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The director of a college orchestra is building a case for donors to continue their support. To start, she gave a survey to 5050 randomly chosen students at the college about their concert attendance. From the survey results, the director calculated a 90%90\% confidence interval of for the mean number of classical music concerts students at her college attended last year.\newlineIs the following conclusion valid?\newlineThere is a 90%90\% chance that the mean number of classical music concerts attended by students last year is in the interval .\newlineChoices:\newline(A)yes\newline(B)no\newline

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Q. The director of a college orchestra is building a case for donors to continue their support. To start, she gave a survey to 5050 randomly chosen students at the college about their concert attendance. From the survey results, the director calculated a 90%90\% confidence interval of for the mean number of classical music concerts students at her college attended last year.\newlineIs the following conclusion valid?\newlineThere is a 90%90\% chance that the mean number of classical music concerts attended by students last year is in the interval .\newlineChoices:\newline(A)yes\newline(B)no\newline
  1. Meaning of Confidence Interval: The confidence interval gives us the range in which the true mean is likely to fall, not the probability of the mean being within that range after the data has been collected.
  2. Calculation of Confidence Level: Since the interval has already been calculated from the sample, the 90%90\% confidence level means that if we were to take many samples and build a confidence interval from each, about 90%90\% of those intervals would contain the true mean.
  3. Common Misunderstanding: The conclusion that there is a 90%90\% chance that the mean number of concerts attended is within the interval is a common misunderstanding of confidence intervals. The interval either contains the true mean or it doesn't for this specific sample.
  4. Correct Interpretation: Therefore, the correct answer is (B) no, because the statement misrepresents what a confidence interval indicates.

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