The five-number summary for the ages of the attendees at the Pawnee Harvest Festival is listed below.

1
20
29
38
80

I. There is at least one attendee whose age is a low potential outlier.

II. There is at least one attendee whose age is a high potential outlier.

Question

The five-number summary for the ages of the attendees at the Pawnee Harvest Festival is listed below.

1
20
29
38
80

I. There is at least one attendee whose age is a low potential outlier.

II. There is at least one attendee whose age is a high potential outlier.

BlackTom AI · Accepted Answer

Let’s parse the five-number summary: minimum = 1, Q1 = 20, median = 29, Q3 = 38, maximum = 80. From these, compute the interquartile range (IQR) = Q3 − Q1 = 38 − 20 = 18.

Next, determine the fences for potential outliers using the common 1.5*IQR rule:
- Lower fence = Q1 − 1.5*IQR = 20 − 1.5×18 = 20 − 27 = −7.
- Upper fence = Q3 + 1.5*IQR = 38 + 1.5×18 = 38 + 27 = 65.

Now evaluate the statements about outliers relative to these fences:
- Statement I says there is at least one attendee whose age is a low potential outlier (below the lower fence). Since the lower fence is −7 and all ages are positive, there cannot be any ages below −7. Therefore, there are no low outliers.
- Statement II says there is at least one attendee whose age is a high potential outlier (above the upper fence). The maximum age given is 80, which is greater than 65, so there is at least one high outlier.

Assessing each option:
- Option 1: “Only I is true.” Given the analysis, Statement I is false and Statement II is true, so this option would be incorrect because it asserts I is true while II is not.
- Option 2: “Neither I nor II are true.” Since II is true (80 exceeds 65), this option is incorrect because it claims no outliers at all.
- Option 3: “Only II is true.” This aligns with the findings: I is false (no low outliers) and II is true (80 is a high outlier). This option matches the computed status of the statements.
- Option 4: “Both I and II are true.” This would require both low and high outliers to exist, which contradicts the lower fence result and the data provided.

In summary, the evaluation shows that the low-outlier statement cannot hold, while the high-outlier statement does. The reasoning hinges on the IQR, the fences derived from 1.5×IQR, and comparing the given ages to those fences. The presence of 80 clearly exceeds the upper bound, establishing a high outlier, whereas no age falls below −7, ruling out a low outlier.

The five-number summary for the ages of the attendees at the Pawnee Harvest Festival is listed below. 1 20 29 38 80 I. There is at least one attendee whose age is a low potential outlier. II. There is at least one attendee whose age is a high potential outlier.

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