TMUA Statistics 3 — Questions and Answers
Question 1: A continuous random variable X has PDF f(x) = kx for 0 ≤ x ≤ 4, and 0 otherwise. What is the value of k?
- 1/8 (Correct answer)
- 1/4
- 1/16
- 1/2
Correct answer: 1/8
Integrating kx from 0 to 4 gives 8k = 1, so k = 1/8.
Question 2: Heights of a population are normally distributed with mean 170 cm and standard deviation 8 cm. What proportion of individuals are taller than 186 cm?
- 0.0228 (Correct answer)
- 0.0455
- 0.1587
- 0.9772
Correct answer: 0.0228
186 cm is 2 standard deviations above the mean; P(Z > 2) ≈ 0.0228.
Question 3: A hypothesis test uses H₀: μ = 100 vs H₁: μ > 100 with significance level α = 0.05. The test statistic is z = 1.60. What is the correct conclusion?
- Reject H₀ because z > 1.645
- Fail to reject H₀ because z < 1.645 (Correct answer)
- Reject H₀ because z > 1.282
- Fail to reject H₀ because z < 1.960
Correct answer: Fail to reject H₀ because z < 1.645
For a one-tailed test at α = 0.05, the critical value is z = 1.645; since 1.60 < 1.645, we fail to reject H₀.
Question 4: Which of the following correctly describes the relationship between variance and standard deviation?
- Variance = Standard deviation / 2
- Standard deviation = Variance²
- Variance = (Standard deviation)² (Correct answer)
- Standard deviation = 2 × Variance
Correct answer: Variance = (Standard deviation)²
By definition, variance = σ² and standard deviation σ = √(variance), so variance = (standard deviation)².
Question 5: A box contains 5 red, 3 blue, and 2 green balls. Two balls are drawn without replacement. What is the probability that both are red?
- 1/4
- 2/9 (Correct answer)
- 4/15
- 1/5
Correct answer: 2/9
P(both red) = (5/10)×(4/9) = 20/90 = 2/9.
Question 6: The regression line for predicting y from x passes through the point (x̄, ȳ). Which statement is always true?
- The regression line has a positive gradient
- The regression line passes through the origin
- The regression line minimizes the sum of squared residuals (Correct answer)
- The regression line has an r² value of 1
Correct answer: The regression line minimizes the sum of squared residuals
The least-squares regression line is defined as the line that minimizes the sum of squared vertical residuals.
Question 7: A Poisson random variable X has mean λ = 3. What is P(X = 0)?
- e⁻³ (Correct answer)
- 3e⁻³
- 1 − e⁻³
- e³
Correct answer: e⁻³
P(X=0) = e^(−λ) × λ⁰ / 0! = e^(−3).
TMUA Statistics 3
A continuous random variable X has PDF f(x) = kx for 0 ≤ x ≤ 4, and 0 otherwise.
What is the value of k?