Lecture 8(1)

Chapter 8: Interval Estimation

8.1 - Population mean, σ known

• Point Estimates: Rarely give exact population parameter.

• Interval Estimators: Provide an interval within which the true population parameter lies with specified confidence.

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

8.2 - Population mean, σ unknown

• General Form: Interval estimate of a population mean is given by ( \bar{x} \pm margin \ of \ error ).

• Margin of Error Calculation:

  • Uses population standard deviation σ or sample standard deviation s.

  • σ is rarely known precisely, but can be estimated.

• Interval Formula:

  • For σ known: ( \bar{x} \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}} )

  • For σ unknown: Uses t-distribution rather than standard normal distribution.

Key Relationships

• Confidence Level (%): Determines α, 1-α, zα/2, and margin of error.

• Probability for Interval Estimate: The likelihood that sample mean falls within the defined margin of error.

Calculating Margin of Error

• Margin of Error Equation: For known σ: ( E = z_{\alpha/2} \frac{\sigma}{\sqrt{n}} )

• For unknown σ, replace with sample standard deviation s and incorporate t distribution: ( E = t_{\alpha/2} \frac{s}{\sqrt{n}} )

8.3 - Determining the Sample Size

• Desired Margin of Error (E): Affecting sample size for achieving specified confidence.

• Calculating Sample Size:

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

  • Sample size typically minimum of 30, or higher if distribution is skewed.

8.4 - Population Proportion

• Interval Estimate for Proportion: ( \hat{p} \pm margin \ of \ error )

• Conditions: Must satisfy ( n\hat{p} \geq 5 ) and ( n(1-\hat{p}) \geq 5 ) for valid usage of normal approximation.

• Sample Size for Proportion:

  • ( n = \frac{(z_{\alpha/2})^2 \hat{p}(1-\hat{p})}{E^2} )

  • Planning value of ( \hat{p} ) can be estimated using previous data or set at 0.5.

Examples

• Discount Sounds: Example illustrating calculation of confidence interval given population mean and standard deviations — margin of error computation provides final interval estimate capturing population mean.

• Apartment Rents: Describes calculating 95% confidence interval estimation using sample data with unknown σ, follows with t-statistics.

• Crop Yield: Further demonstrates calculation of interval using sample size and standard deviations for confidence estimation.

Chapter Recap

• Interval estimation provides mechanisms to infer population parameters based on sample statistics, relying on Central Limit Theorem for distribution approximation.

• Importance of random sampling and implications on bias and accuracy of estimates highlighted.