Key Concepts: Confidence Intervals, t-distribution, and Outliers

Confidence Interval and t-distribution

• Confidence interval expresses the range around a calculated mean that is believed to contain the true mean μ, with a specified probability.
• For a given confidence level (e.g., 95%), use the t-distribution with degrees of freedom df = n − 1.
• Formula (two-sided):
  $\mu \in \bar{x} \pm t_{\alpha/2,\,n-1}\ \frac{s}{\sqrt{n}}.$
• Example (n = 5, \bar{x} = 0.6820 mg/dL, s = 0.00935 mg/dL, t = 2.776):
  $0.6820 \pm \frac{2.776 \times 0.00935}{\sqrt{5}} = 0.6820 \pm 0.0116,$ yielding (0.6704, 0.6936) mg/dL.
• Interpretation: 95% confidence that the interval contains the true mean μ, under the assumption of random sampling and normal distribution.

Mean and Standard Deviation; Confidence Interval Calculation

• Mean:
  $\bar{x} = \frac{1}{n} \sum<em>{i=1}^n x</em>i.$
• Standard deviation (sample):
  $s = \sqrt{\frac{1}{n-1}\ \sum<em>{i=1}^n (x</em>i-\bar{x})^2}.$
• 95% confidence interval for the mean uses the same formula as above with the appropriate t-value for df = n − 1.
• Think: Larger n reduces the margin of error via the factor ( \frac{s}{\sqrt{n}} ).

Grubbs\' Test for Outliers

• Purpose: identify a single outlier in a small data set.
• Statistic:
  $Z = \frac{|x_i - \bar{x}|}{s}.$
• Decision: compare Z to critical values from Table 1.10 for the given n and confidence level (e.g., 95%, 99%).
• If $Z \ge Z_{crit}$, the data point is considered an outlier and may be discarded for analysis.
• Note: Grubbs\' test identifies only one outlier at a time; data context matters (e.g., measurement differences due to material properties, not error).
• Example takeaway: In a set of pennies with one heavier coin, a computed Z may be below the 95% critical value and thus not an outlier at that level; different n or confidence levels may yield different conclusions.

Normal Distribution and Outliers; Ethical Considerations

• Random variability in data is modeled by a normal (bell-shaped) distribution; roughly 68% of data lie within ±1 standard deviation of the mean.
• Outliers are values far from the majority; discarding them without valid justification is unethical.
• Practical note: Use objective tests (like Grubbs') and report the rationale for handling outliers.

Quick Reference: Core Equations

• Mean:
  $\bar{x} = \frac{1}{n}\sum x_i$
• Standard deviation:
  $s = \sqrt{\frac{1}{n-1}\sum (x_i-\bar{x})^2}$
• Confidence interval for the mean (95% example):
  $\mu \in \bar{x} \pm t_{0.025,\,n-1}\ \frac{s}{\sqrt{n}}$
• Grubbs\' Test statistic:
  $Z = \frac{|x_i-\bar{x}|}{s}$
• Critical values for Grubbs\' test depend on n and the desired confidence level and are found in reference tables (e.g., Table 1.10).