Question

question 6
1.5 pts
here is the five - number summary for a group of 100 runners in a 5 - kilometer race. the variable is the time to complete the race.
five - number summary:

• minimum: 15 minutes
• q1: 27 minutes
• median: 31 minutes
• q3: 32 minutes
• maximum: 50 minutes

are there any outliers in the runners’ finish times by the 1.5 * iqr definition?
options:

• the slowest runner is an outlier, but the fastest runner is not an outlier.
• the fastest runner is an outlier, but the slowest runner is not an outlier.
• both the slowest and the fastest runners are outliers.
• neither the slowest or the faster runner is an outlier.

Explanation:

Step1: Calculate IQR

IQR (Interquartile Range) is \( Q3 - Q1 \). Given \( Q1 = 27 \) and \( Q3 = 32 \), so \( IQR = 32 - 27 = 5 \).

Step2: Calculate lower and upper bounds for outliers

Lower bound: \( Q1 - 1.5\times IQR = 27 - 1.5\times5 = 27 - 7.5 = 19.5 \)
Upper bound: \( Q3 + 1.5\times IQR = 32 + 1.5\times5 = 32 + 7.5 = 39.5 \)

Step3: Check minimum and maximum against bounds

Minimum time is 15 minutes, which is less than 19.5 (lower bound). Maximum time is 50 minutes, which is greater than 39.5 (upper bound). So both the slowest (max) and fastest (min) runners are outliers? Wait, no: Wait, minimum is 15 (fastest runner, since lower time is faster), maximum is 50 (slowest runner, higher time is slower). Wait, let's re - check:
Fastest runner's time: 15 minutes. Lower bound is 19.5. 15 < 19.5, so fastest runner is an outlier? Wait no, wait: Wait, the five - number summary: minimum is the smallest (fastest, since less time to finish), maximum is largest (slowest, more time to finish).
Wait, lower bound for outliers (values below which are outliers) is \( Q1 - 1.5IQR = 19.5 \). The minimum is 15, which is less than 19.5, so the fastest runner (with time 15) is an outlier? Wait no, wait I think I mixed up. Wait, the formula for outliers: values less than \( Q1 - 1.5IQR \) or greater than \( Q3 + 1.5IQR \) are outliers.
So minimum time: 15. \( Q1 - 1.5IQR = 27 - 7.5 = 19.5 \). 15 < 19.5, so 15 is an outlier (fastest runner). Maximum time: 50. \( Q3 + 1.5IQR = 32+7.5 = 39.5 \). 50 > 39.5, so 50 is an outlier (slowest runner). Wait, but let's check the options. Wait, the options:
1. The slowest runner is an outlier, but the fastest runner is not an outlier.
2. The fastest runner is an outlier, but the slowest runner is not an outlier.
3. Both the slowest and the fastest runners are outliers.
4. Neither the slowest or the faster runner is an outlier.

Wait, my calculation:
\( IQR = 32 - 27 = 5 \)
Lower fence: \( 27 - 1.5\times5 = 27 - 7.5 = 19.5 \)
Upper fence: \( 32+1.5\times5 = 32 + 7.5 = 39.5 \)

Minimum value: 15 < 19.5 → so the fastest runner (15 minutes) is an outlier? Wait no, 15 is the minimum, which is the fastest time (since less time to finish). Wait, 15 is less than 19.5, so it's an outlier. Maximum value: 50 > 39.5, so the slowest runner (50 minutes) is an outlier. So both are outliers? But let's check the options again. Wait, maybe I made a mistake. Wait, Q3 is 32, Q1 is 27. IQR is 5. 1.5*IQR is 7.5. Lower bound: 27 - 7.5 = 19.5. The minimum is 15, which is below 19.5, so it's an outlier (fastest runner). Upper bound: 32 + 7.5 = 39.5. The maximum is 50, which is above 39.5, so it's an outlier (slowest runner). So both are outliers. But let's check the options. Wait, the third option is "Both the slowest and the fastest runners are outliers." But wait, maybe I messed up Q3 and Q1? Wait, the five - number summary: minimum, Q1, median, Q3, maximum. So Q1 is 27, Q3 is 32. So IQR is Q3 - Q1 = 5. Then lower fence: Q1 - 1.5*IQR = 27 - 7.5 = 19.5. Upper fence: Q3 + 1.5*IQR = 32 + 7.5 = 39.5. Minimum is 15 < 19.5 → outlier (fastest). Maximum is 50 > 39.5 → outlier (slowest). So the answer should be the third option? Wait, but let's check again. Wait, maybe the Q3 is 32 and Q1 is 27, so IQR is 5. 1.5*IQR is 7.5. Lower bound: 27 - 7.5 = 19.5. The minimum is 15, which is less than 19.5, so it's an outlier (fastest runner). Upper bound: 32 + 7.5 = 39.5. The maximum is 50, which is greater than 39.5, so it's an outlier (slowest runner). So both are outliers. So the correct option is "Both the slowest and the fastest runners are outliers."

Wait, but let's check the options again:

Option 1: The slowest runner is an outlier, but the fastest runner is not an outlier. → No, because fastest (15) is below 19.5.

Option 2: The fastest runner is an outlier, but the slowest runner is not an outlier. → No, slowest (50) is above 39.5.

Option 3: Both the slowest and the fastest runners are outliers. → Yes.

Option 4: Neither the slowest or the faster runner is an outlier. → No.

So the correct answer is the third option.

Answer:

C. Both the slowest and the fastest runners are outliers. (Assuming the options are labeled A, B, C, D with C being the third option as per the order: first option is A, second B, third C, fourth D) Wait, in the given options, the third option is "Both the slowest and the fastest runners are outliers." So the answer is the option with text "Both the slowest and the fastest runners are outliers."