AP Statistics Random Variables, Expected Value and Combining Variables Test Solutions and Scoring (PDF)

AP Statistics math test solutions: Random Variables, Expected Value and Combining Variables – Maths PDF

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This random variables test answers for AP Statistics covers random variables, expected value and combining variables (Common Core AP STAT 4.7, AP STAT 4.8, AP STAT 4.9, HSS-MD.A.1, HSS-MD.A.2, HSS-MD.B.5), with a free PDF to print.

Each solution includes a scoring guide. Give credit for correct products, the correct combination rule, and a conclusion that interprets the mean or standard deviation in context.

AP Statistics · Common Core State Standards: AP STAT 4.7, AP STAT 4.8, AP STAT 4.9, HSS-MD.A.1, HSS-MD.A.2, HSS-MD.B.5

Random variables test answers: worked solutions

Question 1 – Mean and SD [4 pts]

Number of siblings.

(a) $0(0.1)+1(0.3)+2(0.4)+3(0.2)=1.7$.

(b) $(0-1.7)^{2}(0.1)+(1-1.7)^{2}(0.3)+(2-1.7)^{2}(0.4)+(3-1.7)^{2}(0.2)=0.289+0.147+0.036+0.338=0.81$, $\sigma=0.9$.

1.7; 0.9; long-run average.

Scoring: 1 point for (a), 2 points for (b) (1 for variance, 1 for sqrt), 1 point for (c).

Question 2 – Linear transformation [2 pts]

$5(12)-20=40$. $5(3)=15$.

Mean 40, standard deviation 15.

Scoring: 1 point each.

Question 3 – Independent sum [3 pts]

Total: mean 50, sd $\sqrt{0.09+0.16}=0.5$. Difference: mean 10, sd 0.5.

Total: 50 cm, 0.5 cm. Difference: 10 cm, 0.5 cm.

Scoring: 1 point for the means, 1 point for the sd of the total, 1 point for the sd of the difference.

Question 4 – Fair game [3 pts]

A carnival game.

(a) $9(0.1)+(-1)(0.9)=0.9-0.9=0$.

(b) The expected gain is 0, so yes.

$0; fair.

Scoring: 1 point for the setup, 1 point for 0, 1 point for fair.

Question 5 – Variances add [2 pts]

$\sqrt{36+64}=10$.

10; variances add, standard deviations do not.

Scoring: 1 point for 10, 1 point for the explanation.

Question 6 – 2X versus X + Y [2 pts]

2X: mean 10, sd 4. X + Y: mean 10, sd $\sqrt{8}\approx 2.83$.

Same mean 10, but sd 4 for 2X and 2.83 for X + Y.

Scoring: 1 point for each of the two sets.

Question 7 – Insurance [2 pts]

Profit: 60 with probability 0.9; 60 – 400 = -340 with probability 0.1. $60(0.9)-340(0.1)=54-34=20$.

$20 per warranty.

Scoring: 1 point for the profit values, 1 point for 20.

Question 8 – Valid distribution [2 pts]

Steps.

$k=1-0.15-0.35-0.25=0.25$. $P(X\ge 3)=0.25+0.25=0.50$.

k = 0.25; P = 0.50.

Scoring: 1 point each.

Total score

Total

20 points

Key facts behind this random variables test answers

Key facts

Probabilities in a distribution are between 0 and 1 and sum to 1.
Mean (expected value): sum of x p; interpret it as the long-run average.
Standard deviation: square root of the sum of (x – mean)^2 p.
For a + bX: mean a + b mu, standard deviation |b| sigma.
For independent X, Y: means add or subtract; variances add (also for X – Y).

How to use this random variables 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.

All AP Statistics chapters: AP Statistics math.

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