Ch. 8 = Confidence Intervals

Introduction to Confidence Intervals

Confidence intervals are a fundamental concept in inferential statistics, providing a range of values within which a population parameter is expected to lie with a specified probability. They are widely used in business statistics to estimate population means and proportions based on sample data.

• Point Estimate: A single value (e.g., sample mean) used as the best estimate of an unknown population parameter.

• Interval Estimate: A range of values, derived from sample statistics, that is likely to contain the population parameter.

• Confidence Level: The probability (expressed as a percentage, such as 90%, 95%, or 99%) that the confidence interval contains the true population parameter.

• Margin of Error (ME): The amount added and subtracted from the point estimate to form the confidence interval.

Example: If a sample mean is 100 and the margin of error is 5, a 95% confidence interval is (95, 105).

Theories Underlying Confidence Intervals

• Empirical Rule: For a normal distribution:

  • Approximately 68% of values fall within ±1 standard deviation from the mean.

  • Approximately 95% within ±2 standard deviations.

  • Approximately 99.7% within ±3 standard deviations.

• Central Limit Theorem (CLT): For large sample sizes (n ≥ 30), the sampling distribution of the sample mean is approximately normal, regardless of the population's distribution.

Critical Values and Confidence Levels

The critical value (zα/2 or tα/2) determines the width of the confidence interval for a given confidence level. Common z-scores for confidence levels are:

Calculating Confidence Intervals for the Population Mean

When Population Standard Deviation (σ) is Known

Use the z-distribution when σ is known and either n ≥ 30 or the population is normally distributed.

• Formula:

• Where:

  • = sample mean

  • = population standard deviation

  • = sample size

  • = critical z-score for the desired confidence level

Example: A sample of 35 flights has a mean of 145.5 and σ = 30.2. For a 90% confidence interval:

• Standard error:

• Critical value:

• Interval:

Finite Population Correction

When the sample size is a significant fraction of the population (n/N > 0.05), apply the finite population correction (FPC):

The confidence interval formulas become:

When Population Standard Deviation (σ) is Unknown

Use the t-distribution when σ is unknown and the population is normally distributed. The sample standard deviation (s) is used in place of σ.

• Formula:

• Where:

  • = critical t-score with degrees of freedom

  • = sample standard deviation

Example: A sample of 15 patients has a mean of 5.11 days and s = 0.85. For a 95% confidence interval (df = 14, ):

• Standard error:

• Interval:

Finite Population Correction for t-Distribution

When using the t-distribution and the sample is a significant fraction of the population, apply the FPC:

Calculating Confidence Intervals for Proportions

Confidence Interval for a Population Proportion

Proportion data often follow the binomial distribution, which can be approximated by the normal distribution if and .

• Formula:

• Where:

  • = sample proportion ()

  • = sample size

  • = critical z-score

Example: In a sample of 100 people, 22 have blue eyes. For a 98% confidence interval ():

• Standard error:

• Interval:

Finite Population Correction for Proportions

When the sample is a significant fraction of the population, apply the FPC to the standard error:

The confidence interval formulas become:

Summary Table: Types of Confidence Intervals

Key Takeaways

• Confidence intervals provide a range of plausible values for population parameters.

• The width of the interval depends on the confidence level, sample size, and variability in the data.

• Finite population correction should be applied when the sample is a significant fraction of the population.

• Use the z-distribution when σ is known, and the t-distribution when σ is unknown and the population is normal.

• For proportions, ensure the sample size is large enough for the normal approximation to be valid.