Interval Estimation Notes

Module 06: Interval Estimation

Overview of Interval Estimation

  • Interval Estimation for Population Mean:
    • When population standard deviation (B3) is known
    • When population standard deviation (B3) is unknown
    • Determining sample size for a desired precision

Point Estimators and Sampling Distribution

  • Point Estimator: A sample statistic that estimates a population parameter.
    • Example: Sample mean (x̄) estimates population mean (μ).
  • Sample Mean (x̄): A random variable with its own probability distribution, known as the sampling distribution of the sample mean.
    • Mean of sampling distribution = μ, standard deviation (standard error) = σ/√n
    • Shape: Normal distribution if n ≥ 30 (Central Limit Theorem - CLT)
  • Unbiased Estimator: x̄ is unbiased for μ if E(x̄) = μ.

Interval Estimates

  • An interval estimate provides information about how close the point estimate is to the actual population parameter.
  • General Form: Point Estimate ± Margin of Error
    • Calculate margin of error using either known or sample standard deviation.

Case 1: Population Standard Deviation Known

  • Formula: μ = x̄ ± z(σ/√n)
    • z = z-value for the confidence level
    • Example: Kohl's customer spending scenario
    • n = 100, σ = $20, x̄ = $82
    • Margin of Error: ± 1.96(20/√100) = ±3.92
    • 95% Confidence Interval: [78.08, 85.92]

Case 2: Population Standard Deviation Unknown

  • Use sample standard deviation (s) to estimate population standard deviation.
  • Formula: μ = x̄ ± t(s/√n)
    • t-value aligns with the confidence level and degrees of freedom (n-1)
    • Larger sample size yields a closer match to the normal distribution.

Confidence Intervals

  • A 95% CI means we are 95% confident the interval includes the true population mean.
  • For a 90% CI, use z = 1.645; for a 99% CI, use z = 2.576.
  • Adjustments in confidence levels affect the margin of error: larger confidence levels increase margin of error.
    • Example from Kohl's:
    • 90% CI: ±3.29, resulting in [78.71, 85.29]
    • 99% CI: ±5.15, resulting in [76.85, 87.15]

Sample Size Determination

  • Formula when B3 is known: n = (z * σ / E)^2
  • If σ is unknown: Use preliminary sample or best estimate.
    • Example: For a desired margin of error of $5 at 95% confidence for Kohl's:
    • n = (1.96 * 20 / 5)^2 ≈ 62
  • Example for Credit Card Debts study:
    • Given a margin of error of $500:
    • n = (1.96 * 4007 / 500)^2 ≈ 247

Summary of Key Formulas

  • Known Population Standard Deviation: μ = x̄ ± z(σ/√n)
  • Unknown Population Standard Deviation: μ = x̄ ± t(s/√n)
  • Sample Size for Desired Margin of Error: n = (z * σ / E)²