Confidence intervals are crucial tools in econometrics, providing a range of plausible values for unknown population parameters. They consist of a point estimate and margin of error, reflecting the uncertainty in estimating population parameters from sample data.

Understanding confidence intervals is key to interpreting statistical results accurately. They're often misinterpreted, but correctly represent the long-run frequency of intervals capturing the true value if sampling were repeated. Confidence levels, construction methods, and factors affecting interval width are essential concepts to grasp.

Definition of confidence intervals

• Confidence intervals provide a range of plausible values for an unknown population parameter based on sample data
• Consist of a point estimate (sample statistic) and a margin of error
• Reflect the uncertainty inherent in estimating population parameters from sample statistics

Interpretation of confidence intervals

• Confidence intervals are often misinterpreted as containing the true population parameter with a certain probability
• Correct interpretation: If the sampling and estimation process were repeated many times, a certain percentage (confidence level) of the resulting intervals would contain the true population parameter
• Do not indicate the probability that the true population parameter lies within the interval for a single sample

Confidence level

Meaning of confidence level

• Represents the proportion of intervals that would contain the true population parameter if the sampling and estimation process were repeated many times
• Reflects the long-run frequency of intervals capturing the true value
• Higher confidence levels result in wider intervals, providing greater assurance that the interval contains the population parameter

Common confidence levels

• 90% confidence level commonly used in social sciences and business
• 95% confidence level widely used in various fields ($\alpha = 0.05$)
• 99% confidence level used when a higher degree of certainty is required ($\alpha = 0.01$)

Constructing confidence intervals

Formula for confidence intervals

• General formula: $\text{Point estimate} \pm \text{Critical value} \times \text{Standard error}$
• Point estimate is the sample statistic (e.g., sample mean, sample proportion)
• Critical value depends on the desired confidence level and the sampling distribution of the statistic
• Standard error measures the variability of the point estimate

Standard error in confidence intervals

• Standard error quantifies the amount of sampling variability in the point estimate
• Calculated differently for different statistics (means, proportions, etc.)
• For means: $\text{Standard error} = \frac{\text{Sample standard deviation}}{\sqrt{\text{Sample size}}}$
• Smaller standard errors lead to narrower confidence intervals

Critical values for confidence intervals

• Critical values determine the width of the confidence interval based on the desired confidence level
• Obtained from the sampling distribution of the statistic (e.g., t-distribution for means, z-distribution for proportions)
• For a 95% confidence interval, the critical value is approximately 1.96 (assuming a large sample size and normal distribution)

Factors affecting confidence intervals

Sample size vs confidence interval width

• Larger sample sizes generally result in narrower confidence intervals
• As sample size increases, the standard error decreases, leading to more precise estimates
• Doubling the sample size reduces the width of the confidence interval by a factor of $\sqrt{2}$

Variability vs confidence interval width

• Greater variability in the data leads to wider confidence intervals
• Higher standard deviations or variances result in larger standard errors
• Reducing variability through better measurement or experimental design can narrow confidence intervals

Confidence intervals for means

Population mean confidence intervals

• Used when the population standard deviation is known
• Formula: $\bar{x} \pm z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$
• $\bar{x}$ is the sample mean, $\sigma$ is the population standard deviation, $n$ is the sample size, and $z_{\alpha/2}$ is the critical value from the standard normal distribution

Sample mean confidence intervals

• Used when the population standard deviation is unknown and estimated from the sample
• Formula: $\bar{x} \pm t_{\alpha/2, n-1} \times \frac{s}{\sqrt{n}}$
• $s$ is the sample standard deviation and $t_{\alpha/2, n-1}$ is the critical value from the t-distribution with $n-1$ degrees of freedom

Confidence intervals for proportions

• Used to estimate population proportions based on sample proportions
• Formula: $\hat{p} \pm z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
• $\hat{p}$ is the sample proportion, $n$ is the sample size, and $z_{\alpha/2}$ is the critical value from the standard normal distribution
• Requires a large enough sample size (usually $n\hat{p} \geq 10$ and $n(1-\hat{p}) \geq 10$) for the normal approximation to be valid

Confidence intervals vs point estimates

• Point estimates provide a single value estimate of a population parameter
• Confidence intervals provide a range of plausible values for the population parameter
• Confidence intervals convey the uncertainty associated with the point estimate
• Point estimates alone can be misleading, as they do not reflect the precision of the estimate

Misinterpreting confidence intervals

Confidence intervals vs probability

• Confidence intervals do not indicate the probability that the true population parameter lies within the interval for a single sample
• They represent the long-run proportion of intervals that would contain the true parameter if the sampling process were repeated many times
• Interpreting a 95% confidence interval as having a 95% probability of containing the true parameter is a common misconception

Confidence intervals vs statistical significance

• Confidence intervals and statistical significance are related but distinct concepts
• Statistical significance testing assesses whether sample results are likely to have occurred by chance, assuming a null hypothesis is true
• Confidence intervals provide a range of plausible values for the population parameter
• A confidence interval that does not contain the null hypothesis value (e.g., 0 for no difference) suggests statistical significance at the corresponding alpha level