AP Statistics Inference for the Difference of Two Means Test Solutions and Scoring (PDF)
This two means test answers for AP Statistics covers inference for the difference of two means (Common Core AP STAT 7.6, AP STAT 7.7, AP STAT 7.8, AP STAT 7.9, HSS-IC.B.5), with a free PDF to print.
AP Statistics · Common Core State Standards: AP STAT 7.6, AP STAT 7.7, AP STAT 7.8, AP STAT 7.9, HSS-IC.B.5
Two means test answers: worked solutions
Question 1 – Two-sample interval [4 pts]
(a) Independent random samples; each under 10% of its population; both n at least 30. (b) \(SE=1.910,\ df\approx 82.8,\ t^{*}=1.989,\ ME=3.798\), interval (0.402, 7.998). (c) We are 95% confident that the true mean weekly study time is between 0.402 and 7.998 hours higher for online students than for in-person students.
(0.402, 7.998) hours.
Scoring: 1 point for conditions, 1 point for the SE and t*, 1 point for the interval, 1 point for the interpretation.
Question 2 – Two-sample test [4 pts]
\(H_0:\mu_1-\mu_2=0,\ H_a:\mu_1-\mu_2\neq 0\). \(t=2.20,\ df\approx 82.8\), p-value \(=0.0306\).
Reject H0: convincing evidence of a difference in the mean weekly study times.
Scoring: 1 point each for hypotheses, t statistic, p-value, and a conclusion in context.
Question 3 – Paired or two-sample [3 pts]
(a) Paired t procedure; parameter mu_d, the true mean change in cholesterol. (b) Two-sample t procedure; parameter mu_drivers – mu_cyclists.
(a) paired; (b) two-sample.
Scoring: 1 point per correct procedure, 1 point for correctly defining both parameters.
Question 4 – Standard error [2 pts]
\(\sqrt{\frac{36}{25}+\frac{81}{36}}=\sqrt{1.44+2.25}=\sqrt{3.69}=1.921\).
1.921.
Scoring: 1 point for the formula, 1 point for 1.921.
Question 5 – Normal condition [2 pts]
Group 2 has a very long upper whisker, indicating strong skew or an outlier, and n is small.
No: the Normal condition is questionable for Group 2.
Scoring: 1 point for the answer, 1 point for the reason.
Question 6 – Interpret p-value [2 pts]
If the two population means were equal, there would be a 0.3% chance of a difference in sample means this large or larger.
Since 0.003 < 0.01, reject H0: convincing evidence the means differ.
Scoring: 1 point for the interpretation, 1 point for the conclusion.
Question 7 – Name the parameter [1 pt]
mu_1 is the true mean weekly study time of all online students; mu_2 that of all in-person students.
mu_1 – mu_2 is the difference between these two population means.
Scoring: 1 point for a definition in context.
Question 8 – Effect of changes [2 pts]
(a) narrower; (b) wider.
Narrower; wider.
Scoring: 1 point each.
Total score
20 points
Key facts behind this two means test answers
- Parameter: mu1 – mu2; statistic: x-bar1 – x-bar2.
- SE = sqrt(s1^2/n1 + s2^2/n2); df from the calculator, or min(n1 – 1, n2 – 1) conservatively.
- Interval: (x-bar1 – x-bar2) +/- t* SE; test: t = (x-bar1 – x-bar2)/SE with H0: mu1 – mu2 = 0.
- Conditions: independent random samples or random assignment; 10%; each group normal or n >= 30.
- Same subjects measured twice or matched pairs: use a one-sample t on the differences.
How to use this two means test answers
Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.
Download the two means test answers as a PDF with the button below and keep it with your notes.
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AP Statistics Inference for the Difference of Two Means Test Solutions and Scoring
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