Confidence Interval Formulas

Confidence intervals are essential for estimating population parameters based on sample data. They provide a range of values where the true parameter likely falls, helping in decision-making across various fields like business, science, and social research.

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) ).
   • Requires knowledge of the population standard deviation ((\sigma)).
   • Provides a range where the true population mean is likely to fall with a specified confidence level.

2. Confidence Interval for Population Mean (unknown population standard deviation)

   • Uses the t-distribution due to the unknown standard deviation.
   • Formula: ( \bar{x} \pm t_{\alpha/2, df} \left( \frac{s}{\sqrt{n}} \right) ).
   • Requires sample standard deviation ((s)) and degrees of freedom ((df = n - 1)).
   • More conservative than the Z-interval, especially for small sample sizes.

3. Confidence Interval for Population Proportion

   • Uses the normal approximation for the sampling distribution of proportions.
   • Formula: ( \hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} ).
   • Requires a sufficiently large sample size to ensure normality (np and n(1-p) both > 5).
   • Provides a range for the true population proportion with a specified confidence level.

4. Confidence Interval for Difference Between Two Population Means (independent samples)

   • Compares means from two independent groups.
   • Formula: ( (\bar{x}_1 - \bar{x}2) \pm Z{\alpha/2} \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} ) (known (\sigma)) or t-distribution if unknown.
   • Assumes independent samples and normality of the sampling distribution.
   • Useful for hypothesis testing and comparing group differences.

5. Confidence Interval for Difference Between Two Population Proportions

   • Compares proportions from two independent groups.
   • 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}} ).
   • Requires both sample sizes to be large enough for normal approximation.
   • Helps assess the difference in proportions between two groups.

6. Confidence Interval for Paired Differences

   • Used for dependent samples (e.g., before-and-after studies).
   • Formula: ( \bar{d} \pm t_{\alpha/2, df} \left( \frac{s_d}{\sqrt{n}} \right) ), where (\bar{d}) is the mean of the differences.
   • Assumes the differences are normally distributed.
   • Focuses on the mean difference rather than individual group means.

7. Confidence Interval for Population Variance

   • Estimates the variance of a population based on sample data.
   • Formula: ( \left( \frac{(n-1)s^2}{\chi^2_{\alpha/2, df}}, \frac{(n-1)s^2}{\chi^2_{1-\alpha/2, df}} \right) ).
   • Uses the chi-squared distribution for the interval.
   • Requires a normally distributed population for valid results.

8. 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 F_{\alpha/2, df_1, df_2}, \frac{s_1^2}{s_2^2} \cdot F_{1-\alpha/2, df_1, df_2} \right) ).
   • Assumes both populations are normally distributed.
   • Useful for assessing the equality of variances in hypothesis testing.

9. Confidence Interval for Correlation Coefficient

   • Estimates the range of the true correlation between two variables.
   • Uses Fisher's z-transformation for the interval.
   • Formula: ( z' \pm Z_{\alpha/2} \cdot \frac{1}{\sqrt{n-3}} ), where ( z' ) is the Fisher transformation of the correlation coefficient.
   • Assumes a linear relationship and normally distributed variables.

10. Confidence Interval for Regression Slope

    • Estimates the slope of the regression line in a linear relationship.
    • Formula: ( b_1 \pm t_{\alpha/2, df} \cdot SE(b_1) ), where ( SE(b_1) ) is the standard error of the slope.
    • Assumes linearity, independence, and normally distributed residuals.
    • Provides insight into the strength and direction of the relationship between variables.