Question

a student was asked to find a 98% confidence interval for widget width using data from a random sample of size ( n = 21 ). which of the following is a correct interpretation of the interval ( 13.2<mu<27.9 )?
there is a 98% chance that the width of the widget is between 13.2 and 27.9.
with 98% confidence, the width of a randomly selected widget will be between 13.2 and 27.9.
with 98% confidence, the mean width of all widgets is between 13.2 and 27.9.
there is a 98% chance that the mean of a sample of 21 widgets will be between 13.2 and 27.9.
the width of all widgets is between 13.2 and 27.9, 98% of the time. we know this is true because the mean of our sample is between 13.2 and 27.9.

Explanation:

A confidence interval is an interval estimate for an unknown population parameter. In this case, the parameter is the population mean \(\mu\) (mean width of all widgets). A 98% confidence interval means that if we were to take many samples and construct confidence intervals in the same way, about 98% of those intervals would contain the true population mean.

• The first option is incorrect because it refers to the width of a single widget (not the population mean).
• The second option is incorrect as it also refers to a single widget's width (not the population mean).
• The fourth option is incorrect because it refers to the mean of a sample (a confidence interval is for the population mean).
• The fifth option is incorrect because the fact that the sample mean is in the interval does not imply the population mean is in the interval 98% of the time in the way described.

The third option “With 98% confidence, the mean width of all widgets is between 13.2 and 27.9” is the correct interpretation of a confidence interval for the population mean.

Answer:

With 98% confidence, the mean width of all widgets is between 13.2 and 27.9.