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A student was asked to find a 99% confidence interval for widget width using data from a random sample of size $n=26$. Which of the following is a correct interpretation of the interval $10<\mu<21.6$?


AWith 99% confidence, the mean width of all widgets is between 10 and 21.6.
BThere is a 99% chance that the mean of the population is between 10 and 21.6.
CThere is a 99% chance that the mean of a sample of 26 widgets will be between 10 and 21.6.
DThe mean width of all widgets is between 10 and 21.6, 99% of the time. We know this is true because the mean of our sample is between 10 and 21.6.
EWith 99% confidence, the mean width of a randomly selected widget will be between 10 and 21.6.

Verified Answer :

  • The population mean \(\mu\) is a fixed parameter, not a random variable, so it cannot have a probability of being within an interval.
  • Confidence intervals relate to the reliability of the process of sampling and construction, not the specific interval calculated.
  • The correct statement is: "With 99% confidence, the mean width of all widgets is between 10 and 21.6".
  • Final Answer: \(\boxed{\text{With 99\% confidence, the mean width of all widgets is between 10 and 21.6.}}\)

Explanation

  1. Understanding Confidence Intervals

A confidence interval, such as the one calculated for the mean width of widgets (\(10 < \mu < 21.6\)), is a statistical tool used to estimate a population parameter. The key to understanding this concept is to recognize that the population mean, \(\mu\), is a fixed, albeit unknown, value. It is not a random variable. Therefore, it does not have a "probability" of falling into a range. Instead, the confidence interval is a result of a sampling process that has a certain reliability.

  1. Frequentist Interpretation Explained

When we say we have a \(99\%\) confidence interval, we mean that if we were to repeat the sampling process of size \(n=26\) many times, and construct an interval from each sample using the same method, approximately \(99\%\) of those intervals would contain the true population mean \(\mu\). The confidence refers to the reliability of the estimation procedure, not to the specific interval \((10, 21.6)\) containing the parameter with \(99\%\) probability. It is either inside or outside, but we are \(99\%\) confident in the method that produced this specific interval.

  1. Evaluating the Options

Let's examine the provided statements based on the frequentist definition:

  • "There is a 99% chance that the mean of the population is between 10 and 21.6": Incorrect. The population mean is a constant, not a random variable.
  • "There is a 99% chance that the mean of a sample of 26 widgets will be between 10 and 21.6": Incorrect. This describes a prediction interval for a sample mean, not the population mean confidence interval.
  • "The mean width of all widgets is between 10 and 21.6, 99% of the time": Incorrect. This implies the population mean changes over time or fluctuates.
  • "With 99% confidence, the mean width of a randomly selected widget will be between 10 and 21.6": Incorrect. This refers to an individual observation (prediction interval), not the population mean \(\mu\).
  • "With 99% confidence, the mean width of all widgets is between 10 and 21.6": Correct. This is the standard, accepted way to interpret a confidence interval, expressing our level of confidence in the estimation process.

Examples

Imagine you are quality-testing components for a high-precision machine. You measure a sample of 26 parts and calculate that their mean width is between 10mm and 21.6mm with 99% confidence. This doesn't mean the true mean has a 99% chance of being there; it means that if you tested thousands of batches of 26 parts, 99% of those batch intervals would successfully capture the true average width of all parts manufactured. This helps engineers set standards for manufacturing reliability.

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