TMUA Statistics 5 — Questions and Answers
Question 1: A linear regression model gives ŷ = 3 + 2x. For x = 5, the observed value is y = 14. What is the residual?
- 1 (Correct answer)
- −1
- 3
- 14
Correct answer: 1
Predicted value ŷ = 3 + 2(5) = 13; residual = observed − predicted = 14 − 13 = 1.
Question 2: A fair six-sided die is rolled. Given that the result is even, what is the probability it is greater than 4?
- 1/6
- 1/3 (Correct answer)
- 1/2
- 2/3
Correct answer: 1/3
Even outcomes are {2, 4, 6}; of these, outcomes greater than 4 are {6}, giving P = 1/3.
Question 3: A frequency distribution has class intervals of width 5. The frequency for the interval [20, 25) is 12 and the total frequency is 60. What is the relative frequency for that class?
- 0.12
- 0.20 (Correct answer)
- 0.25
- 0.50
Correct answer: 0.20
Relative frequency = frequency / total = 12/60 = 0.20.
Question 4: X ~ N(μ, σ²). If Z = (X − μ)/σ, which distribution does Z follow?
- N(μ, σ²)
- N(0, σ²)
- N(0, 1) (Correct answer)
- t-distribution
Correct answer: N(0, 1)
Standardizing a normal variable by subtracting its mean and dividing by its standard deviation produces a standard normal Z ~ N(0,1).
Question 5: The cumulative distribution function (CDF) of a random variable X gives F(x). Which statement correctly describes F(x)?
- F(x) = P(X > x)
- F(x) = P(X = x)
- F(x) = P(X ≤ x) (Correct answer)
- F(x) = f(x) × x
Correct answer: F(x) = P(X ≤ x)
By definition, the CDF F(x) gives the probability that the random variable X takes a value less than or equal to x.
Question 6: A data set has n = 9 values with Σx = 45 and Σx² = 261. What is the variance?
- 4 (Correct answer)
- 6
- 29
- 2
Correct answer: 4
Mean = 45/9 = 5; variance = Σx²/n − (mean)² = 261/9 − 25 = 29 − 25 = 4.
Question 7: A Type I error in hypothesis testing occurs when:
- H₀ is true and we fail to reject it
- H₀ is false and we fail to reject it
- H₀ is true and we reject it (Correct answer)
- H₀ is false and we reject it
Correct answer: H₀ is true and we reject it
A Type I error (false positive) is rejecting H₀ when H₀ is actually true; its probability equals the significance level α.
TMUA Statistics 5
A linear regression model gives ŷ = 3 + 2x.
For x = 5, the observed value is y = 14.
What is the residual?