Diego is a wildlife researcher. They were analyzing the mean and median lengths of $9$ whales their team had observed. The whales all had different lengths between $23 \mathrm{~m}$ and $27 \mathrm{~m}$ . $\newline$ Diego found out that they were misreading the shortest length. It was actually $20 \mathrm{~m}$ , not $23 \mathrm{~m}$ . $\newline$ How will this length decreasing affect the mean and median? $\newline$ Choose $1$ answer: $\newline$ (A) Both the mean and median will decrease. $\newline$ (B) The mean will decrease, and the median will increase. $\newline$ (C) The mean will decrease, and the median will stay the same. $\newline$ (D) The mean will stay the same, and the median will decrease.

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Q. Diego is a wildlife researcher. They were analyzing the mean and median lengths of

9

whales their team had observed. The whales all had different lengths between

23 \mathrm{~m}

and

27 \mathrm{~m}

\newline

Diego found out that they were misreading the shortest length. It was actually

20 \mathrm{~m}

, not

23 \mathrm{~m}

\newline

How will this length decreasing affect the mean and median?

\newline

Choose

1

answer:

\newline

(A) Both the mean and median will decrease.

\newline

(B) The mean will decrease, and the median will increase.

\newline

\newline

(D) The mean will stay the same, and the median will decrease.

Calculate Original Mean: Calculate the original mean with the incorrect shortest length ( $23\,\text{m}$ ). Since the lengths are between $23\,\text{m}$ and $27\,\text{m}$ and all different, the sum of lengths is $23 + 24 + 25 + 26 + 27 + (3$ more values within the range). For simplicity, assume the $3$ missing values are $23, 24,$ and $25$ (the actual values don't matter for this problem). The mean is then $(23 + 24 + 25 + 26 + 27 + 23 + 24 + 25 + 26) / 9$ .
Calculate New Mean: Perform the calculation for the original mean: $(23 + 24 + 25 + 26 + 27 + 23 + 24 + 25 + 26) / 9 = 223 / 9 = 24.777...$ (rounded to $24.778$ ).
Determine Original Median: Calculate the new mean with the corrected shortest length $20m$ . Replace one of the $23m$ values with $20m$ and recalculate: $\frac{20 + 24 + 25 + 26 + 27 + 23 + 24 + 25 + 26}{9}$ .
Determine New Median: Perform the calculation for the new mean: $(20 + 24 + 25 + 26 + 27 + 23 + 24 + 25 + 26) / 9 = 220 / 9 = 24.444\ldots$ (rounded to $24.444$ ).
Compare Mean and Median: Determine the original median. With the lengths sorted, the median is the middle value, which is the $5^{\text{th}}$ value in the sorted list. The original median is $25\,\text{m}$ .
Compare Mean and Median: Determine the original median. With the lengths sorted, the median is the middle value, which is the $5^{\text{th}}$ value in the sorted list. The original median is $25\,\text{m}$ .Determine the new median after the shortest length is corrected to $20\,\text{m}$ . The new sorted list is $20, 23, 24, 25, 26, 27$ , and three more values within the range. The median is still the $5^{\text{th}}$ value, which remains $25\,\text{m}$ .
Compare Mean and Median: Determine the original median. With the lengths sorted, the median is the middle value, which is the $5^{\text{th}}$ value in the sorted list. The original median is $25\,\text{m}$ .Determine the new median after the shortest length is corrected to $20\,\text{m}$ . The new sorted list is $20, 23, 24, 25, 26, 27$ , and three more values within the range. The median is still the $5^{\text{th}}$ value, which remains $25\,\text{m}$ .Compare the original and new means and medians to answer the question. The mean decreased from $24.778$ to $24.444$ , and the median remained the same at $25\,\text{m}$ .