Confidence Intervals Updated

Confidence Intervals Overview

  • Confidence intervals are important statistical tools used for estimating population parameters based on sample statistics.

Learning Objectives

  • Familiarity with point estimators.

  • Construction and interpretation of confidence intervals for means and proportions.

  • Determining necessary sample sizes for appropriate confidence interval estimates.

  • Applications of confidence intervals in auditing.

Estimation Concept

Objective of Estimation

  • Estimation aims to approximate a population parameter based on a sample statistic.

Definitions

  • Point Estimator: An estimator that provides a single value as an estimate of a population parameter from a random sample.

    • Example: Sample Mean (

(\bar{X}) = (\frac{\sum X}{n}))

  • Interval Estimator: An estimator that provides an interval which is likely to contain the true population parameter.

Notation

  • Generic notation (\theta_{hat}) will denote a point estimator of a population parameter (\theta).

    • Common parameters:

      • Population Mean: (\mu)

      • Population Standard Deviation: (\sigma)

      • Population Proportion: (p)

Properties of Estimators

Unbiased Estimator

  • An estimator (\theta_{hat}) is unbiased if its expected value equals the parameter being estimated:

    • (E(\theta_{hat}) = \theta)

    • Sample mean (\bar{X}) is unbiased for mean (\mu).

  • Sample variance (S^2) is unbiased for (\sigma^{2}).

Consistent Estimator

  • An estimator is consistent if it gets closer to the population parameter as the sample size increases.

  • The standard error tends to zero with increasing sample size.

Confidence Intervals

Construction of Confidence Intervals

  • We estimate a population mean by finding an interval (LCL, UCL) such that the true mean (\mu) falls within that interval with a certain level of confidence (1-(\alpha)).

  • LCL (Lower Confidence Limit) and UCL (Upper Confidence Limit) need to be determined:

    • (\text{Confidence level} = 1 - \alpha)

Finding the Limits

  1. Obtain the sample mean (\bar{x}).

  2. Calculate the standard error of the sampling distribution.

  3. Identify the critical z-score based on the confidence level:

    • Common z-scores: 1.645 (90%), 1.960 (95%), 2.576 (99%).

  4. Calculate LCL and UCL using:

    • (LCL = \bar{x} - z_{\alpha/2} \cdot \text{SE})

    • (UCL = \bar{x} + z_{\alpha/2} \cdot \text{SE})

Example - Assembly Line

  • A sample of 75 completion times leads to a 99% confidence interval for mean assembly time.

Interval Estimator When Population Standard Deviation is Unknown

  • Use the sample standard deviation and the Student's t-distribution.

  • Effective for small samples or when population standard deviation is unknown.

Student's t Distribution

  • Developed by William Gossett, it accounts for additional uncertainty in estimating the population mean when using a sample standard deviation.

  • The critical values depend on the sample size (degrees of freedom = n - 1).

Applications in Auditing

Practical Use

  • Auditors often utilize probability sampling and confidence intervals to make inferences about population parameters (e.g., voucher amounts).

Example - Vouchers

  • Using sample data for estimating population mean voucher amounts results in a confidence interval, helping auditors assess total expected value.

Ethical Considerations

  • Report confidence intervals alongside point estimates to reflect sampling uncertainty. Provide interpretations, the sample size, and confidence levels.

Additional Exercises

Practice Problems

  1. Exercise: Karaoke – Evaluate sample size adjustments based on confidence intervals for reported time spent on a site.

  2. Exercise: Sweet Snacks – Analyze given studies on per capita consumption with confidence intervals for further insights.