Interval estimation gives us a range of likely values for population parameters, offering more insight than single-point estimates. It uses confidence intervals to show uncertainty, helping us make better decisions based on data. This approach is crucial in statistics.

Confidence intervals are built using point estimates and standard errors. They tell us how sure we can be about our guesses. Factors like sample size and data variability affect how wide these intervals are, impacting the precision of our estimates.

Interval Estimation vs Point Estimation

Advantages of Interval Estimation

• Provides a range of plausible values for a population parameter, rather than a single point estimate
• Expressed as a confidence interval, which is a range of values likely to contain the true population parameter with a specified level of confidence
• Offers a measure of the uncertainty or precision associated with the estimate
• More informative than point estimates, indicating the range of plausible values for the parameter
• Allows for more robust conclusions and decision-making based on the data (hypothesis testing, determining sample size)

Point Estimation Limitations

• Only provides a single value estimate for a population parameter, without any indication of the uncertainty or precision
• Does not account for sampling variability or the possibility of the estimate differing from the true parameter value
• Limited in its ability to support robust conclusions or decision-making, as it lacks information about the range of plausible values
• May lead to overconfidence in the accuracy of the estimate, especially with small sample sizes

Confidence Intervals for Population Parameters

Constructing Confidence Intervals

• Constructed using the point estimate (sample mean, proportion) and the standard error of the estimate
• General formula: point estimate ± (critical value) × (standard error)
• Critical value determined by the desired confidence level and the sampling distribution of the estimator (z-score for normal distribution, t-score for t-distribution)
• Requires knowledge of the appropriate sampling distribution and its parameters (mean, standard deviation)

Interpreting Confidence Intervals

• Represents the long-run probability of the interval containing the true parameter value
• A 95% confidence interval means that if the sampling process were repeated many times, approximately 95% of the resulting intervals would contain the true population parameter
• Provides a range of plausible values for the population parameter, rather than a single point estimate
• Indicates the precision of the estimate, with narrower intervals suggesting greater precision and less uncertainty

Factors Affecting Confidence Interval Width

Sample Size and Variability

• Sample size: As the sample size increases, the width of the confidence interval decreases, indicating greater precision in the estimate
• Variability of the data: Greater variability in the sample data leads to wider confidence intervals, as there is more uncertainty in the estimate
• Larger sample sizes tend to reduce the impact of extreme values or outliers on the interval estimate

Confidence Level and Population Standard Deviation

• Confidence level: Higher confidence levels (99% vs. 95%) result in wider intervals, as they require a larger range of values to achieve the desired level of confidence
• Population standard deviation: If known, it directly affects the width of the confidence interval; a larger standard deviation results in a wider interval
• Unknown population standard deviation must be estimated using the sample standard deviation, which may introduce additional uncertainty

Sample Size, Confidence Level, and Precision

Relationships and Trade-offs

• Increasing sample size while holding confidence level constant results in a narrower confidence interval, indicating greater precision in the estimate
• Increasing confidence level while holding sample size constant results in a wider confidence interval, as a larger range of values is needed to achieve the higher level of confidence
• To obtain a more precise interval estimate, one can either increase the sample size or decrease the confidence level, assuming all other factors remain constant

Inverse Proportionality and Square Root Rule

• The relationship between sample size and the width of the confidence interval is inversely proportional
• Doubling the sample size will reduce the width of the interval by a factor of $\sqrt{2}$, assuming all other factors remain constant
• This inverse proportionality is due to the standard error being proportional to $\frac{1}{\sqrt{n}}$, where $n$ is the sample size
• The square root rule provides a quick way to estimate the impact of sample size changes on the precision of the interval estimate