11.4 Student's t and chi-square distributions

The Student's t-distribution is crucial for analyzing small samples when population standard deviation is unknown. It's used to construct confidence intervals and perform hypothesis tests, adapting to situations where the normal distribution falls short.

Chi-square distribution is key for goodness-of-fit tests and variance estimation. It helps assess if data fits an expected distribution and creates confidence intervals for population variance, making it vital for various statistical analyses.

Student's t-distribution

Origin of Student's t-distribution

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• Developed by William Sealy Gosset, published under the pseudonym "Student"
• Used when sample size is small ($n < 30$) and population standard deviation is unknown
• Similar to standard normal distribution but with heavier tails, especially for smaller degrees of freedom ($df$)
• Approaches standard normal distribution as $df$ increases
• Characterized by a single parameter: $df = n - 1$, where $n$ is the sample size

Confidence intervals for small samples

• Construct confidence intervals for population mean when sample size is small and population standard deviation is unknown
• Confidence interval formula: $\bar{x} \pm t_{\alpha/2, df} \cdot \frac{s}{\sqrt{n}}$
  • $\bar{x}$: sample mean
  • $t_{\alpha/2, df}$: critical value from t-distribution with $df$ degrees of freedom and confidence level $1 - \alpha$
  • $s$: sample standard deviation
  • $n$: sample size
• Steps to construct confidence interval:
  1. Calculate sample mean ($\bar{x}$) and sample standard deviation ($s$)
  2. Determine desired confidence level ($1 - \alpha$) and $df = n - 1$
  3. Find critical value ($t_{\alpha/2, df}$) from t-distribution table or statistical software
  4. Substitute values into confidence interval formula and calculate lower and upper bounds

Chi-square distribution

Chi-square distribution fundamentals

• Continuous probability distribution arising from sum of squares of independent standard normal random variables
• Characterized by a single parameter: degrees of freedom ($df$), the number of independent standard normal random variables being summed
• Skewed to the right, with skewness decreasing as $df$ increases
• Non-negative, with range from 0 to infinity
• Approaches normal distribution as $df$ increases

Applications of chi-square distribution

• Goodness-of-fit tests:
  • Determine if sample data comes from a population with a specific distribution
  • Compare observed frequencies with expected frequencies under assumed distribution
  • Steps to perform chi-square goodness-of-fit test:
    1. State null and alternative hypotheses
    2. Calculate expected frequencies for each category under assumed distribution
    3. Calculate chi-square test statistic: $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$, where $O_i$ and $E_i$ are observed and expected frequencies for category $i$
    4. Determine $df = k - 1 - m$, where $k$ is number of categories and $m$ is number of parameters estimated from data
    5. Find critical value from chi-square distribution table or statistical software
    6. Compare test statistic to critical value and make decision to reject or fail to reject null hypothesis
• Variance estimation:
  • Construct confidence intervals for population variance using chi-square distribution
  • Confidence interval formula: $\frac{(n-1)s^2}{\chi^2_{\alpha/2, df}} \leq \sigma^2 \leq \frac{(n-1)s^2}{\chi^2_{1-\alpha/2, df}}$
    • $n$: sample size
    • $s^2$: sample variance
    • $\chi^2_{\alpha/2, df}$ and $\chi^2_{1-\alpha/2, df}$: critical values from chi-square distribution with $df = n - 1$ and confidence level $1 - \alpha$
    • $\sigma^2$: population variance