Confidence Intervals

Point and Interval Estimates

• Point Estimate: A single number that estimates a population parameter.

• Confidence Interval: Provides a range of values, giving information about variability.

• Estimation Relationships:

  • Population Parameter: µ (population mean) or π (population proportion).

  • Sample Statistic: x̄ (sample mean) or p (sample proportion).

Confidence Intervals

• Uncertainty in Point Estimates:

  • Interval estimates convey more information about population characteristics than point estimates.

• Key Characteristics:

  • Accounts for variation in sample statistics from sample to sample.

  • Based on observations from one sample.

  • Stated with a confidence level (e.g. 95%, 99%).

  • Cannot achieve 100% confidence.

Example: Cereal Fill

• Population Parameters:

  • µ = 368

  • σ = 15

• Sample Size: n = 25

  • Confidence interval derived:

    • Formula: µ ± Z × σ_x̄ = 368 ± 1.96 × √(15/25) = (362.12, 373.88)

    • 95% intervals formed will contain µ.

  • If X̄ = 362.3:

    • Interval calculation: 362.3 ± 1.96 × √(15/25) = (356.42, 368.18)

    • Validity of interval confirmed as it contains µ.

General Formula for Confidence Intervals

• Formula: Point Estimate ± (Critical Value)(Standard Error)

  • Components:

    • Point Estimate: Sample statistic estimating the population parameter.

    • Critical Value: Table value based on the desired confidence level.

    • Standard Error: Standard deviation of the point estimate.

Confidence Level Interpretation

• Example with 95% confidence level:

  • Written as (1 - α) = 0.95; thus α = 0.05.

  • Interpretation: 95% of all confidence intervals will contain the true parameter; specific intervals will either contain or not contain the true parameter (no probability).

Confidence Interval for µ (σ Known)

• Assumptions:

  • Population standard deviation (σ) known.

  • Population is normally distributed or n > 30.

• Confidence Interval Estimate:

  • Formula: X̄ ± Zα/2(σ/√n).

Common Levels of Confidence

• Common Confidence Levels and Coefficients:

  • 80%: Z = 1.280

  • 90%: Z = 1.645

  • 95%: Z = 1.960

  • 99%: Z = 2.580

• Example: 95% confidence interval for population mean resistance:

  • Sample Mean (X̄) = 2.22 ohms, σ = 0.35 ohms.

  • Interval: 2.22 ± 1.96(0.35/√11) = (2.0132, 2.4268).

Confidence Interval for µ (σ Unknown)

• If σ is unknown, substitute sample standard deviation (S) using t distribution.

• Confidence Interval Estimate:

  • Formula: X̄ ± tα/2(S/√n).

  • Assumptions remain: Population is normally distributed, n > 30 if not normal.

Understanding Degrees of Freedom

• Defined as n-1 (observations free to vary).

Confidence Intervals for Population Proportion, π

• Uses sample proportion (p) with added uncertainty allowance:

  • Formula: p ± Zα/2√(p(1 - p)/n).

Example for Proportion

• Random sample of 100 reveals 25 left-handed individuals:

  • p = 0.25.

  • Confidence interval calculation: 0.25 ± 1.96√(0.25*0.75/100) = (0.1651, 0.3349).

Determining Sample Size

• Sampling Error defined as margin of error (e):

  • e = Zα/2(σ/√n).

• Rearrangement for sample size:

  • n = (Z²α/2 * σ²) / e².

• Example for sample size calculation:

  • σ = 45, desired error ± 5 with 90% confidence:

  • n = (1.645² * 45²) / 5² = 220.

Ethical Issues

• Reporting requirements:

  • Include confidence interval, level of confidence, sample size, and interpretation.