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Bryan is convinced that the German novel he just finished reading has some of the longest sentences he has ever encountered. To prove this to his friends, he randomly selected $100$ sentences in the novel and noted the number of words in each. Bryan found a $95\%$ confidence interval of for the mean number of words in sentences from the novel. $\newline$ Is the following conclusion valid? $\newline$ If Bryan takes another random sample, there is a $95\%$ chance that the mean number of words in sentences from the novel will be in the new sample's $95\%$ confidence interval. $\newline$ Choices: $\newline$ (A)yes $\newline$ (B)no $\newline$

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Interpret confidence intervals for population means

Full solution

Q. Bryan is convinced that the German novel he just finished reading has some of the longest sentences he has ever encountered. To prove this to his friends, he randomly selected

100

sentences in the novel and noted the number of words in each. Bryan found a

95\%

confidence interval of for the mean number of words in sentences from the novel.

\newline

Is the following conclusion valid?

\newline

If Bryan takes another random sample, there is a

95\%

chance that the mean number of words in sentences from the novel will be in the new sample's

95\%

confidence interval.

\newline

Choices:

\newline

(A)yes

\newline

(B)no

\newline

Confidence Interval Explanation: Bryan's $95\%$ confidence interval is based on the sample he took, which means he can be $95\%$ confident that the population mean falls within this interval.
Variability in New Samples: However, this does not mean that a new sample's mean will fall within the original confidence interval. Each new sample could have its own confidence interval.
Interpreting Confidence Intervals: The conclusion that there is a $95\%$ chance that the mean number of words in sentences from the novel will be in the new sample's $95\%$ confidence interval is incorrect. The correct interpretation is that we can be $95\%$ confident that the population mean falls within the original interval, not that a new sample mean will fall within this interval.

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