Chapter 8: Confidence Interval Estimation

Overview and Key Concepts

Confidence interval estimation is a fundamental statistical technique used to infer population parameters based on sample data. Unlike point estimates, confidence intervals account for sampling variability and provide a range within which the true parameter is likely to fall.

• Confidence Interval Estimates consider variation in sample statistics from sample to sample.

• They provide information about the closeness to unknown population parameters.

• Intervals are always stated with a level of confidence (e.g., 95%), which is less than 100%.

Why We Need Confidence Interval Estimates

• Point Estimates (e.g., sample mean) do not account for sampling variability.

• Confidence Intervals provide a range, reflecting uncertainty and allowing for more informed decisions.

• They quantify the probability that the interval contains the true population parameter.

Confidence Interval Estimate for the Mean (Population Variance Known)

When the population variance is known, the confidence interval for the mean is constructed using the normal distribution.

• Assumptions:

  • Population variance is known.

  • Population is normally distributed or sample size is large (Central Limit Theorem applies).

• Point Estimate: Sample mean .

• Confidence Interval Formula:

• Where:

  • is the critical value from the standard normal distribution corresponding to the desired confidence level.

  • is the sample size.

Elements of Confidence Interval Estimate

• Level of Confidence: Probability that the interval contains the true parameter (e.g., 95%).

• Precision (Range): The width of the interval; narrower intervals indicate greater precision.

• Cost: Resources required to obtain a sample of size .

Factors Affecting Interval Width

• Data Variation (): Greater variance leads to wider intervals.

• Sample Size (): Larger sample sizes yield narrower intervals.

• Level of Confidence: Higher confidence levels (e.g., 99%) produce wider intervals.

Interpretation of Confidence Interval

• If all possible samples of size are taken, % of the intervals will contain the true population mean.

• For example, a 95% confidence interval means that 95% of such intervals would contain the true mean.

Confidence Interval Estimate for the Mean (Population Variance Unknown)

When the population variance is unknown, the sample standard deviation is used, and the t-distribution is applied.

• Assumptions:

  • Population variance is unknown.

  • Population is normally distributed or sample size is large.

• Confidence Interval Formula:

• Where:

  • is the critical value from the Student's t-distribution with degrees of freedom.

  • is the sample standard deviation.

Confidence Interval Estimate for the Proportion

Confidence intervals for proportions are used when the variable of interest is categorical (e.g., success/failure).

• Assumptions:

  • Two categorical outcomes (e.g., yes/no).

  • Population follows a Binomial distribution.

  • Normal approximation is valid if and .

• Point Estimate: , where is the number of successes.

• Confidence Interval Formula:

Determining Sample Size

Sample size determination is crucial for achieving desired precision in confidence interval estimation.

• For Estimating the Mean:

  • Where is the acceptable sampling error, and is estimated from past data, an educated guess, or a pilot study.

• For Estimating the Proportion:

  • Where is the acceptable sampling error, and is estimated from prior data or a pilot study.

Example: Constructing a Confidence Interval for the Mean

• Suppose a sample of yields a mean and known population standard deviation .

• For a 95% confidence level, .

• Confidence interval:

• Calculation:

• Interval: (95.84, 104.16)

Example: Determining Sample Size for Proportion

• Suppose desired margin of error , estimated proportion , and 95% confidence ().

• Sample size:

• Calculation:

Summary Table: Confidence Interval Formulas

Additional info: Some formulas and explanations were expanded for clarity and completeness, including explicit LaTeX formatting and example calculations.