Question

1. here are the exam scores for the 15 students in mr. kirk’s statistics class: 90 95 72 75 75 78 81 83 85 89 90 90 90 91 95 95 98. karen was at the 20th percentile of the distribution. what score did karen earn on the exam? a. 75 b. 78 c. 81 d. 83

Explanation:

Step1: Order the scores

First, we order the exam scores from least to greatest. The scores are: 72, 75, 75, 78, 81, 83, 85, 89, 90, 90, 90, 91, 95, 95, 98. Wait, no, wait, the original data: let's list all 15 scores. Wait, the given scores: wait, the problem says "Here are the exam scores for the 15 students": let's check the numbers. Wait, the user's image: let's parse the scores. Wait, the numbers are: 95, 98, 72, 75, 75, 78, 81, 83, 85, 89, 90, 90, 90, 91, 95? Wait, no, maybe I miscounted. Wait, 15 scores. Let's list them properly. Let's see: the first line (maybe) has 95, 98? Wait, no, the text: "95 98 72 75 75 78 81 83 85 89 90 90 90 91 95" – no, that's 14? Wait, maybe a typo. Wait, the problem says 15 students. Wait, maybe the original scores are: 72, 75, 75, 78, 81, 83, 85, 89, 90, 90, 90, 91, 95, 95, 98 – no, that's 14. Wait, maybe I missed one. Wait, the first number: maybe 95, 98, 72, 75, 75, 78, 81, 83, 85, 89, 90, 90, 90, 91, 95 – no, 15? Wait, 95 (1), 98 (2), 72 (3), 75 (4), 75 (5), 78 (6), 81 (7), 83 (8), 85 (9), 89 (10), 90 (11), 90 (12), 90 (13), 91 (14), 95 (15). Yes, 15. Now, order them from least to greatest: 72, 75, 75, 78, 81, 83, 85, 89, 90, 90, 90, 91, 95, 95, 98. Wait, no: 72 (1), 75 (2), 75 (3), 78 (4), 81 (5), 83 (6), 85 (7), 89 (8), 90 (9), 90 (10), 90 (11), 91 (12), 95 (13), 95 (14), 98 (15). Now, to find the 20th percentile. The formula for percentile is: \( i = \frac{p}{100} \times n \), where \( p \) is the percentile (20), \( n \) is the number of data points (15). So \( i = \frac{20}{100} \times 15 = 3 \). Since \( i \) is an integer, the 20th percentile is the average of the \( i \)-th and \( (i+1) \)-th values? Wait, no, different methods. Wait, the percentile formula: if \( i \) is an integer, some methods take the average of the \( i \)-th and \( (i+1) \)-th, but sometimes, for discrete data, we use the \( i \)-th value when \( i \) is a whole number. Wait, let's check the formula. The position \( i \) is calculated as \( i = \frac{p}{100} \times (n + 1) \) for some methods, but the most common in education is \( i = \frac{p}{100} \times n \), and if \( i \) is not an integer, we round up. Wait, let's confirm. The number of data points \( n = 15 \). For the 20th percentile, \( i = 0.20 \times 15 = 3 \). So the 3rd value in the ordered list. Wait, ordered list: 72 (1), 75 (2), 75 (3), 78 (4), 81 (5), 83 (6), 85 (7), 89 (8), 90 (9), 90 (10), 90 (11), 91 (12), 95 (13), 95 (14), 98 (15). Wait, no, wait, the ordered list should be from lowest to highest. Let's reorder correctly: 72, 75, 75, 78, 81, 83, 85, 89, 90, 90, 90, 91, 95, 95, 98. So positions: 1:72, 2:75, 3:75, 4:78, 5:81, 6:83, 7:85, 8:89, 9:90, 10:90, 11:90, 12:91, 13:95, 14:95, 15:98. Now, \( i = 0.20 \times 15 = 3 \). So the 3rd value is 75? But wait, the options include 75 (a), 78 (b), 81 (c), 83 (d). Wait, maybe I ordered wrong. Wait, maybe the original scores are different. Wait, the user's image: let's re-express the scores. The problem says: "Here are the exam scores for the 15 students in Mr. Kirk’s statistics class: 95, 98, 72, 75, 75, 78, 81, 83, 85, 89, 90, 90, 90, 91, 95" – no, that's 14. Wait, maybe a typo, and one more score. Wait, maybe the first score is 95, 98, 72, 75, 75, 78, 81, 83, 85, 89, 90, 90, 90, 91, 95, 95? No, 15. Wait, maybe I made a mistake in counting. Let's count: 95 (1), 98 (2), 72 (3), 75 (4), 75 (5), 78 (6), 81 (7), 83 (8), 85 (9), 89 (10), 90 (11), 90 (12), 90 (13), 91 (14), 95 (15). Yes, 15. Now, ordered from least to greatest: 72, 75, 75, 78, 81, 83, 85, 89, 90, 90, 90, 91, 95, 95, 98. Wait, no: 72 (1), 75 (2), 75 (3), 78 (4), 81 (5), 83 (6), 85 (7), 89 (8), 90 (9), 90 (10), 90 (11), 91 (12), 95 (13), 95 (14), 98 (15). Now, \( i = 0.20 \times 15 = 3 \). So the 3rd value is 75. But wait, the options have 75 as option a. But let's check another method. The percentile formula: \( i = \frac{p}{100} \times (n + 1) = 0.20 \times 16 = 3.2 \). Then we round up to 4, so the 4th value. The 4th value is 78. Wait, now I'm confused. Different percentile methods. Let's check the standard method for percentiles in a dataset. The most common method for finding the \( p \)-th percentile is:

1. Order the data from smallest to largest.
2. Calculate the index \( i = \frac{p}{100} \times n \), where \( n \) is the number of data points.
3. If \( i \) is an integer, the \( p \)-th percentile is the average of the \( i \)-th and \( (i+1) \)-th values.
4. If \( i \) is not an integer, round \( i \) up to the next integer, and the \( p \)-th percentile is the value at that position.

Wait, no, actually, there are multiple methods (e.g., Excel uses a different method). Let's check with \( n = 15 \), \( p = 20 \).

Method 1: \( i = \frac{20}{100} \times 15 = 3 \). Since \( i \) is an integer, the 20th percentile is the average of the 3rd and 4th values. The 3rd value is 75, the 4th is 78. Average is \( \frac{75 + 78}{2} = 76.5 \), but that's not an option.

Method 2: \( i = \frac{20}{100} \times (15 + 1) = 3.2 \). Round up to 4, so the 4th value is 78. Which is option b.

Ah, maybe the method used here is the one where \( i = \frac{p}{100} \times (n + 1) \). Let's confirm. In some textbooks, the percentile formula is \( i = \frac{p}{100} \times (n + 1) \). So with \( n = 15 \), \( i = 0.20 \times 16 = 3.2 \). We take the ceiling of 3.2, which is 4. So the 4th value in the ordered list. The ordered list is: 72 (1), 75 (2), 75 (3), 78 (4), 81 (5), 83 (6), 85 (7), 89 (8), 90 (9), 90 (10), 90 (11), 91 (12), 95 (13), 95 (14), 98 (15). So the 4th value is 78, which is option b.

Step2: Confirm the ordered list

Wait, let's reorder the scores correctly. The given scores: 95, 98, 72, 75, 75, 78, 81, 83, 85, 89, 90, 90, 90, 91, 95. Wait, no, that's 14 scores. Wait, maybe the first score is 95, 98, 72, 75, 75, 78, 81, 83, 85, 89, 90, 90, 90, 91, 95, 95 – 15. Then ordered: 72, 75, 75, 78, 81, 83, 85, 89, 90, 90, 90, 91, 95, 95, 98. Yes, 15. So positions 1-15.

Using the formula \( i = \frac{p}{100} \times (n + 1) \), \( i = 0.20 \times 16 = 3.2 \). So we take the 4th value (since 3.2 rounded up is 4). The 4th value is 78. So Karen's score is 78.

Answer:

b. 78