Confidence Interval Calculations

Confidence intervals help estimate population parameters based on sample data. They provide a range of values likely containing the true parameter, which is crucial in data science, biostatistics, and decision-making across various fields. Understanding these calculations enhances data interpretation and inference.

1. Confidence interval for population mean (known population standard deviation)

   • Uses the Z-distribution to calculate the interval.
   • Formula: (\bar{x} \pm Z_{\alpha/2} \left(\frac{\sigma}{\sqrt{n}}\right)), where (\bar{x}) is the sample mean, (\sigma) is the known population standard deviation, and (n) is the sample size.
   • Provides a range of values that likely contains the true population mean.
   • The width of the interval decreases with larger sample sizes.

2. Confidence interval for population mean (unknown population standard deviation)

   • Utilizes the t-distribution due to the unknown standard deviation.
   • Formula: (\bar{x} \pm t_{\alpha/2} \left(\frac{s}{\sqrt{n}}\right)), where (s) is the sample standard deviation.
   • The t-distribution accounts for additional uncertainty, especially with smaller sample sizes.
   • As sample size increases, the t-distribution approaches the normal distribution.

3. Confidence interval for population proportion

   • Based on the normal approximation of the binomial distribution.
   • Formula: (\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}), where (\hat{p}) is the sample proportion.
   • Assumes a sufficiently large sample size for the normal approximation to be valid.
   • Useful for estimating the proportion of a characteristic in a population.

4. Confidence interval for difference between two population means

   • Compares means from two independent samples.
   • Formula: ((\bar{x}_1 - \bar{x}2) \pm Z{\alpha/2} \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}) for known variances, or use t-distribution for unknown variances.
   • Helps determine if there is a statistically significant difference between the two means.
   • Assumes independent samples and normality of the distributions.

5. Confidence interval for difference between two population proportions

   • Compares proportions from two independent samples.
   • Formula: ((\hat{p}_1 - \hat{p}2) \pm Z{\alpha/2} \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}).
   • Useful for assessing differences in proportions across two groups.
   • Requires large sample sizes for the normal approximation to be valid.

6. Confidence interval for population variance

   • Based on the chi-squared distribution.
   • Formula: (\left(\frac{(n-1)s^2}{\chi^2_{\alpha/2, n-1}}, \frac{(n-1)s^2}{\chi^2_{1-\alpha/2, n-1}}\right)), where (s^2) is the sample variance.
   • Provides a range for the true population variance.
   • Assumes the underlying data is normally distributed.

7. Confidence interval for ratio of two population variances (F-distribution)

   • Compares variances from two independent samples.
   • Formula: (\left(\frac{s_1^2}{s_2^2} \cdot \frac{1}{F_{1-\alpha/2, n_1-1, n_2-1}}, \frac{s_1^2}{s_2^2} \cdot F_{\alpha/2, n_1-1, n_2-1}\right)).
   • Useful for testing the equality of variances between two groups.
   • Assumes both samples are independent and normally distributed.

8. Confidence interval for correlation coefficient

   • Assesses the strength and direction of a linear relationship between two variables.
   • Uses Fisher's z-transformation for the interval calculation.
   • Formula: (z' \pm Z_{\alpha/2} \cdot \frac{1}{\sqrt{n-3}}), where (z' = \frac{1}{2} \ln\left(\frac{1+r}{1-r}\right)).
   • Transforms the correlation coefficient to stabilize variance and improve interval estimation.

9. Confidence interval for regression coefficients

   • Evaluates the uncertainty around estimated coefficients in a regression model.
   • Formula: (\hat{\beta} \pm t_{\alpha/2, n-k} \cdot SE(\hat{\beta})), where (SE(\hat{\beta})) is the standard error of the coefficient.
   • Helps determine the significance of predictors in the model.
   • Assumes linearity, independence, and normally distributed errors.

10. Bootstrap confidence intervals

    • Non-parametric method that resamples the data to estimate the sampling distribution.
    • Involves repeatedly drawing samples with replacement and calculating the statistic of interest.
    • Provides a way to construct confidence intervals without relying on normality assumptions.
    • Useful for small sample sizes or complex estimators where traditional methods may not apply.