Final answer:
The confidence interval for a sample mean of 15 with an error bound of 3.2 is (11.8, 18.2). Confidence levels affect the width of confidence intervals, with 95% giving a wider range than 90%. Increasing the sample size tends to narrow the confidence interval.
Step-by-step explanation:
If the sample mean is 15 and the error bound for the mean is 3.2, then the confidence interval estimate for the population mean can be calculated by subtracting and adding the error bound from the sample mean. Thus, the confidence interval is (15 - 3.2, 15 + 3.2), which gives us the interval (11.8, 18.2).
It's important to understand that different confidence levels, such as 90% and 95%, will yield different widths for confidence intervals. A higher confidence level, such as 95%, will result in a wider interval than a lower confidence level, like 90%, when we're estimating the same parameter. This is because a higher confidence level means that we want to be more certain that our interval contains the true population parameter, thus we allow for a larger range of possible values.
When a sample size changes, it can also affect the confidence interval. For instance, if the sample size were to increase from 30 to 50, and we were still looking for a 95 percent confidence interval with a sample mean of 41, the interval would likely become narrower. This happens because a larger sample size generally provides more reliable information about the population, thus reducing the necessary range for our estimate.