Question

1. make a box-and-whisker plot for the data. what is the upper quartile value? 56 32 48 52 51 53 48 38 35 42 40 46 54 50

52
56
48
40

Explanation:

Step1: Order the data

First, we order the given data set: \(32, 35, 38, 40, 42, 46, 48, 48, 50, 51, 52, 53, 54, 56\)

Step2: Find the median (Q2)

The number of data points \(n = 14\) (even). The median is the average of the 7th and 8th values. The 7th value is \(48\) and the 8th value is \(48\), so the median \(Q2=\frac{48 + 48}{2}=48\)

Step3: Find the upper half of the data

The upper half of the data (values above the median) is: \(50, 51, 52, 53, 54, 56\)

Step4: Find the upper quartile (Q3)

For the upper half, \(n = 6\) (even). The upper quartile is the average of the 3rd and 4th values in this upper half. The 3rd value is \(52\) and the 4th value is \(53\)? Wait, no, wait. Wait, the upper half after median: the original data after the 8th term (since median is between 7th and 8th, the upper half is from 9th term? Wait, no, let's correct. When \(n\) is even, the data is split into two halves: first \(n/2 = 7\) values? Wait, no, for \(n = 14\), the first half is the first 7 values: \(32, 35, 38, 40, 42, 46, 48\) and the second half is the last 7 values: \(48, 50, 51, 52, 53, 54, 56\). Wait, I made a mistake earlier. So the upper half (second half) has 7 values. Then the median of the upper half (Q3) is the 4th value of the upper half. Let's list the upper half: \(48, 50, 51, 52, 53, 54, 56\). The 4th value is \(52\). Wait, let's re - order the data correctly:

Original data: \(32, 35, 38, 40, 42, 46, 48, 48, 50, 51, 52, 53, 54, 56\)

Number of data points \(n = 14\). The median (Q2) is the average of the \(\frac{n}{2}=7\)th and \(\frac{n}{2}+ 1=8\)th terms. The 7th term is \(48\), the 8th term is \(48\), so \(Q2 = 48\)

The lower half (data below Q2) is the first 7 terms: \(32, 35, 38, 40, 42, 46, 48\)

The upper half (data above Q2) is the last 7 terms: \(48, 50, 51, 52, 53, 54, 56\)

To find Q3, we find the median of the upper half. Since the upper half has \(n = 7\) (odd) terms, the median is the \(\frac{7 + 1}{2}=4\)th term of the upper half.

The upper half ordered: \(48, 50, 51, 52, 53, 54, 56\)

The 4th term is \(52\).

Answer:

The upper quartile value is \(52\), so the correct option is the one with \(52\) (the first option: \(52\))