Question

question 19
consider the boxplot below.
24 27 29 36 42
a. what quarter has the smallest spread of data? second
b. what is that spread? 2
c. what quarter has the largest spread of data? third
d. what is that spread? 7
e. find the inter quartile range (iqr): 9
f. which interval has the most data in it? 30 - 36
g. what value could represent the 55th percentile? ?

Explanation:

Step1: Recall the properties of box - plots

In a box - plot, the spread of a quarter (inter - quartile spread for each quarter) can be calculated as the difference between the upper and lower bounds of that quarter. The first quarter (Q1 - minimum) spread: \(27 - 24=3\). The second quarter (Q2 - Q1) spread: \(29 - 27 = 2\). The third quarter (Q3 - Q2) spread: \(36 - 29=7\). The fourth quarter (maximum - Q3) spread: \(42 - 36 = 6\).

Step2: Calculate the inter - quartile range (IQR)

The inter - quartile range \(IQR=Q3 - Q1\). Given \(Q1 = 27\) and \(Q3 = 36\), then \(IQR=36 - 27=9\).

Step3: Determine the 55th percentile

The 50th percentile is \(Q2 = 29\) and the 75th percentile is \(Q3 = 36\). Since \(55\%\) is between \(50\%\) and \(75\%\), we can use a linear interpolation formula. Let \(x\) be the 55th percentile. The formula for linear interpolation between two points \((p_1,y_1)\) and \((p_2,y_2)\) is \(y=y_1+\frac{p - p_1}{p_2 - p_1}(y_2 - y_1)\). Here, \(p_1 = 50\), \(y_1=29\), \(p_2 = 75\), \(y_2 = 36\), and \(p = 55\).

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Answer:

a. Second
b. \(2\)
c. Third
d. \(7\)
e. \(9\)
f. \(30 - 36\)
g. \(30.4\)