Question

A student was asked to find a 98% confidence interval for widget width (in mm) using data from a random sample of size n = 24. Which of the following is a correct interpretation of the interval 11.1 < p < 28.82? Check all that are correct.

a. With 98% confidence, the mean width of all widgets is between 11.1 mm and 28.8 mm.
b. The mean width of all widgets is between 11.1 mm and 28.8 mm, 98% of the time. We know this is true because the mean of our sample is between 11.1 mm and 28.8 mm.
c. There is a 98% chance that the mean of the population is between 11.1 mm and 28.8 mm.
d. With 98% confidence, the mean width of a randomly selected widget will be between 11.1 mm and 28.8 mm.
e. There is a 98% chance that the mean of a sample of 24 widgets will be between 11.1 mm and 28.8 mm.

Answer 1

The correct interpretation of a 98% confidence interval of 11.1 mm to 28.82 mm for the mean widget width is that we're 98% confident that this interval contains the true mean width of all widgets. Other interpretations that suggest the interval reflects the sample mean or the width of a particular widget are incorrect.

Step-by-step explanation:

When asked to interpret a 98% confidence interval for widget width (in mm), one must understand how to interpret confidence intervals in statistics. A confidence interval provides a range of values within which we believe the true population parameter (in this case, the mean width of widgets) lies, with a given level of confidence. In the interval 11.1 < p < 28.82, we can correctly interpret the results as follows:

• a. With 98% confidence, the mean width of all widgets is between 11.1 mm and 28.82 mm. This means that if we were to take many repeated samples and compute confidence intervals from those samples, 98% of those confidence intervals would be expected to contain the true population mean.
• c. There is a 98% chance that the confidence interval of 11.1 mm to 28.82 mm will contain the true mean width of the widget population. This is a common interpretation of a confidence interval.

Options b, d, and e are incorrect because confidence intervals pertain to the population mean, rather than the mean of a sample or of a single widget and do not define a 'chance' for the mean itself to be within certain limits.

The correct interpretation is crucial for making decisions or predictions based on the data and its confidence interval.