CourseKata Chapter 12

Confidence Interval (1)

Mansour Abdoli, PhD

Overview / Goals

  • Sampling Dist. & Test of Hypothesis
  • Plausible Parameters Values
  • Confidence Interval
  • CI & t-Distribution
  • Assumptions and Interpretation

Sampling Dist. & Test of Hypothesis

Sampling & Null Distributions

  • Fixed Population \(\to\) Fixed Parameter
  • Random Sample \(\to\) Random Statistic
    • Sampling Distribution:
      The distribution of the random statistic
    • Null Distribution:
      The sampling distribution for a given parameter

Sampling & Null Distributions

  • DGP with \(\beta_0=\mu=0\)
  • DGP with \(\beta_0=\mu=.5\)

Estimating Null Dist.

Given Sample \(\{0, 1, 5, 1, 3\}\)

  • DGP with Sample Mean
  • DGP with Claimed Mean

Plausible Parameters Values

  • Is \(\mu=0\) plausible?
    • Run a test on \(H_0: \mu=0\):
      \(\mu=0\) is plausible if we fail to reject.
  • process:
    • Define a Sataitic
    • Find Null Distribution
    • Reject \(H_0\) for extreme observed statistics.

Confidence Interval

Interval Estimate vs. Point Estimate

  • \(\bar y\) is a point estimate (Best Guess) for \(\mu\)
  • An interval estimate is all plausible values at \(\alpha\).

Distanced Based C.I.

  • For symmetric sampling distributions: \[ C.I. = \text{Point Estimate} \pm \text{Margin of Error}\]

CI & t-Distribution

Theoretical Sampling Distribution

  • When
    • \(Y\sim N(\mu,\sigma)\), or
    • \(n\) is large
  • \(t=\frac{\bar Y-\mu}{SE}\sim t(\text{df}=n-1)\)
    • \(SE=\frac{s}{\sqrt{n}}\)
  • For \(100(1-\alpha)\)% Confidence Level: \[ME=t^* \cdot SE\] \(t^*=t_{\alpha/2}\): top \(\alpha/2\) cut-off point

Assumptions and Interpretation

Assumptions

  • Independent Observations
  • For Theoretical C.I.
  • Is \(Y\) normal?
  • Is \(n\) large?
    • \(n>=30\) without outliers
    • Check with/use bootstrap

Interpretation

  • We are \(100(1-\alpha)\)% confident that the mean is in \((\bar y - ME, \bar y + ME)\).

Caution

  • C.I is for the population mean, not observations.
  • C.I is a random variable, the populaiton mean is fixed.

  2.5%  97.5% 
17.075 63.650 
    2.5%    97.5% 
26.90852 33.50057