AP Statistics Inference for the Slope of a Regression Line Worksheet Solutions (PDF)
This inference slope answers for AP Statistics covers inference for the slope of a regression line (Common Core AP STAT 9.1, AP STAT 9.2, AP STAT 9.3, AP STAT 9.4, AP STAT 9.5, AP STAT 9.6, HSS-ID.B.6), with a free PDF to print.
AP Statistics · Common Core State Standards: AP STAT 9.1, AP STAT 9.2, AP STAT 9.3, AP STAT 9.4, AP STAT 9.5, AP STAT 9.6, HSS-ID.B.6
Inference slope answers: exercises 1 to 10
Exercise 1 – Parameters
\(\beta\) is the true slope of the population regression line of free throws made on weekly practice hours: the change in the mean number made per additional hour.
\(H_0:\beta=0\) and \(H_a:\beta\neq 0\).
Exercise 2 – LINER conditions
Linear, Independent, Normal, Equal variance, Random.
Equal variance is checked on the residual plot: the vertical spread of residuals should be about the same for all x (no fan shape).
Exercise 3 – Residual plot
The residuals scatter randomly around 0 with no curve and no change in spread.
Yes: no pattern supports Linear, and the roughly constant spread supports Equal variance.
Exercise 4 – Read the output
Constant 1.003, slope 1.133, SE of the slope 0.104.
\(\hat{y}=1.003+1.133x\); SE_b = 0.104.
Exercise 5 – Interpret the slope
The slope is a predicted change per unit of x.
For each additional hour of weekly practice, the predicted number of free throws made increases by about 1.13.
Exercise 6 – Degrees of freedom and t*
\(df=n-2=10\).
df = 10; t* = 2.228.
Exercise 7 – Confidence interval
\(b\pm t^{*}SE_b=1.133\pm 2.228(0.104)\).
(0.901, 1.365).
Exercise 8 – Interpret the interval
State the parameter and the units.
We are 95% confident that the true slope is between 0.901 and 1.365: for each additional hour of weekly practice, the mean number of free throws made increases by between 0.90 and 1.36.
Exercise 9 – t statistic
\(t=\frac{1.133}{0.104}=10.89\) with df = 10.
t = 10.89.
Exercise 10 – Conclusion
0.0000 < 0.05.
Reject H0. There is convincing evidence of a linear relationship between weekly practice hours and free throws made.
Exercises 11 and up
Exercise 11 – Interval and test
0 is inside the interval.
Fail to reject H0: not convincing evidence of a linear relationship.
Exercise 12 – Temperature and sales
\(\hat{y}=59.45-1.60x\). \(t=-11.73,\ df=8\), p-value \(=0.0000\).
Reject H0: convincing evidence of a negative linear relationship between temperature and hot cocoa sales.
Exercise 13 – Interval from output
\(df=8,\ t^{*}=3.355,\ ME=0.459\).
(-2.061, -1.144) cups per degree.
Exercise 14 – R squared and r
The slope is negative, so \(r=-\sqrt{0.945}=-0.972\).
r = -0.972; about 94.5% of the variation in cocoa sales is explained by the linear relationship with temperature.
Exercise 15 – One-sided test
\(t=\frac{0.62}{0.30}=2.07\) is positive, in the direction of Ha.
One-sided p-value \(=0.059/2=0.0295\).
0.0295 < 0.05: reject H0. There is convincing evidence that the true slope is positive (more rainfall goes with more mud).
Exercise 16 – Residual plot warning
The U shape means the relationship is curved.
No: the Linear condition fails, so the t procedure is not appropriate until the data are transformed.
Exercise 17 – Choose a procedure
(a) t interval for the slope. (b) Two-proportion z test. (c) Chi-square goodness-of-fit.
(a) slope t interval; (b) two-proportion z test; (c) goodness-of-fit.
Exercise 18 – Standard error of the slope
\(SE_b=\frac{s}{s_x\sqrt{n-1}}=\frac{2.4}{3\sqrt{19}}=0.1835\).
0.1835.
Exercise 19 – Multi-step
\(t=\frac{-1.35}{0.21}=-6.43,\ df=16\), p-value \(p < 0.0001\).
\(t^{*}=2.120,\ ME=0.445\).
Reject H0; 95% interval (-1.795, -0.905): each extra year of age lowers the mean price by between about 0.90 and 1.79 thousand dollars.
Exercise 20 – Predict and caution
\(\hat{y}=1.00+1.133(8)=10.07\).
About 10.1 free throws. x = 40 is far outside the data (extrapolation), so the line cannot be trusted.
Key facts behind this inference slope answers
- Parameter: true slope beta; statistic: sample slope b; df = n – 2.
- Interval: b +/- t* SE_b; test: t = (b – 0)/SE_b for H0: beta = 0.
- LINER: Linear, Independent, Normal residuals, Equal variance, Random.
- Computer output gives b, SE_b, t, and the two-sided P in the row for the explanatory variable.
- If the interval contains 0, there is no convincing evidence of a linear relationship.
How to use this inference slope answers
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
Download the inference slope answers as a PDF with the button below and keep it with your notes.
Download the inference slope answers as a PDF with the button below and keep it with your notes.
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