A fair die is rolled 90 times. Let X be the number of times a "6" appears. Calculate E(X) and Var(X).
02
보통주관식
Normal Distribution — Standardization
IQ scores follow N(100, 15²). What is P(85 < X < 130)? Express as P(a < Z < b) and find the z-values.
03
보통주관식
Hypothesis Testing — P-value Decision
A researcher claims a new drug reduces blood pressure by more than 10mmHg. A sample of n=64 gives x̄=12, s=8. Compute the test statistic z and state the conclusion at α=0.05.
04
심화주관식
Confidence Interval Calculation
A sample of n=100 from N(μ, 4) gives x̄=50. Construct a 95% confidence interval for μ. (z₀.₀₂₅ = 1.96)
05
심화주관식
Type I & Type II Error
In a hypothesis test with α=0.05, define Type I error and Type II error. If the power of the test is 0.80, what is β?
🔬 Unit 2
T-Test · Statistical Inference
t-분포, 소표본 검정, 두 집단 비교의 핵심 문제
🔬
핵심 개념 & 암기 공식
One-sample t-test: 모분산 미지, 소표본(n<30)
t = (x̄ - μ₀) / (s/√n) ～ t(n-1)
Two-sample t-test: 두 독립 집단 평균 비교
t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂) ～ t(df)
Paired t-test: 대응 표본 (전후 비교)
t = d̄ / (sᵈ/√n) where d̄ = mean of differences
자유도(df) = n-1 (단일표본), n₁+n₂-2 (독립 이표본)
n≥30이면 t분포 → 정규분포 근사 가능
등분산 가정: Pooled variance 사용
📝 예제
n=16, x̄=22, s=4, μ₀=20에서 t-통계량을 구하라.
t = (22-20)/(4/√16) = 2/1 = 2.0 (df=15)
06
기본주관식
One-Sample t-Test Statistic
A sample of n=25 students has mean score x̄=78 and s=10. Test H₀: μ=80 vs H₁: μ≠80. Compute t and degrees of freedom.
07
보통주관식
Paired t-Test — Before & After
8 patients' blood pressure before and after treatment: differences d = {-5,-3,-8,-2,-6,-4,-7,-5}. Compute d̄ and sᵈ (to 2 decimal places).
08
보통주관식
Two-Sample t-Test Setup
Group A: n₁=10, x̄₁=85, s₁=6. Group B: n₂=12, x̄₂=80, s₂=8. Assuming equal variance, compute pooled variance s²p and test statistic t.
09
심화주관식
t-Distribution vs Normal Distribution
Explain why the t-distribution has heavier tails than the standard normal distribution, and state when it is appropriate to use z instead of t. (키워드 3개 포함)
10
심화주관식
t-Test Conclusion & Effect Size
n=20, t=2.45, df=19. At α=0.05 (two-tailed, t_crit=2.093), state the conclusion. Also compute Cohen's d if x̄-μ₀=4.9 and s=10.
∫ Unit 3
Calculus · Differentiation & Integration
극한, 미분법, 정적분, 넓이 계산의 핵심 문제
∫
핵심 개념 & 암기 공식
도함수 정의 & 기본 공식
d/dx[xⁿ] = nxⁿ⁻¹ d/dx[eˣ] = eˣ d/dx[ln x] = 1/x
d/dx[sin x] = cos x d/dx[cos x] = -sin x
연쇄법칙 (Chain Rule)
d/dx[f(g(x))] = f'(g(x)) · g'(x)
정적분 & 넓이
∫ₐᵇ f(x)dx = [F(x)]ₐᵇ = F(b) - F(a)
두 곡선 사이 넓이 = ∫ₐᵇ |f(x) - g(x)| dx