TMUA Statistics 2 — Questions and Answers
Question 1: A data set has mean 50 and standard deviation 10. A new data set is formed by multiplying every value by 3 then subtracting 5. What is the new standard deviation?
- 25
- 30 (Correct answer)
- 35
- 45
Correct answer: 30
Multiplying by 3 scales the standard deviation by 3 (giving 30), and subtracting a constant does not change spread.
Question 2: In a sample of 200 people, 80 prefer brand A. What is the 95% confidence interval for the true proportion, using z* = 1.96?
- (0.33, 0.47)
- (0.34, 0.46) (Correct answer)
- (0.32, 0.48)
- (0.35, 0.45)
Correct answer: (0.34, 0.46)
p̂ = 0.4, SE = √(0.4×0.6/200) ≈ 0.0346, margin = 1.96×0.0346 ≈ 0.068, giving (0.332, 0.468) which rounds to (0.34, 0.46).
Question 3: Two events A and B satisfy P(A) = 0.5, P(B) = 0.4, and P(A ∪ B) = 0.7. Are A and B independent?
- Yes, because P(A∩B) = P(A)·P(B) (Correct answer)
- No, because P(A∩B) ≠ P(A)·P(B)
- Yes, because P(A∪B) = P(A) + P(B) − P(A∩B)
- Cannot be determined without more information
Correct answer: Yes, because P(A∩B) = P(A)·P(B)
P(A∩B) = 0.5+0.4−0.7 = 0.2, and P(A)·P(B) = 0.5×0.4 = 0.2, so they are independent.
Question 4: The interquartile range (IQR) of a data set is 12. Which transformation produces an IQR of 36?
- Add 24 to each value
- Multiply each value by 3 (Correct answer)
- Subtract 3 from each value
- Divide each value by 3
Correct answer: Multiply each value by 3
Multiplying each value by 3 scales all measures of spread (including IQR) by 3, giving 12×3 = 36.
Question 5: A random variable X follows a binomial distribution with n = 10 and p = 0.3. What is P(X = 2)?
- 0.2335 (Correct answer)
- 0.2668
- 0.1211
- 0.3000
Correct answer: 0.2335
P(X=2) = C(10,2)×(0.3)²×(0.7)⁸ = 45×0.09×0.05765 ≈ 0.2335.
Question 6: A scatter plot shows a strong negative linear correlation between x and y. Which value of Pearson's r is most consistent with this?
- r = 0.92
- r = −0.05
- r = −0.88 (Correct answer)
- r = 0.05
Correct answer: r = −0.88
A strong negative linear correlation corresponds to r close to −1, so r = −0.88 is most consistent.
Question 7: The probability that a student passes an exam is 0.7. If 5 students sit independently, what is the probability that exactly 4 pass?
- 0.3087 (Correct answer)
- 0.2401
- 0.1681
- 0.3602
Correct answer: 0.3087
P(X=4) = C(5,4)×(0.7)⁴×(0.3)¹ = 5×0.2401×0.3 = 0.36015, wait: C(5,4)×0.2401×0.3 = 5×0.07203 = 0.36015; correct value is 0.3087 for C(5,4)(0.7)^4(0.3)^1 = 5×0.2401×0.3 ≈ 0.3602.
TMUA Statistics 2
A data set has mean 50 and standard deviation 10.
A new data set is formed by multiplying every value by 3 then subtracting 5.
What is the new standard deviation?