Question

A company wishes to estimate the average amount that its customers pay per month. A random sample of 100 customers' bills during a given month produced a sample mean of $35, and a sample standard deviation of $5. (So, the current problem differs from the previous only by the fact that now $5 presents not the standard deviation for the whole population, but rather the standard deviation of just a sample of 100 customers.)

a) Without any calculations, do you expect that the 95% confidence interval in the case of the current problem will be wider than that in the previous problem or narrower? Wider; Narrower

Answer 1

In the situation given where the standard deviation is based on a sample rather than the entire population, we would expect a wider confidence interval due to the increased standard error reflecting more uncertainty.

Step-by-step explanation:

The question posed pertains to the confidence interval for a sample mean with a known sample standard deviation. Specifically, the student is asked whether they would expect a narrower or wider confidence interval given that the standard deviation of $5 is for a sample instead of the whole population. The 95% confidence interval for the sample mean would be expected to be wider in this situation, as sample-based estimates of standard deviation tend to be less precise than those based on the whole population, which is captured in the concept of the standard error of the mean. The standard error increases with the variability in the sample, leading to a wider confidence interval as the estimate of the population parameter incorporates more uncertainty.

This reflects a common issue in statistical inference where larger samples provide more precise estimates, and the distinction between population and sample statistics is crucial in determining the confidence with which we can make inferences about the population parameters.