As a domain expert in statistical analysis, I often deal with confidence intervals which are a crucial part of inferential statistics. Confidence intervals provide an estimated range of values that likely contain an unknown population parameter. The width of this interval is influenced by several factors, and understanding these can help in interpreting the results more accurately. **Factors That Increase the Width of the Confidence Interval:** 1. Level of Confidence: The higher the level of confidence, the wider the interval. For instance, a 99% confidence interval will be wider than a 95% interval because it aims to capture a larger proportion of the variability in the data. 2. Sample Size (n): Contrary to the statement provided, a smaller sample size actually increases the width of the confidence interval. This is because with fewer observations, there is more uncertainty about the population parameter, leading to a larger margin of error. 3. Population Standard Deviation (σ): A larger population standard deviation means that the data points are more spread out, which increases the variability and thus the width of the interval. 4. Margin of Error (E): A larger margin of error directly translates to a wider confidence interval. The margin of error is often chosen based on the precision required for the study. 5. Variability Within the Sample: Greater variability or dispersion within the sample data results in a wider confidence interval, as it reflects greater uncertainty about the population parameter. 6. Estimation of the Standard Deviation: If the standard deviation is estimated from the sample rather than known from the population, this adds additional uncertainty, increasing the width of the interval. 7. Non-Normal Distribution: If the population distribution is not normal and is skewed or has outliers, the confidence interval may need to be wider to account for the non-normal distribution. 8. Complexity of the Model: In regression analysis, the more predictors a model includes, the wider the confidence intervals may become, especially if the predictors are correlated. 9. Measurement Error: If there is significant measurement error in the data collection process, this can also increase the width of the confidence interval. It's important to note that while a wider confidence interval indicates greater uncertainty, it does not necessarily mean that the study is of lower quality. It may simply reflect a more cautious approach or a decision to cover a wider range of possible values for the population parameter. Now, let's address the second part of your question regarding the interpretation of a confidence interval. The statement "the 95% confidence interval for the population mean is (350, 400)" is often misunderstood. It does not mean that there is a 95% probability that the population mean is between 350 and 400. Instead, it means that if we were to take many samples and construct a confidence interval from each, 95% of those intervals would contain the true population mean. The probability is about the method, not the specific interval for the mean. **read more >>

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".read more >>