
Legacy mastery case 5 centers on causation limits. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating causation limits as a required 2027 inferential procedure.
Current-Course Status mastery extension 6
Legacy mastery case 6 centers on current-course status. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating current-course status as a required 2027 inferential procedure.
Population Slope Parameter mastery extension 7
Legacy mastery case 7 centers on population slope parameter. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating population slope parameter as a required 2027 inferential procedure.
Legacy T Statistic mastery extension 8
Legacy mastery case 8 centers on legacy t statistic. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating legacy t statistic as a required 2027 inferential procedure.
Slope Units mastery extension 9
Legacy mastery case 9 centers on slope units. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating slope units as a required 2027 inferential procedure.
Confidence Interval mastery extension 10
Legacy mastery case 10 centers on confidence interval. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating confidence interval as a required 2027 inferential procedure.
Causation Limits mastery extension 11
Legacy mastery case 11 centers on causation limits. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating causation limits as a required 2027 inferential procedure.
Current-Course Status mastery extension 12
Legacy mastery case 12 centers on current-course status. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating current-course status as a required 2027 inferential procedure.
Population Slope Parameter mastery extension 13
Legacy mastery case 13 centers on population slope parameter. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating population slope parameter as a required 2027 inferential procedure.
Legacy T Statistic mastery extension 14
Legacy mastery case 14 centers on legacy t statistic. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating legacy t statistic as a required 2027 inferential procedure.
Slope Units mastery extension 15
Legacy mastery case 15 centers on slope units. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating slope units as a required 2027 inferential procedure.
Confidence Interval mastery extension 16
Legacy mastery case 16 centers on confidence interval. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating confidence interval as a required 2027 inferential procedure.
Causation Limits mastery extension 17
Legacy mastery case 17 centers on causation limits. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating causation limits as a required 2027 inferential procedure.
Current-Course Status mastery extension 18
Legacy mastery case 18 centers on current-course status. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating current-course status as a required 2027 inferential procedure.
Population Slope Parameter mastery extension 19
Legacy mastery case 19 centers on population slope parameter. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating population slope parameter as a required 2027 inferential procedure.
Legacy T Statistic mastery extension 20
Legacy mastery case 20 centers on legacy t statistic. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating legacy t statistic as a required 2027 inferential procedure.
Slope Units mastery extension 21
Legacy mastery case 21 centers on slope units. Start with a hypothetical sample slope b, its standard error, and a sample size. Name β as the population parameter before performing any arithmetic, and attach response-per-explanatory units to the slope. For a historical H₀:β=0 test, compute t=b/SE(b) and df=n−2, then describe what a two-sided tail area would measure under the legacy model. Separate statistical evidence from practical magnitude and from causation. Next imagine rescaling x and explain why the numerical slope changes while the underlying fitted predictions can remain equivalent after unit conversion. Finally mark the procedure as enrichment for revised AP Statistics: the former formal slope-inference unit was removed, so current exam study should use this exercise to reinforce parameter thinking and regression interpretation without treating slope units as a required 2027 inferential procedure.
Legacy regression-slope inference FAQs
Is regression-slope inference on the revised 2027 AP Statistics exam? It is not current-core required content; the former slope-inference unit was removed in the revised course.
Why keep this page? Older AP materials and many college introductory statistics courses still teach the procedure, so a clearly labeled legacy guide remains useful.
Does a significant slope prove causation? No. Causal interpretation comes from design, not from statistical significance alone.
What df is used in the simple legacy procedure? n−2.
What should current AP students study instead? Current regression analysis, residuals, predictions, statistical problem solving, and the revised inference content in the five-unit framework.
Continue with related AP Statistics resources
Inference For Regression Slope review focus
The phrase inference for regression slope names this page’s specific purpose. Use inference for regression slope as the focus when deciding which workflow, conditions, interpretation, or legacy-status guidance belongs here rather than on a neighboring AP Statistics page.
For final review, return to the opening explanation and verify that you can explain inference for regression slope in context without relying on a memorized label alone.
