Question

below are wing flaps-per-second for a sample of amethyst wood-star hummingbirds.
{81.1, 82.1, 75.4, 83.0, 81.3, 80.0, 77.3, 81.0, 78.9, 81.4, 74.1, 82.9, 79.5, 78.9, 83.3, 82.0, 78.1, 78.4, 80.0, 81.6, 79.7, 81.2, 78.7, 81.1, 82.2}.
enter the five-number summary below. do not round any answers in this problem.

minimum	q₁	q₂	q₃	maximum

part 2 of 4
hint 5 hint 6

the five point summary is as follows: min = 74.1, q₁ = 78.8, median = 81, q₃ = 81.8, max = 83.3. the interquartile range is iqr = 3.

give the outlier fences below.

lower fence	upper fence

list any outliers below. if there is more than 1, separate them by commas.

74.1

part 3 of 4
hint 7 hint 8

slide the elements of the boxplot below into their correct positions.

boxplot with points: 72.10, 75.40, 82.00, 85.30, 78.70; axis from 72 to 85

Explanation:

To determine the outlier fences and identify outliers, we use the interquartile range (IQR) method. The formula for the lower fence is \( Q_1 - 1.5 \times \text{IQR} \) and the upper fence is \( Q_3 + 1.5 \times \text{IQR} \).

Step 1: Recall the values of \( Q_1 \), \( Q_3 \), and IQR

From the problem:

• \( Q_1 = 78.8 \)
• \( Q_3 = 81.8 \)
• \( \text{IQR} = 3 \)

Step 2: Calculate the Lower Fence

The formula for the lower fence is:
\[
\text{Lower Fence} = Q_1 - 1.5 \times \text{IQR}
\]
Substitute the values:
\[
\text{Lower Fence} = 78.8 - 1.5 \times 3
\]
\[
\text{Lower Fence} = 78.8 - 4.5 = 74.3
\]

Step 3: Calculate the Upper Fence

The formula for the upper fence is:
\[
\text{Upper Fence} = Q_3 + 1.5 \times \text{IQR}
\]
Substitute the values:
\[
\text{Upper Fence} = 81.8 + 1.5 \times 3
\]
\[
\text{Upper Fence} = 81.8 + 4.5 = 86.3
\]

Step 4: Identify Outliers

An outlier is a data point that lies below the lower fence or above the upper fence.

The data set is: \( \{81.1, 82.1, 75.4, 83.0, 81.3, 80.0, 77.3, 81.0, 78.9, 81.4, 74.1, 82.9, 79.5, 78.9, 83.3, 82.0, 78.1, 78.4, 80.0, 81.6, 79.7, 81.2, 78.7, 81.1, 82.2\} \)

• Check values below the lower fence (\( 74.3 \)): Only \( 74.1 \) is below \( 74.3 \).
• Check values above the upper fence (\( 86.3 \)): No values in the data set exceed \( 86.3 \).

Final Answers

• Lower Fence: \( 74.3 \)
• Upper Fence: \( 86.3 \)
• Outliers: \( 74.1 \)

Boxplot Elements (for reference)

• The minimum non - outlier value (near the lower fence) is around \( 75.4 \) (left end of the box).
• The first quartile (\( Q_1 \)) is \( 78.8 \), but the vertical line in the box (median) is at \( 81.0 \).
• The third quartile (\( Q_3 \)) is \( 81.8 \), and the right end of the box is around \( 82.0 \).
• The maximum non - outlier value is \( 83.3 \), but the upper whisker goes to \( 85.30 \) (though no outlier above the upper fence). The outlier \( 74.1 \) is plotted as a separate point to the left of the lower whisker (which starts at \( 72.10 \) or similar, but the key outlier is \( 74.1 \)).

Answer:

• Lower Fence: \( 74.3 \)
• Upper Fence: \( 86.3 \)
• Outliers: \( 74.1 \)

For the boxplot (description of correct positions):

• The outlier (74.1) is plotted as a single point to the left of the lower whisker (which extends from the minimum non - outlier value, around 75.4, to the lower fence - related whisker start, e.g., 72.10).
• The box starts at \( Q_1 = 78.8 \) (left side of the box), has a median line at \( 81.0 \), and ends at \( Q_3 = 81.8 \) (right side of the box).
• The upper whisker extends from the box (right side, around 82.0) to the maximum non - outlier value (around 83.3 or up to the upper fence - related whisker end, e.g., 85.30).