7.2 Confidence Intervals for Means

Confidence intervals help estimate population means using sample data. They provide a range of likely values for the true population average, accounting for sampling variability and uncertainty.

For large samples or known population standard deviations, we use z-distributions. With small samples or unknown standard deviations, t-distributions are applied. Understanding sample size requirements and interpreting results are crucial for accurate business insights.

Confidence Intervals for Population Means

Confidence intervals with z-distribution

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• Used when sample size is large ($n \geq 30$) or population is normally distributed and population standard deviation ($\sigma$) is known
• Confidence interval formula: $\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$
  • $\bar{x}$ represents sample mean
  • $z_{\alpha/2}$ represents critical value from standard normal distribution
  • $\alpha$ represents significance level and $1 - \alpha$ represents confidence level (95%, 99%)
  • $n$ represents sample size
• Example: Estimating average customer spending (\sigma = \20$,$n = 100$,$\bar{x} = $50$, 95% confidence level)

Confidence intervals with t-distribution

• Used when sample size is small ($n < 30$), population is normally distributed, and population standard deviation is unknown
• Sample standard deviation ($s$) used as estimate for population standard deviation
• Confidence interval formula: $\bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}$
  • $t_{\alpha/2, n-1}$ represents critical value from t-distribution with $n-1$ degrees of freedom
• t-distribution has heavier tails compared to standard normal distribution accounting for additional uncertainty when using sample standard deviation
• Example: Estimating average employee satisfaction score ($n = 25$, $\bar{x} = 3.8$, $s = 0.6$, 90% confidence level)

Sample size for confidence intervals

• Margin of error ($E$) represents maximum acceptable difference between sample mean and population mean
• Sample size formula for known population standard deviation: $n = (\frac{z_{\alpha/2} \cdot \sigma}{E})^2$
• Sample size formula for unknown population standard deviation: $n = (\frac{t_{\alpha/2, n-1} \cdot s}{E})^2$
  • Iterative process or software often used to solve for $n$ since sample size appears on both sides of equation
• Example: Determining sample size for estimating average customer wait time ($E = 2$ minutes, 95% confidence level, $\sigma = 10$ minutes)

Interpretation of confidence intervals

• Provides range of plausible values for population mean
• Confidence level (95%, 99%) represents long-run probability that interval will contain true population mean
• Business applications:
  • Estimating average sales, revenue, or customer satisfaction score for product or service
  • Comparing mean performance of different business units or strategies
  • Determining if process change has resulted in significant improvement in key metric
• Consider width of confidence interval and practical significance of results when making decisions based on interval estimate
• Example: Comparing average sales between two store locations (\bar{x}_1 = \1000$,$\bar{x}_2 = $1200$, 95$$50$to$$350$)

Additional Considerations

Assumptions and limitations

• Assumes sample is randomly selected from population
• Assumes population is normally distributed or sample size is large enough for Central Limit Theorem to apply
• Violations of assumptions may lead to inaccurate confidence intervals
• Provides estimate of population mean but does not prove causality between variables
• Example: Non-random sampling may result in biased estimate of population mean