Question

Suppose that a random sample of 50 bottles of a particular brand of cough medicine is selected, and the alcohol content of each bottle is determined. Let \(\mu\) denote the mean alcohol content for the population of all bottles of this brand. Suppose that this sample of 50 results in a \(95 \%\) confidence interval for \(\mu\) of (7.8,9.4) a. Would a \(90 \%\) confidence interval have been narrower or wider than the given interval? Explain your answer. b. Consider the following statement: There is a \(95 \%\) chance that \(\mu\) is between 7.8 and 9.4 . Is this statement correct? Why or why not? c. Consider the following statement: If the process of selecting a random sample of size 50 and then calculating the corresponding \(95 \%\) confidence interval is repeated 100 times, exactly 95 of the resulting intervals will include \(\mu .\) Is this statement correct? Why or why not?

Short answer

a. A \(90%\) confidence interval would have been narrower. b. No, the interpretation of the confidence interval is incorrect. It should be: We are \(95%\) confident that the interval (7.8, 9.4) contains the true mean alcohol content \(\mu\). c. No, the statement is incorrect. Confidence level means that \(95%\) of the intervals from repeated sampling would contain the population mean. However it doesn't provide any guarantee about exact number out of 100.

Step-by-step solution

Understanding Confidence Level Impact on Interval Width

When you decrease the confidence level, from \(95%\) to \(90%\) for instance, the width of the confidence interval decreases as well. The confidence level is a measure of certainty. If we are less certain (\(90%\) compared to \(95%)\), we need less precision, hence a narrower confidence interval.

Identifying Errors in Confidence Interval Interpretation

The statement 'There is a \(95%\) chance that the mean alcohol content \(\mu\) is between 7.8 and 9.4' is incorrect. Confidence intervals do not work that way. The correct interpretation should be: We are \(95%\) confident that the interval (7.8, 9.4) contains the true mean alcohol content \(\mu\). It is not about probability of \(\mu\), but about the level of confidence in our method.

Understanding Probabilistic Interpretation of Confidence Intervals

Similarly, the statement 'If the process is repeated 100 times, exactly 95 of the intervals will include \(\mu\)' is incorrect as well. This is a common misunderstanding. What the \(95%\) confidence level means is that \(95%\) of random samples of the same size from the same population will result in confidence intervals that contain the population parameter. However, it does not state any guarantee about the exact number out of 100.