KA

Lecture_8_1_

Chapter 8: Interval Estimation

8.1: Population Mean, σ Known

  • Introduction to Interval Estimation

    • Point estimates usually do not provide exact values of population parameters.

    • Interval estimators provide an interval where true parameters likely lie, with a specified degree of confidence.

    • Formula: Interval estimate = Point estimate ± Margin of error.

Interval Estimation of Population Mean:

  • Form:

    • Interval estimate of mean: ( \bar{x} \pm \text{margin of error} ).

    • Margin of error is calculated using:

      • Population standard deviation (σ)

      • Sample standard deviation (s).

    • Note: σ is often not known but can be estimated.

Confidence Levels and Margins of Error

  • 1 − α is the probability that a sample mean will provide a specific margin of error of z({\alpha/2})σ({\bar{x}}).

  • Fixed confidence level determines α, 1 − α, z(_{\alpha/2}), influencing the margin of error.

  • Formula for interval estimate:

    • ( \bar{x} \pm z_{\alpha/2} \left( \frac{\sigma}{\sqrt{n}} \right) )

8.2: Population Mean, σ Unknown

  • When σ is unknown, we use sample standard deviation (s).

  • This introduces the application of Student's t-distribution, which is relevant for smaller sample sizes.

  • The degrees of freedom (df) for t-distribution equals n − 1.

  • Larger n leads to t-distribution approximating normal distribution.

  • Interval Estimation Formula when σ is Unknown:

    • ( \bar{x} \pm t_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right) )

8.3: Determining the Sample Size

  • Margin of Error (E):

    • E is defined as the upper and lower bounds of the interval estimate.

    • Formulae:

      • ( E = z_{\alpha/2} \frac{\sigma}{\sqrt{n}} )

      • Necessary sample size:

        • ( n = \left(\frac{z_{\alpha/2} \sigma}{E}\right)^{2} )

8.4: Population Proportion

  • Interval Estimate of Population Proportion:

    • General form: ( \bar{p} \pm \text{margin of error} ).

    • Sampling distribution approximated as normal given conditions np ≥ 5 and n(1 − p) ≥ 5 are met.

  • Interval Estimate Formula for Proportion:

    • ( \bar{p} \pm z_{\alpha/2} \sqrt{\frac{\bar{p}(1 - \bar{p})}{n}} )

8.5: Examples and Problem Solving

Example: Discount Sounds

  • Sample size n = 36, mean income = $41,100, σ = $4,500.

  • At 95% confidence, the margin of error is calculated leading to the interval estimate of mean income.

Example: Apartment Rents

  • Sample of 16 one-bedroom apartments, mean rent = $750, s = $55, 95% confidence.

  • Derived confidence interval using t-distribution due to unknown σ.

Chapter Recap

  • Key takeaway: Interval estimation allows assessment of how close sample estimates (mean and proportion) are to their true population parameters using predetermined levels of confidence based on the Central Limit Theorem.

  • Ensure random sampling to avoid bias in estimates and confidence intervals.