Normal Distributions

목차

1. Normal Distribution

• has 2 parameters (=number describing whole population): mean (mu; µ) and varaince (sigma squared; σ^2)
  • they are constants (same value for all observations in the population
  • X ~ N (µ, σ^2)
  • probability of getting a single point X is 0; it is area!
• Standard Normal Random Variable
  • variable X is transformed to "Z score"
  • mean= 0 and variance = 1
  • Z ~ N (0,1)
• X (RV) → Z → probability 
  • z= (X-µ) / σ 
• probability → Z → X (RV) 
  • X = (z * σ)  + µ

2. Normal Approximation to the Bionomial (NAB)

• when n is large, we can approximate the bionomial probabilities to normal distribution
• De Moivre's Theorem
  • µ = np, σ^2 = npq
    • (q = 1-p)
  • n is large and p is close to .5; when np ≥5 AND nq ≥5 
  • Y ~ N(µ= np, σ^2= npq)

3. Chi-Square Distribution (χ2)

• it is for categorical data, continuous random variable, not symmetric
• one-parameter distribution: (nu; v) degree of freedom
  • v = n-1
• mean = E(χ2) = v
• varaince = Var(χ2)=2v
• three applications of X^2 (Chi-Square)
  • 1) Testing hypotheses about the value of the variance
    • for variance from the sample data, use s^2 (estimate of σ^2)
    • a. set null and alternative hypotheses
    • b. choose α level (if α=0.05 then α/2=0.025)
    • c. set up rejection regions using chi-square table, find critical values of χ2
      • Critical upper and lower value
    • d. calculate value of test statistics
      • χ2 observed value = [(n-1) * s^2] / σ^2
    • e. compare test statistics to critical value
  • 2) Chi-Square Goodness of Fit Test
  • 3) Chi-Square Test of Independence (of 2 categorical RV's)
    
    • test of "homogeneity" (= equality) of two proportions

4. Sampling and Estimation

• Why Xbar (sample statistics) better than the median?
  • 1) unbiased: estimator is parameter 
  • 2) consistent: as n gets to infinity, the value of the estimator gets closer to the true parameter value (Law of Large Numbers)
  • 3) Efficient
• Mean: µ
• Variance: σ^2/n
• Standard Error of the Mean (standard deviation): σ/Sqrt(n)

5. Confidence Interval

• When we have large sample (Central Limit Theorem - distribution Xbar will be approximately normal)
• When µ and σ is known (then use z distribution) (Law of Large Numbers- when n is large we can assume that σ^2 is known)
• CI: (Xbar - (Zα/2)*σxbar, Xbar + (Zα/2)*σxbar)
  • σxbar= σ/Sqrt(n)

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