AP Statistics Practice Quiz: Combining Random Variables
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Test your understanding with short quizzes. This quiz has 15 questions to check your progress.
Question 1 of 15
All Questions (15)
A) 80
B) 95
C) 65
D) 15
Correct Answer: C
For a linear transformation Y = a + bX, the new mean is a + b*mean(X). In this case, a = -15 and b = 1. The new mean is -15 + 1*(80) = 65.
A) 17
B) 41
C) 12
D) 36
Correct Answer: D
For a linear transformation Y = a + bX, the new standard deviation is |b|*sd(X). Adding a constant 'a' does not affect the spread. Here, b = 3, so the new standard deviation is |3| * 12 = 36.
A) The shape becomes symmetric.
B) The shape remains skewed to the left.
C) The shape becomes skewed to the right.
D) The shape cannot be determined from the information given.
Correct Answer: B
According to the rules for linear transformations of the form Y = a + bX, the shape of the probability distribution of the transformed variable Y is the same as the shape of the original variable X.
A) 2 hours
B) 15 hours
C) 8 hours
D) Cannot be determined without knowing if the variables are independent.
Correct Answer: C
The mean of a sum of random variables is the sum of their means, regardless of independence. Mean(T) = Mean(X) + Mean(Y) = 5 + 3 = 8 hours.
A) 10 minutes
B) 52 minutes
C) 2 minutes
D) 7.21 minutes
Correct Answer: D
For independent random variables, variances add. First, find the variances: Var(X) = 4^2 = 16 and Var(Y) = 6^2 = 36. The variance of the total time is Var(T) = Var(X) + Var(Y) = 16 + 36 = 52. The standard deviation is the square root of the variance, which is sqrt(52) ≈ 7.21 minutes.
A) 7
B) 17
C) 23
D) 161
Correct Answer: B
For independent random variables, variances add even when finding the difference. First, find the variances: Var(X) = 15^2 = 225 and Var(Y) = 8^2 = 64. The variance of the difference is Var(D) = Var(X) + Var(Y) = 225 + 64 = 289. The standard deviation is the square root of the variance, which is sqrt(289) = 17.
A) They have the same probability distribution.
B) They cannot have the same value.
C) Knowing the outcome of one variable provides no information about the probability distribution of the other.
D) The sum of their means must be zero.
Correct Answer: C
The definition of independent random variables is that knowing the value or outcome of one variable does not change the probability distribution (the likelihood of any given outcome) of the other variable.
A) $120
B) $290
C) $310
D) $250
Correct Answer: B
The mean of a linear combination aX + bY is calculated as a*mean(X) + b*mean(Y). Here, a=3 and b=2. The mean profit is 3*($50) + 2*($70) = $150 + $140 = $290.
A) 22
B) 172
C) 116
D) 8
Correct Answer: B
For independent random variables, the variance of aX + bY is a^2*var(X) + b^2*var(Y). In this case, a=4 and b=-2. So, Var(Z) = (4^2)*Var(X) + ((-2)^2)*Var(Y) = 16*(9) + 4*(7) = 144 + 28 = 172.
A) Mean = 59, SD = 39.2
B) Mean = 47, SD = 7.2
C) Mean = 59, SD = 4
D) Mean = 59, SD = 7.2
Correct Answer: D
Using the rules for linear transformations Y = a + bX: The new mean is a + b*mean(X) = 32 + 1.8*(15) = 32 + 27 = 59. The new standard deviation is |b|*sd(X) = |1.8| * 4 = 7.2.
A) $18,500
B) $1.176
C) $1,500
D) Cannot be determined because revenue and costs are not independent.
Correct Answer: C
The mean of a difference of random variables is the difference of their means. This rule applies regardless of whether the variables are independent. Mean(P) = Mean(R) - Mean(C) = $10,000 - $8,500 = $1,500.
A) X and Y must have the same mean.
B) X and Y must be independent.
C) X and Y must follow a normal distribution.
D) The rule is always valid for any two random variables.
Correct Answer: B
The variance of a sum (or difference) of two random variables is the sum of their individual variances only if the two random variables are independent.
A) Mean = 450g, SD = 14g
B) Mean = 450g, SD = 10g
C) Mean = 450g, SD = 100g
D) Mean = 50g, SD = 10g
Correct Answer: B
The mean of the total weight is the sum of the means: 250g + 200g = 450g. Since the variables are independent, the variances add: Var(Total) = Var(A) + Var(C) = 8^2 + 6^2 = 64 + 36 = 100. The standard deviation is the square root of the total variance: sqrt(100) = 10g.
A) 25
B) 35
C) 125
D) 5
Correct Answer: A
For a linear transformation Y = a + bX, the standard deviation is |b|*sd(X). Here, a=10 and b=1. The standard deviation is |1|*sd(X) = sd(X). Since the standard deviation does not change, the variance (which is the standard deviation squared) also does not change. Adding a constant shifts the distribution but does not affect its spread.
A) 20
B) 14.14
C) 200
D) 11.18
Correct Answer: B
First, find the variances: Var(X) = 5^2 = 25 and Var(Y) = 10^2 = 100. Next, use the rule for the variance of a linear combination: Var(Z) = 2^2*Var(X) + 1^2*Var(Y) = 4*(25) + 1*(100) = 100 + 100 = 200. The standard deviation of Z is the square root of its variance: sqrt(200) ≈ 14.14.