AP Statistics t Intervals and t Tests for a Mean Test Solutions and Scoring (PDF)

AP Statistics math test solutions: t Intervals and t Tests for a Mean – Maths PDF

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This inference one mean test answers for AP Statistics covers t intervals and t tests for a mean (Common Core AP STAT 7.1, AP STAT 7.2, AP STAT 7.3, AP STAT 7.4, AP STAT 7.5, HSS-IC.B.4), with a free PDF to print.

Each solution includes a scoring guide. Give credit for correct conditions and df, the interval or test statistic, and a conclusion that names the parameter and the population.

AP Statistics · Common Core State Standards: AP STAT 7.1, AP STAT 7.2, AP STAT 7.3, AP STAT 7.4, AP STAT 7.5, HSS-IC.B.4

Inference one mean test answers: worked solutions

Question 1 – t interval [4 pts]

Interval.

(a) Random sample (given); 10 is less than 10% of all packages; the dotplot shows no strong skew or outliers, so Normal is reasonable. (b) \(\overline{x}=12.75,\ s=0.599,\ SE=0.189,\ t^{*}=2.262\), ME 0.428, interval (12.322, 13.178). (c) We are 95% confident that this interval captures the true mean weight of all packages.

(12.322, 13.178) ounces.

Scoring: 1 point for conditions (random, 10%, Normal from the graph), 1 point for SE/df/t*, 1 point for the interval, 1 point for the interpretation in context.

Question 2 – t test [4 pts]

Two-sided p-value.

\(H_0:\mu=15,\ H_a:\mu\neq 15\). Conditions: random, 32 is less than 10% of all commuters, n at least 30. \(t=1.41,\ df=31\), p-value \(=0.1685\).

Fail to reject H0: not convincing evidence that the mean travel time differs from 15 minutes.

Scoring: 1 point for hypotheses with the parameter, 1 point for conditions, 1 point for t and the p-value, 1 point for the conclusion.

Question 3 – Matched pairs setup [2 pts]

The same students are measured twice, so the two measurements are dependent.

Use differences; the parameter is mu_d, the true mean difference in puzzle time (without music minus with music) for all students.

Scoring: 1 point for pairing, 1 point for defining mu_d.

Question 4 – t* and z* [2 pts]

Normal and t.

The t distribution has heavier tails, so a larger multiplier is needed for the same central area.

t* is larger; as n grows, t* approaches z*.

Scoring: 1 point for heavier tails, 1 point for the limit.

Question 5 – Effect of changes [2 pts]

ME = t* s/sqrt(n).

(a) smaller; (b) larger.

Scoring: 1 point each.

Question 6 – Conditions [2 pts]

(a) No: small n with an outlier. (b) Yes: n is at least 30, so the CLT applies.

(a) no; (b) yes.

Scoring: 1 point each.

Question 7 – Interpret the p-value [2 pts]

The p-value is computed assuming H0 is true.

If the true mean time were 20 minutes, there would be a 3% chance of getting a sample mean at least as large as the one observed.

Scoring: 1 point for assuming H0 true, 1 point for context.

Question 8 – Degrees of freedom [2 pts]

df = 17 and t* = 2.110.

df = 17; t* = 2.110.

Scoring: 1 point for df, 1 point for t*.

Total score

Total

20 points

Key facts behind this inference one mean test answers

Key facts

t distribution: bell-shaped, centered at 0, heavier tails; df = n – 1.
Interval: x-bar +/- t* s/sqrt(n); test: t = (x-bar – mu0)/(s/sqrt(n)).
Conditions: random, 10% condition, and Normal/Large Sample (population normal, n >= 30, or no strong skew or outliers).
As df grows, t* approaches z*.
Matched pairs: take differences and do a one-sample t procedure.

How to use this inference one mean 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 inference one mean test answers as a PDF with the button below and keep it with your notes.

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