AP Statistics Binomial and Geometric Distributions Test Solutions and Scoring (PDF)

AP Statistics math test solutions: Binomial and Geometric Distributions – Maths PDF

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This binomial geometric test answers for AP Statistics covers binomial and geometric distributions (Common Core AP STAT 4.10, AP STAT 4.11, AP STAT 4.12, HSS-MD.A.3, HSS-CP.B.9), with a free PDF to print.

Each solution includes a scoring guide. Give credit for naming the distribution and parameters, using the correct command or formula, and answering in context.

AP Statistics · Common Core State Standards: AP STAT 4.10, AP STAT 4.11, AP STAT 4.12, HSS-MD.A.3, HSS-CP.B.9

Binomial geometric test answers: worked solutions

Question 1 – Binomial probability [4 pts]

Binomial with n = 6 and p = 0.4.

(a) \(0.3110\). (b) \(1-P(X\le 3)=1-0.8208=0.1792\). (c) \(6(0.4)=2.4\).

0.3110; 0.1792; 2.4.

Scoring: 1 point for (a), 2 points for (b) (1 for complement, 1 for value), 1 point for (c).

Question 2 – BINS [3 pts]

(a) Binary: made or missed. Independent: shots do not affect each other. Number: 15 fixed. Same p = 0.45.

(b) \(\text{binompdf}(15,0.45,7)=0.2013\).

Binomial with n = 15 and p = 0.45; P = 0.2013.

Scoring: 1 point for each pair of BINS conditions correctly checked (2 points), 1 point for the value.

Question 3 – Mean and sd [2 pts]

\(\mu=20(0.2)=4\). \(\sigma=\sqrt{20(0.2)(0.8)}=\sqrt{3.2}=1.789\).

Mean 4, standard deviation 1.789.

Scoring: 1 point each.

Question 4 – Geometric [3 pts]

Geometric with p = 0.3.

(a) \(0.7^{2}(0.3)=0.1470\). (b) \(0.7^{4}=0.2401\). (c) \(\frac{1}{0.3}\approx 3.33\).

0.1470; 0.2401; 3.33.

Scoring: 1 point per part.

Question 5 – Unusual outcome [2 pts]

\(P(X\le 12)=\text{binomcdf}(20,0.85,12)=0.0059\).

0.0059 is below 0.05, so the result is unusual and casts doubt on the 85% claim.

Scoring: 1 point for the value, 1 point for the conclusion.

Question 6 – Fixed number [1 pt]

BINS check.

The number of trials is not fixed in advance (and without replacement the trials are dependent).

The fixed Number condition fails.

Scoring: 1 point for naming the fixed number of trials condition.

Question 7 – Choose the model [2 pts]

(a) Binomial with n = 12 and p = 1/6. (b) Geometric with p = 1/6.

(a) binomial, n = 12, p = 1/6; (b) geometric, p = 1/6.

Scoring: 1 point each.

Question 8 – 10 percent condition [3 pts]

(a) The sample (5) is 12.5% of the population (40), more than 10%, so the trials are not independent enough for a binomial model.

(b) \(\text{binompdf}(5,0.25,2)=0.2637\), only a rough approximation.

Not binomial (10% condition fails); approximately 0.2637.

Scoring: 2 points for the condition check, 1 point for the value.

Total score

Total

20 points

Key facts behind this binomial geometric test answers

Key facts

Binomial conditions: Binary, Independent, fixed Number, Same probability.
Binomial: mean = np, standard deviation = sqrt(np(1-p)); use binompdf and binomcdf.
At least k successes: 1 – binomcdf(n, p, k – 1).
Geometric: trials until the first success; P(X = n) = (1-p)^(n-1) p; mean 1/p.
Without replacement, treat trials as independent only if n is at most 10% of the population.

How to use this binomial geometric 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.

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