Day 17

Math 216: Statistical Thinking

Bastola

Recap: Steel Sheets Thickness

  • Context: We revisit the scenario from Section 6.1 where a steel manufacturer’s rolling machine produces sheets of varying thickness, distributed uniformly between 150 and 200 millimeters.
  • Experiment Setup: Random sampling of 11 steel sheets to measure thickness, repeated across 1000 samples.

Example 6.1

Sampling distribution for \(\bar{x}\)

Uniform Distribution Parameters

  • Distribution Type: Uniform, with each value between 150 and 200 mm equally likely.
  • Calculated Mean: \(\mu = \frac{150 + 200}{2} = 175\) mm.
  • Calculated Standard Deviation: \(\sigma = \sqrt{\frac{(200 - 150)^2}{12}} \approx 14.43\) mm.

Sample Mean and Estimation

  • Sample Mean: \(\bar{x} = \frac{\sum x}{11}\)
  • Role of Sample Mean: Serves as a better statistic to approximate the population mean, especially useful when population parameters are unknown.
  • Unbiased Estimator: \(\bar{X} = \frac{1}{n} \sum X_i\) is unbiased, meaning \(E(\bar{x}) = \mu\).

Properties of the Sampling Distribution

  • Mean of Sampling Distribution: Equals the population mean, i.e., \(E(\bar{x}) = \mu\).
  • Standard Error of the Mean (SE): \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\) for sample size (n).
  • Implication: The standard error quantifies the spread of the sampling distribution around the population mean.

Central Limit Theorem (CLT)

  • Theorem Statement: For any population with mean \(\mu\) and standard deviation \(\sigma\), the sampling distribution of the sample mean \(\bar{x}\) will be approximately normal if the sample size \(n\) is sufficiently large (typically \(n \geq 30\)).
  • Application: This theorem applies regardless of the original population’s distribution, making it a powerful tool in statistical inference.

Impact of Sample Size

  • Visualization: Tells us how increasing sample size narrows the spread of the sampling distribution, aligning closer to a normal distribution.
  • Practical Implication: Larger sample sizes lead to more precise estimations of the population mean, emphasizing the importance of sample size in experimental design.

Normal Approx. for Poisson

  • Poisson Properties: For \(X \sim \text{Poisson}(\lambda)\):
    • Mean: \(\mu = \lambda\)
    • Variance: \(\sigma^2 = \lambda\)
    • Shape: Right-skewed for small \(\lambda\), but becomes approximately normal as \(\lambda\) grows (\(\lambda > 10\) is often sufficient).
  • Central Limit Theorem (CLT):
    For large sample sizes \(n\), the sampling distribution of \(\bar{X}\) is approximately normal:
    \[ \bar{X} \approx N\left(\mu_{\bar{x}} = \lambda,\ \sigma_{\bar{x}} = \sqrt{\frac{\lambda}{n}}\right) \]
    Even if the population is Poisson, the sample mean becomes normal when \(n\) is large.