Suppose that a random sample of 50 bottles of a particular brand of cough medici

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. Data from this sample of 50 bottles was used to calculate a 95% confidence interval for the mean alcohol content (in percent) for all bottles of this brand of cough medicine. The interval was (7.8%, 9.4%) (1 point) would a 90% confidence interval have been narrower or wider than the given interval? Explain your answer. a. b. (1 point) Consider the following statement: There is a 95% chance that the mean alcohol content for all bottles of this brand of cough medicine is between 7.8% and 9.4%. Is this statement correct? Why or why not? (1 point) Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding 95% confidence interval is repeated 100 times, 95 of the resulting intervals will include the mean alcohol content for all bottles of this brand of cough medicine. Is this statement correct? Why or why not? C.

Explanation / Answer

(a) As we decrease the level of confidence from 95% to 90%, the critical value in the formula of the confidence interval becomes less. Hence, the value to be subtracted from the mean, in the formula of the confidence interval, becomes less. As a result, the gap between the upper limit and the lower limit of the confidence interval shortens and the confidence interval becomes narrower.

(b) This statement is incorrect. Although we can assign a subjective probability of 0.95 that a 95% confidence interval contains the population mean, this is not the best interpretation of the choices. Other factors come into play, such as knowledge from prior studies and other ways to compute confidence intervals, that make a probability assessment of 0.95 not the most straightforward or accurate interpretation of a 95% confidence interval.

(c) This statement is correct. The correct interpretation of a 95% confidence interval is - If repeated samples were taken and the 95% confidence interval was computed for each sample, 95% of the intervals would contain the population mean.

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