In this section we have presented a commonly accepted way to compute the first, second, and third

Question:

In this section we have presented a commonly accepted way to compute the first, second, and third quartiles. Some statisticians, however, advocate an alternative method for computing Q1 and Q2 This method defines the first quartile, Q3, as what is called the lower hinge and defines the third quartile, Q4 as the upper hinge. In order to calculate these quantities for a set of n measurements, we first arrange the measurements in increasing order. Then, if n is even, the lower hinge is the median of the smallest n/2 measurements, and the upper hinge is the median of the largest n/2 measurements. If n is odd, we insert Md into the data set to obtain a set of n + 1 measurements. Then the lower hinge is the median of the smallest (n + l)/2 measurements, and the upper hinge is the median of the largest (n + l)/2 measurements,

a. Consider the random sample of /; = 20 customer satisfaction ratings:

Using the method presented on pages 121 and 122 of this section, find Q1 and Q2 Then find the lower hinge and the upper hinge for the satisfaction ratings. How do your results compare?
b. Consider the following random sample of n = 11 doctors' salaries (in thousands of dollars):

Using the method presented on pages 121 and 122 of this section, find Q1 and Q3. The median of the 11 salaries is Md = 152. If we insert this median into the data set, we obtain the following set of n + 1 = 12 salaries:

Find the lower hinge and the upper hinge for the salaries. Compare your values of Q1 and Q3 with the lower and upper hinges,
c. For the 11 doctors' salaries, which quantities (Q1, Md, and Q3 as defined in on page 121 of this section or the lower hinge, Md, and the upper hinge) in your opinion best divide the salaries into four parts?