AP Statistics 9.3 Justifying a Claim About the Slope of a Regression Model Based on a Confidence Interval Study Notes
AP Statistics 9.3 Justifying a Claim About the Slope of a Regression Model Based on a Confidence Interval Study Notes- New syllabus
AP Statistics 9.3 Justifying a Claim About the Slope of a Regression Model Based on a Confidence Interval Study Notes -As per latest AP Statistics Syllabus.
LEARNING OBJECTIVE
- An interval of values should be used to estimate parameters, in order to account for uncertainty
Key Concepts:
- Interpreting a Confidence Interval for the Slope (\(\beta\))
- Justifying a Claim Based on a Confidence Interval for the Slope (\(\beta\))
- Effects of Sample Size on the Width of a Confidence Interval for the Slope (\(\beta\))
Interpreting a Confidence Interval for the Slope (\(\beta\))
Interpreting a Confidence Interval for the Slope (\(\beta\))
A confidence interval for the slope of a regression line provides a range of plausible values for the true population slope \(\beta\), which quantifies the relationship between the explanatory variable (\(x\)) and response variable (\(y\)).
Key points:
- The slope \(\beta\) represents the average change in the response variable (\(y\)) for a one-unit increase in the explanatory variable (\(x\)).
- The confidence interval estimates the range of values within which the true population slope is likely to fall with a given level of confidence (e.g., 95%).
- If the interval does not include 0, this provides evidence that there is a statistically significant linear relationship between \(x\) and \(y\).
- Interpretation should be in the context of the data and variables.
Example
A study examines the effect of hours of exercise per week (\(x\)) on resting heart rate (\(y\)) in a sample of 12 adults. The 95% confidence interval for the slope is calculated as \((-2.56, -0.44)\).
Interpret the confidence interval in context.
▶️ Answer / Explanation
Step 1 — Direction of relationship:
The interval is entirely negative (\(-2.56\) to \(-0.44\)), indicating that as exercise hours increase, resting heart rate decreases on average.
Step 2 — Magnitude of change:
We are 95% confident that each additional hour of exercise per week is associated with a decrease in resting heart rate between 0.44 and 2.56 beats per minute, on average, in the population.
Step 3 — Statistical significance:
Since 0 is not included in the interval, there is evidence of a significant negative linear relationship between exercise hours and resting heart rate.
Step 4 — Contextual interpretation:
The confidence interval provides a plausible range for the population slope, quantifying how resting heart rate changes with exercise in the population from which the sample was drawn.
Justifying a Claim Based on a Confidence Interval for the Slope (\(\beta\))
Justifying a Claim Based on a Confidence Interval for the Slope (\(\beta\))
A confidence interval for the slope provides a range of plausible values for the population slope \(\beta\). This interval can be used to evaluate claims about the effect of the explanatory variable (\(x\)) on the response variable (\(y\)).
Steps to justify a claim:
- Examine the interval: Determine whether the interval includes the value stated in the claim (often 0 for no effect).
- Direction of relationship: If the interval is entirely positive, the effect is positive; if entirely negative, the effect is negative.
- Statistical evidence: If the interval does not include 0, there is evidence to support a claim of a significant linear relationship between \(x\) and \(y\).
- Contextual conclusion: State the claim in terms of the actual variables and population studied.
Important:
- The confidence interval is based on sample data; it does not “prove” the claim, but provides evidence consistent with it.
- The width of the interval indicates precision: narrower intervals give stronger evidence for the claim.
Example
A researcher claims that more hours of study per week (\(x\)) increase exam scores (\(y\)) in college students. A 95% confidence interval for the slope of the regression line is calculated as \((1.43, 3.57)\).
Can we justify the researcher’s claim based on the confidence interval?
▶️ Answer / Explanation
Step 1 — Examine the interval:
The entire interval is positive (1.43 to 3.57), indicating a positive relationship between study hours and exam scores.
Step 2 — Compare to the claim:
The claim is that study hours increase exam scores. Since the interval does not include 0 and is entirely above 0, the confidence interval supports this claim.
Step 3 — Contextual conclusion:
We are 95% confident that each additional hour of study per week is associated with an increase in exam scores between 1.43 and 3.57 points, on average, in the population. This provides evidence consistent with the researcher’s claim.
Step 4 — Note on precision:
The interval width (3.57 − 1.43 = 2.14) reflects the precision of the estimate; narrower intervals give stronger evidence for the effect.
Effects of Sample Size on the Width of a Confidence Interval for the Slope (\(\beta\))
Effects of Sample Size on the Width of a Confidence Interval for the Slope (\(\beta\))
The width of a confidence interval for the slope depends on the standard error of the slope and the critical t-value. Sample size (\(n\)) directly affects the standard error:
Formula for standard error of the slope:
\( SE_b = \dfrac{s}{\sqrt{\sum (x_i – \bar{x})^2}} \)
Where \(s\) is the residual standard deviation, and \(\sum (x_i – \bar{x})^2\) grows with more variability in \(x\) and larger \(n\).
Effects of increasing sample size:
- As \(n\) increases, the degrees of freedom \(df = n-2\) increase → \(t^*\) slightly decreases.
- With larger \(n\), \(\sum (x_i – \bar{x})^2\) tends to increase → \(SE_b\) decreases.
- Smaller \(SE_b\) → narrower confidence interval → more precise estimate of \(\beta\).
Effects of decreasing sample size:
- Smaller \(n\) → smaller \(df\) → larger \(t^*\)
- Smaller \(\sum (x_i – \bar{x})^2\) → larger \(SE_b\)
- Larger \(SE_b\) and \(t^*\) → wider confidence interval → less precise estimate of \(\beta\).
Summary: Increasing sample size leads to more precise estimates of the slope because the confidence interval becomes narrower, while smaller sample sizes produce wider intervals and less precise estimates.
Example
A researcher studies the effect of weekly study hours (\(x\)) on exam scores (\(y\)). The residual standard deviation is \(s = 4.0\), and \(\sum (x_i – \bar{x})^2 = 80\).
Consider two different sample sizes:
- Scenario 1: \(n = 10\) students → \(df = n-2 = 8\)
- Scenario 2: \(n = 30\) students → \(df = n-2 = 28\)
Determine the 95% confidence interval width for the slope in both cases and explain the effect of sample size.
▶️ Answer / Explanation
Step 1 — Compute standard error of slope:
\( SE_b = \dfrac{s}{\sqrt{\sum (x_i – \bar{x})^2}} = \dfrac{4.0}{\sqrt{80}} = \dfrac{4.0}{8.944} \approx 0.447 \)
Step 2 — Determine t* values:
- For \(n=10\), \(df=8\), \(t^*_{0.975,8} \approx 2.306\)
- For \(n=30\), \(df=28\), \(t^*_{0.975,28} \approx 2.048\)
Step 3 — Compute margin of error (ME):
- \( ME_{n=10} = t^* \cdot SE_b = 2.306 \cdot 0.447 \approx 1.03 \)
- \( ME_{n=30} = t^* \cdot SE_b = 2.048 \cdot 0.447 \approx 0.92 \)
Step 4 — Interpretation:
- 95% CI width for \(n=10\): \(2 \cdot 1.03 = 2.06\)
- 95% CI width for \(n=30\): \(2 \cdot 0.92 = 1.84\)
- As sample size increases from 10 to 30, the CI becomes narrower → more precise estimate of the slope.
Conclusion: Increasing sample size reduces the standard error and t*, which narrows the confidence interval, giving a more precise estimate of the population slope \(\beta\).