Question

Exercises 2.137 to 2.140 each describe a sample. The information given includes the five number summary, the sample size, and the largest and smallest data values in the tails of the distribution. In each case: (a) Clearly identify any outliers, using the IQR method. (b) Draw a boxplot. Five number summary: \((210,260,270,300,\) 320)\(; n=500\) Tails: \(210,215,217,221,225, \ldots, 318,319,319,319,\) 320,320

Short answer

The Interquartile Range (IQR) is 40. There are no outliers in the data as all points are within the Lower and Upper boundaries, which are 200 and 360, respectively. The boxplot will have a box from 260 to 300, a line at the median (270), and whiskers extending to the minimum (210) and maximum (320).

Step-by-step solution

Compute the Interquartile Range (IQR)

The IQR is calculated as the difference between the third quartile (Q3) and the first quartile (Q1). From the five-point summary in the question, Q1 = 260 Q3 = 300 So, IQR = \(Q3 - Q1 = 300 - 260 = 40\)

Identify potential outliers

To identify potential outliers, we need to calculate the lower boundary and the upper boundary. Any data point that lies below the lower boundary or above the upper boundary is considered an outlier. The lower boundary is defined as Q1 - 1.5*IQR and the upper boundary is Q3 + 1.5*IQR. Lower Boundary = Q1 - 1.5*IQR = 260 - 1.5*40 = 200 Upper Boundary = Q3 + 1.5*IQR = 300 + 1.5*40 = 360 Looking at the tails, we can see that there is no value falling outside these boundaries. Hence, there are no outliers.

Draw a boxplot

A boxplot is a graphical representation of the five number summary. The 'box' is drawn from Q1 to Q3 with a line inside the box at Q2 (the median). The 'whiskers' extend from Q1 to the minimum value and from Q3 to the maximum value. If there are outliers, they're represented as separate points outside the whiskers. In this case, there are no outliers.All data points are within these limits, so the data can be represented with a box stretching from 260 to 300, with a line at 270 (the median), and whiskers extending to 210 (the minimum) and 320 (the maximum). Details for the plotting would not be possible in a JSON format or within this text-based solution, so the content stops here.