How does an increase in sample size affect the confidence interval?

As a statistical expert with extensive experience in data analysis and statistical modeling, I have a deep understanding of how sample size impacts the confidence interval in statistical inference. Let's delve into the details of this topic. Confidence Intervals and Sample Size A confidence interval is a range that is likely to contain the value of an unknown population parameter. It is constructed from a statistical model where the model's parameters are estimated from the data. The confidence level, often denoted as (1-α)100%, indicates the frequency (or proportion) of possible confidence intervals that contain the true value of the unknown parameter. For instance, a 95% confidence interval suggests that if we were to take many samples and construct a confidence interval from each, then 95% of these intervals would contain the true population mean. The Role of Sample Size The sample size plays a crucial role in determining the width of the confidence interval. Here's how it works: 1. Standard Error: The standard error (SE) is a measure of the variability in the sampling distribution of the estimator (e.g., the sample mean). It is calculated as the standard deviation of the population (σ) divided by the square root of the sample size (n), SE = σ/√n. As the sample size increases, the standard error decreases, assuming the population standard deviation remains constant. 2. Margin of Error: The margin of error (E) is the distance between the confidence interval boundaries and the sample statistic (e.g., the sample mean). It is calculated as a multiple of the standard error, E = Z * SE, where Z is the Z-score corresponding to the desired confidence level. A smaller standard error results in a smaller margin of error. 3. Width of the Confidence Interval: The width of the confidence interval is the difference between the upper and lower bounds. As the margin of error decreases with an increase in sample size, the width of the confidence interval also decreases, assuming the population standard deviation is known or estimated with reasonable accuracy. 4. Precision: A larger sample size leads to greater precision in estimating the population parameter. This is because larger samples are more likely to represent the population from which they are drawn, thus reducing the impact of random sampling error. 5. Statistical Power: In hypothesis testing, an increased sample size also enhances the statistical power, which is the probability of correctly rejecting a false null hypothesis. Common Misconceptions It's important to clarify a common misconception regarding the interpretation of confidence intervals: - The statement "the 95% confidence interval for the population mean is (350, 400)" is not equivalent to saying "there is a 95% probability that the population mean is between 350 and 400." This is a misunderstanding of how confidence intervals are constructed. Instead, it means that if we were to repeat the sampling process and construct a confidence interval from each sample, 95% of those intervals would contain the true population mean. Conclusion Increasing the sample size is a powerful tool in statistics for narrowing the confidence interval and increasing the precision of our estimates. It's a fundamental concept in designing experiments and surveys, as it directly affects the reliability and validity of the results.

Increasing the sample size decreases the width of confidence intervals, because it decreases the standard error. c) The statement, "the 95% confidence interval for the population mean is (350, 400)", is equivalent to the statement, "there is a 95% probability that the population mean is between 350 and 400".