What happens to the width of the confidence interval as the sample size increases?

As a domain expert in statistical analysis, I have spent considerable time studying the intricacies of confidence intervals and their relationship with sample size. Confidence intervals are a fundamental concept in statistics, providing an estimated range for an unknown population parameter, such as the population mean, based on sample data. They are used to express the uncertainty associated with the estimation of the parameter. The width of the confidence interval is a critical aspect, as it indicates the precision of the estimate. When we increase the sample size, several things happen that affect the width of the confidence interval: 1. Standard Error Reduction: The most immediate effect is a reduction in the standard error of the estimate. The standard error is the standard deviation of the sampling distribution of the estimate. It is calculated as the standard deviation of the population (σ) divided by the square root of the sample size (n). As n increases, the denominator of this fraction becomes larger, leading to a smaller standard error (\(SE = \frac{\sigma}{\sqrt{n}}\)). 2. Decreased Sampling Fluctuations: With a larger sample size, the sample mean becomes a more stable and reliable estimate of the population mean. Larger samples are less susceptible to sampling fluctuations and random errors, which can skew the results when the sample size is small. 3. Narrower Confidence Intervals: Because the margin of error is calculated based on the standard error, a smaller standard error leads to a narrower confidence interval. The formula for the margin of error is typically \(ME = Z \times SE\), where Z is the z-value corresponding to the desired confidence level (e.g., 1.96 for a 95% confidence interval). A smaller SE results in a smaller ME, which means the confidence interval does not extend as far above and below the sample mean. 4. Increased Precision: A narrower confidence interval indicates greater precision in the estimate of the population parameter. This means that with a larger sample size, we can be more confident that our estimate is closer to the true population value. 5. Statistical Power: Larger sample sizes also increase the statistical power of a study, which is the probability of correctly rejecting a false null hypothesis. This is another benefit of increasing sample size, as it makes it more likely that significant effects will be detected if they truly exist. 6. Practical Considerations: While increasing sample size has these statistical benefits, it also comes with practical considerations. Larger samples require more resources, time, and effort to collect and analyze. Additionally, there may be diminishing returns in terms of the proportionate decrease in the width of the confidence interval as the sample size becomes very large. 7. Limitations and Assumptions: It's important to remember that the benefits of increasing sample size are contingent on the assumptions underlying the statistical analysis being met. For example, the central limit theorem, which allows us to use normal distribution theory for sample means even when the population distribution is not normal, requires a sufficiently large sample size. Now, regarding the statement about the 95% confidence interval for the population mean, it is a common misconception to equate the confidence interval directly with a probability statement about the population parameter itself. The correct interpretation is that if we were to take an infinite number of samples and construct a 95% confidence interval from each, then 95% of those intervals would contain the true population mean. It is not correct to say that there is a 95% probability that the population mean lies within a particular interval based on a single sample. To summarize, increasing the sample size leads to a decrease in the width of the confidence interval due to the reduction in standard error, which in turn increases the precision and reliability of the estimate of the population parameter.

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".