Question

to gather data for a statistics project, a student asked 10 friends how many hours of sleep they got on the previous night. the data are shown in the following list: 7, 6, 5, 9, 3, 4, 7, 9, 5, 8. what is the interquartile range (iqr) of the number of hours of sleep shown in the list? options: a 3 hours, b 4 hours, c 6 hours, d 6.3 hours, e 6.5 hours.

Explanation:

Step1: Order the data

First, we order the given data: \(3, 4, 5, 5, 6, 7, 7, 8, 9, 9\) (wait, wait, original data: let's check the list again. The data is: 7, 6, 5, 9, 3, 4, 7, 9, 5, 8. So ordering them: \(3, 4, 5, 5, 6, 7, 7, 8, 9, 9\). Wait, no, count the numbers: 3,4,5,5,6,7,7,8,9,9? Wait, original data points: 7,6,5,9,3,4,7,9,5,8. Let's list them: 3,4,5,5,6,7,7,8,9,9? Wait, no, 3,4,5,5,6,7,7,8,9,9? Wait, 3 (1), 4 (2), 5 (3), 5 (4), 6 (5), 7 (6), 7 (7), 8 (8), 9 (9), 9 (10). Yes, 10 data points.

Step2: Find Q1 and Q3

For a set with \(n = 10\) (even), the median is the average of the 5th and 6th terms. The 5th term is 6, 6th term is 7, so median \(M=\frac{6 + 7}{2}=6.5\). But for IQR, we split the data into lower half and upper half. Lower half: first 5 terms: \(3, 4, 5, 5, 6\). Upper half: last 5 terms: \(7, 7, 8, 9, 9\).

Q1 is the median of the lower half. Lower half has 5 terms (odd), so Q1 is the 3rd term: \(5\).

Q3 is the median of the upper half. Upper half has 5 terms (odd), so Q3 is the 3rd term: \(8\).

Step3: Calculate IQR

IQR is \(Q3 - Q1\). So \(IQR = 8 - 5 = 3\)? Wait, no, wait, wait, I think I made a mistake. Wait, original data: let's re - order correctly. The data is: 3,4,5,5,6,7,7,8,9,9? Wait, no, the original data list is: 7,6,5,9,3,4,7,9,5,8. Let's sort them: 3,4,5,5,6,7,7,8,9,9. Yes. So lower half: first 5 numbers: 3,4,5,5,6. Median of lower half (Q1) is the 3rd number: 5. Upper half: last 5 numbers: 7,7,8,9,9. Median of upper half (Q3) is the 3rd number: 8. Then IQR = 8 - 5 = 3? But the options have A as 3 hours. Wait, but wait, maybe I messed up the splitting. Wait, when \(n\) is even, the lower half is the first \(n/2\) terms and upper half is the last \(n/2\) terms. So \(n = 10\), lower half: first 5 terms (positions 1 - 5): 3,4,5,5,6. Upper half: positions 6 - 10: 7,7,8,9,9. So Q1 is median of lower half: (5th term + 4th term)/2? Wait, no, for odd number of terms in the half, it's the middle term. Wait, lower half has 5 terms (odd), so median is the 3rd term: 5. Upper half has 5 terms (odd), median is 3rd term: 8. So IQR = 8 - 5 = 3. So the answer should be A. 3 hours. Wait, but let's check again. Wait, maybe I made a mistake in the data. Let's list all data points: 3,4,5,5,6,7,7,8,9,9. So Q1: the median of the first 5 numbers (3,4,5,5,6) is 5. Q3: median of last 5 numbers (7,7,8,9,9) is 8. IQR = 8 - 5 = 3. So the answer is 3 hours, which is option A.

Answer:

A. 3 hours