As a domain expert in statistics, I understand the significance of confidence intervals in inferential statistics. Let's delve into the impact of increasing the sample size on confidence intervals.
When we talk about confidence intervals, we're referring to a range that we can be confident contains the population parameter. The confidence level is a key component here, which is typically set at 95% in many studies. This means that if we were to repeat the sampling process an infinite number of times, 95% of the confidence intervals we calculate would contain the true population parameter.
Now, let's consider the effect of increasing the sample size. There are a few key points to understand:


1. Standard Error Reduction: The standard error of the statistic is inversely proportional to the square root of the sample size (\( n \)). As the sample size increases, the standard error decreases. This is because with more data, the estimate of the population parameter becomes more precise.


2. Width of the Interval: The width of the confidence interval is directly influenced by the standard error. A smaller standard error means that the confidence interval will be narrower. This is because the margin of error, which is calculated as the critical value from the distribution (often the normal or t-distribution) multiplied by the standard error, will be smaller.


3. Precision: A larger sample size leads to greater precision in the estimate of the population parameter. This is because the variability of the sample mean decreases as more data is included.


4. Confidence Level: It's important to note that increasing the sample size does not change the confidence level itself. The confidence level remains a fixed probability (e.g., 95%), but the interval is calculated in such a way that it becomes narrower, reflecting the increased precision.


5. Practical Considerations: While increasing the sample size can improve the precision of the estimate, it also comes with practical considerations such as increased cost, time, and effort. Researchers must balance the need for precision with these practical constraints.


6. Statistical Power: In hypothesis testing, a larger sample size also increases the statistical power of the test, which is the probability of correctly rejecting a false null hypothesis.

7.
Population Variance: The effect of increasing the sample size assumes that the population variance is known or can be accurately estimated. If the population variance is large, even a larger sample size may not significantly narrow the confidence interval.

Now, let's address the two statements provided:

a) The statement "Increasing the sample size decreases the width of confidence intervals, because it decreases the standard error." is True. As explained, the larger the sample size, the smaller the standard error, leading to a narrower confidence interval.

b) 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." The correct interpretation is that we are 95% confident that our calculated interval contains the true population mean. It is a misunderstanding to say that the population parameter itself has a probability distribution.

In conclusion, increasing the sample size is a critical tool in statistical analysis for improving the precision of estimates and narrowing the width of confidence intervals. However, it must be applied with an understanding of its implications and the context in which it is used.

read more >>

False. 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 >>