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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.