Constructing a 99 Percent Confidence Interval

With a sample size of 85, construct a 99% confidence interval for the population mean.

In this problem, we are tasked with constructing a 99 percent confidence interval for a population mean based on a sample size of 85. Confidence intervals are a fundamental concept in inferential statistics, providing a range of values within which we expect the true population parameter to lie, given a certain level of confidence. This particular problem focuses on a large sample size, which allows us to apply the Central Limit Theorem. This theorem tells us that the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution, given the relatively large sample size.

When constructing a confidence interval, one of the primary components to consider is the level of confidence desired, in this case, 99 percent. This high level of confidence requires a wider interval compared to a lower confidence level, such as 95 percent, because we are being more cautious about containing the true population mean within our interval. The construction involves calculating the margin of error, which is influenced by the sample standard deviation, the sample size, and the critical value associated with the desired confidence level from the standard normal distribution or t-distribution, depending on whether the population standard deviation is known.

As you work through constructing confidence intervals, it's important to understand the implications of your confidence level and sample size on the precision of your interval. A larger sample size generally leads to a more precise interval, while a higher confidence level broadens the interval. This balance between precision and confidence is crucial in statistical inference, as it affects how we interpret the reliability and usefulness of our interval estimates.