Question

Jimmy, a very promising 8th grader, is taking an AP Statistics class. The teacher has generated on a computer a large set of numbers that are approximately normally distributed and are being considered as the population. Each student is asked to select a random sample from the population and use the sample to construct a 95 confidence interval for the population mean. Jimmy goes ahead and does this. However, when the teacher announces the true population mean to the class, Jimmy notices that the population mean does not lie within his confidence interval. Which of the following is true?

(A) Jimmy's work could have been correct. Confidence intervals are designed to be narrow, and therefore, for most random samples, the population parameter being estimated will not lie within the interval.

(B) Jimmy's work could have been correct. For about 5% of students doing this correctly the population mean will not lie within the confidence interval calculated.

(C) Jimmy's work could have been correct. It will always be the case that the population mean is not within the confidence interval when the population mean doesn't happen to be in the sample selected.

(D) There must be an error in Jimmy's work. The whole point of confidence intervals is that they should capture the population parameter that is being estimated.

(E) There must be an error in Jimmy's work. He must have calculated an interval that was too narrow, and it was this that caused the interval not to capture the population mean.

Answer 1

(B) Jimmy's work could have been correct. For about 5% of students doing this correctly, the population mean will not lie within the confidence interval calculated.

A 95% confidence interval is a range of values that you can be 95% certain contains the true mean of the population.

So, there can be 5% chances that the true values does not lie in the interval.