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5.2.2 Confidence Interval for Population Mean - 𝜎 Unknown

Introduction

  • Overview: Discusses the approach to estimating confidence intervals for the population mean when the population standard deviation (σ) is not known.
  • Context: This is a common scenario in practical statistics, where precise information about the population's variability is lacking.

Theoretical Background

  • Student T-Distribution: Unlike scenarios where σ is known and the normal distribution can be used, the unknown σ scenario necessitates using the sample standard deviation (s) and the t-distribution.
  • Rationale: The t-distribution is used because it accommodates the extra uncertainty introduced by estimating σ from the sample itself. This distribution becomes particularly relevant when sample sizes are small, providing a more accurate reflection of the uncertainty in the interval estimation.

Confidence Interval Calculation

  • Formula: The confidence interval for the population mean (μ) when σ is unknown is calculated as: [ \bar{x} \pm t_{\alpha/2, n-1} \left(\frac{s}{\sqrt{n}}\right) ] where:
  • \( \bar{x} \) is the sample mean.
  • \( t_{\alpha/2, n-1} \) is the critical value from the t-distribution for \( n-1 \) degrees of freedom.
  • \( s \) is the sample standard deviation.
  • \( n \) is the sample size.

Practical Application

  • Scenario Example: A business analyst might use this method to estimate the mean spending of customers when only a sample of transactions is available without prior knowledge of the overall spending variability.

Advantages of Using the T-Distribution

  • Flexibility: The t-distribution is more adaptable to situations with less information and smaller samples, making it ideal for preliminary studies or when high-cost data gathering is not feasible.
  • Behavior: As the sample size increases, the t-distribution approaches the normal distribution, illustrating its suitability for a wide range of sample sizes.

Conclusion

  • Summary: When σ is unknown, the t-distribution provides a robust method for constructing confidence intervals, ensuring that estimates remain reliable despite the lack of complete information about the population's variability.
  • Next Steps: Further modules will explore how these techniques can be applied to more complex statistical analyses and decision-making processes.
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