#1. Parts have weights normally distributed with μ = 100 g and σ = 8 g. A sample of 16 parts is taken. What is the probability that the sample mean exceeds 104 g?
#2. For a normal distribution, the middle 90% of data lies between z = -1.645 and z = 1.645. If μ = 200 and σ = 30, what is the upper boundary?
#3. A process mean must not exceed 50 with probability 0.99. If σ = 5 and z₀.₉₉ = 2.33, what should the target mean be set at?
#4. Two independent normal variables have X ~ N(40, 16) and Y ~ N(60, 9). What is the standard deviation of X + Y?
#5. If X ~ N(100, 225), what is P(85 < X < 130)?
#6. A manufacturing specification requires that 99.73% of products fall within ±0.15 mm of the target. What must the process standard deviation be?
#7. The difference between two independent sample means has standard error 4. If n₁ = n₂ = 25 and σ₁² = σ₂² = σ², what is σ?
#8. For X ~ N(μ, σ²), P(X < 60) = 0.25 and P(X < 80) = 0.75. What is μ?
#9. A sample of 100 is taken from N(μ, 400). For P(|X̄ – μ| < 2) = 0.6826, verify this is approximately correct. What is σ/√n?
#10. If Z is standard normal, what is E[Z²]?
#11. X follows a Poisson distribution with λ = 4. What is P(X = 2 | X ≤ 4)?
#12. The number of arrivals in a 1-hour period is Poisson with λ = 6. What is the probability of at least 2 arrivals in 30 minutes?
#13. A negative binomial distribution models the number of trials until the rth success. If p = 0.3 and we want 4 successes, what is the expected number of trials?
#14. For an exponential distribution, the median is ln(2)/λ. If the mean is 10, what is the median?
#15. A gamma distribution has shape parameter k = 3 and rate λ = 0.5. What is the expected value?
#16. In multiple regression with 3 predictors and 50 observations, what are the degrees of freedom for the error term?
#17. A regression model has R² = 0.72 with 5 predictors and n = 30. What is the adjusted R²?
#18. If the correlation between residuals and fitted values is significantly different from zero, this indicates:
#19. In a regression with SSR = 800, SSE = 200, and n = 50 with 2 predictors, what is the F-statistic?
#20. The slope coefficient in a simple regression is b₁ = r(sᵧ/sₓ). If r = 0.8, sᵧ = 12, and sₓ = 4, what is b₁?
#21. A 95% CI for μ is (45, 55) based on n = 36. If n increases to 144 with the same sample statistics, the new CI width will be:
#22. To estimate μ with 99% confidence (z = 2.576) and margin of error 2 when σ = 10, what sample size is needed?
#23. A sample of 25 has s² = 36. Using χ²₀.₀₂₅,₂₄ = 39.364 and χ²₀.₉₇₅,₂₄ = 12.401, what is the upper bound of the 95% CI for σ²?
#24. For a proportion, a 95% CI is (0.42, 0.58). What was the sample proportion p̂?
#25. The width of a CI for μ is inversely proportional to √n. If doubling the width is acceptable, the sample size can be reduced by what factor?
#26. Testing H₀: μ = 100 vs H₁: μ > 100 at α = 0.05, if the true μ = 105, σ = 10, and n = 25, what is the power? (Use z₀.₀₅ = 1.645)
#27. A Type I error occurs when we reject H₀ when it is true. If α = 0.01 and we conduct 5 independent tests, what is the probability of at least one Type I error?
#28. For a two-sample t-test with unequal variances (Welch’s test), the degrees of freedom formula gives df = 23.7. What df should be used?
#29. In testing H₀: σ² = 25 vs H₁: σ² ≠ 25 with n = 20 and s² = 40, the chi-square statistic is:
#30. A test has power 0.90 against a specific alternative. What is P(correctly rejecting H₀ when this alternative is true)?
#31. A diagnostic test has sensitivity 0.95 and specificity 0.90. If disease prevalence is 2%, what is P(disease | positive test)?
#32. Machine A produces 60% of parts with 3% defect rate; Machine B produces 40% with 5% defect rate. Given a defective part, what is P(from A)?
#33. P(A) = 0.3, P(B|A) = 0.8, P(B|A’) = 0.4. What is P(A|B)?
#34. Three factories produce 30%, 50%, and 20% of items with defect rates 2%, 3%, and 5% respectively. What is the overall defect rate?
#35. If P(A|B) = 0.6, P(B) = 0.5, and P(A) = 0.4, are A and B independent?
#36. In how many ways can 12 people be divided into 3 groups of 4?
#37. How many distinct permutations are there of the letters in MISSISSIPPI?
#38. A committee of 5 is selected from 6 men and 4 women. How many committees have at least 3 women?
#39. How many ways can 10 identical balls be distributed into 4 distinct boxes?
#40. The number of ways to arrange n objects where p are alike is n!/p!. How many ways can ARRANGE be arranged?
#41. A k-out-of-n system requires at least k of n components working. For a 2-out-of-4 system with component reliability 0.9, what is the system reliability?
#42. A component has failure rate λ(t) = 0.001t (increasing). This represents which part of the bathtub curve?
#43. Two components have exponential failure distributions with λ₁ = 0.01 and λ₂ = 0.02 per hour. In parallel, what is the system’s mean time to failure?
#44. A standby system has one active component (R = 0.9) and one standby (R = 0.9) with perfect switching. What is system reliability?
#45. If component MTTF = 1000 hours and mission time = 200 hours, assuming exponential distribution, what is the reliability?
#46. The moment generating function of a random variable is M(t) = e^(3t+2t²). What is E[X]?
#47. If X ~ Exp(λ), then M_X(t) = λ/(λ-t) for t < λ. What is E[X²] if λ = 2?
#48. For a standard normal Z, E[Z⁴] = 3. This is known as the:
#49. If X and Y are independent with MGFs M_X(t) and M_Y(t), the MGF of X + Y is:
#50. The coefficient of variation is undefined when:
#51. For n independent uniform(0,1) random variables, the expected value of the maximum is:
#52. In a sample of 5 from a continuous distribution, what is the probability that the median exceeds the population median?
#53. The range of a sample is defined as max – min. For n = 2 from Uniform(0,1), what is E[Range]?
#54. For a sample of size n, the sampling distribution of the sample median is approximately normal for large n with variance:
#55. The sample minimum X₍₁₎ of n exponential(λ) variables is exponential with parameter:
#56. In two-way ANOVA with 3 rows, 4 columns, and 2 replications per cell, what are the error degrees of freedom?
#57. The total sum of squares in ANOVA equals 500. If SSB = 150 and SSW = 350, what is the variance explained by group differences?
#58. Tukey’s HSD test is used for:
#59. In one-way ANOVA, if F = 3.5 with df₁ = 3 and df₂ = 36, and F₀.₀₅,₃,₃₆ = 2.87, the conclusion is:
#60. The assumption of homoscedasticity in ANOVA means:
#61. The Mann-Whitney U test is the nonparametric alternative to:
#62. In a sign test with n = 20 observations above and below a hypothesized median, the test statistic follows a:
#63. Spearman’s rank correlation coefficient is preferred over Pearson’s when:
#64. The Kruskal-Wallis test is a nonparametric alternative to:
#65. A runs test is used to test for:
#66. The MLE of λ for a Poisson distribution based on observations x₁,…,xₙ is:
#67. An unbiased estimator has:
#68. The sample variance s² = Σ(xᵢ – x̄)²/(n-1) uses (n-1) in the denominator to:
#69. If θ̂ is a consistent estimator of θ, then as n → ∞:
#70. The Cramér-Rao lower bound gives:
#71. An X̄ chart has UCL = 52, CL = 50, LCL = 48. A point at 53 indicates:
#72. For a p-chart monitoring defective proportion, if p̄ = 0.05 and n = 100, the upper control limit (3σ) is approximately:
#73. The capability index Cpk = min[(USL-μ)/(3σ), (μ-LSL)/(3σ)]. If μ = 50, σ = 2, LSL = 44, USL = 56, what is Cpk?
#74. A process with Cp = 1.33 and Cpk = 0.67 is:
#75. The average run length (ARL) of a control chart when the process is in control should be:
#76. A time series with constant mean and variance over time exhibits:
#77. The autocorrelation at lag k measures:
#78. Exponential smoothing with α = 0.2 gives how much weight to the most recent observation?
#79. A 3-period moving average smooths a series by:
#80. In ARIMA(1,1,1), the middle ‘1’ indicates:
#81. If X and Y are jointly continuous with joint PDF f(x,y) = 6xy for 0 < x < 1, 0 < y < 1, what is P(X < Y)?
#82. The convolution of two independent uniform(0,1) random variables produces a:
#83. If X ~ Exp(2) and Y ~ Exp(3) are independent, P(X < Y) equals:
#84. For the birthday problem, the minimum number of people needed for P(at least 2 share birthday) > 0.5 is:
#85. The law of large numbers states that as n → ∞, the sample mean:
#86. In stratified sampling with proportional allocation from two strata (N₁ = 1000, N₂ = 500) with sample size 150, how many from stratum 1?
#87. Cluster sampling is most efficient when:
#88. The finite population correction factor is √[(N-n)/(N-1)]. For N = 1000 and n = 100, FPC equals:
#89. Systematic sampling with k = 10 means selecting:
#90. The design effect (DEFF) compares variance of a complex sample to:
#91. A residual plot showing a funnel shape (wider spread at higher fitted values) indicates:
#92. Cook’s distance measures:
#93. A VIF (Variance Inflation Factor) of 10 for a predictor suggests:
#94. The Durbin-Watson statistic tests for:
#95. Leverage hᵢᵢ measures how far observation i is from:
#96. In Bayesian inference, the posterior is proportional to:
#97. A bootstrap confidence interval is constructed by:
#98. The jackknife method estimates bias and variance by:
#99. Cross-validation is used primarily to:
#100. The Akaike Information Criterion (AIC) balances:
