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Tom owned an orange grove and wanted to know how productive his trees were. He randomly selected $175$ trees in the grove and kept track of how many oranges each tree produced during one season. Tom found a $90\%$ confidence interval of for the mean number of oranges produced by his $175$ trees that season. $\newline$ Is the following conclusion valid? $\newline$ There is a $90\%$ chance that the mean number of produced by his trees that season is in the interval . $\newline$ Choices: $\newline$ (A)yes $\newline$ (B)no

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Interpret confidence intervals for population means

Full solution

Q. Tom owned an orange grove and wanted to know how productive his trees were. He randomly selected

175

trees in the grove and kept track of how many oranges each tree produced during one season. Tom found a

90\%

confidence interval of for the mean number of oranges produced by his

175

trees that season.

\newline

Is the following conclusion valid?

\newline

There is a

90\%

chance that the mean number of produced by his trees that season is in the interval .

\newline

Choices:

\newline

(A)yes

\newline

(B)no

Understanding Confidence Intervals: Tom found a $90\%$ confidence interval for the mean number of oranges produced by his trees. This means that if Tom were to repeat his sampling process many times, $90\%$ of the confidence intervals calculated from those samples would contain the true mean.
Common Misconception: The conclusion that there is a $90\%$ chance that the mean number of oranges produced is in the interval is a common misunderstanding of confidence intervals. The interval either contains the true mean or it doesn't; the ＂ $90\%$ chance＂ refers to the confidence level of the interval estimation method, not the probability of the interval containing the mean.
Correct Interpretation: Therefore, the correct answer is (B) no, because the statement misinterprets what a confidence interval represents.

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