Question

The article "Determination of Most Representative Subdivision" (Journal of Energy Engineering [1993]: 43 - 55) gave data on various characteristics of subdivisions that could be used in deciding whether to provide electrical power using overhead lines or underground lines. Data on the variable \(x=\) total length of streets within a subdivision are as follows: \(\begin{array}{rrrrrrrr}1280 & 5320 & 4390 & 2100 & 1240 & 3060 & 4770 & 1050 \\\ 360 & 3330 & 3380 & 340 & 1000 & 960 & 1320 & 530 \\ 3350 & 540 & 3870 & 1250 & 2400 & 960 & 1120 & 2120 \\ 450 & 2250 & 2320 & 2400 & 3150 & 5700 & 5220 & 500 \\ 1850 & 2460 & 5850 & 2700 & 2730 & 1670 & 100 & 5770 \\ 3150 & 1890 & 510 & 240 & 396 & 1419 & 2109 & \end{array}\) a. Construct a stem-and-leaf display for these data using the thousands digit as the stem. Comment on the various features of the display. b. Construct a histogram using class boundaries of 0 to 1000,1000 to 2000 , and so on. How would you describe the shape of the histogram? c. What proportion of subdivisions has total length less than 2000 ? between 2000 and 4000 ?

Short answer

In order to answer these questions precisely, the raw data needs to be processed. It's important not to rush into plotting the data or calculating proportions as without properly handling the data set, the insights drawn from these steps may be faulty. After carrying out each step carefully, one can see the characteristics of the data and make meaningful interpretations.

Step-by-step solution

Construct the Stem-and-Leaf Display

A stem-and-leaf display consists of a stem, typically on the left, which contains the leading digit(s), and a leaf, on the right, which contains the trailing digit(s). In this case, the stem is the thousands digit and the leaf is all remaining digits. Order the leaf digits in each row once you have separated the leading and trailing digits.

Comment on the Features of the Display

Look for patterns and trends in the stem-and-leaf plot. These can include peak values, outliers, symmetry, skewness, etc.

Construct the Histogram

A histogram graph uses bars to represent frequency distribution. Here the classes would be 0-1000, 1000-2000, 2000-3000 and so on. For each class, count how many data values fall into each one and represent this on the y-axis. The x-axis will contain the classes ranges.

Describe the Shape of the Histogram

Examine the histogram for features such as symmetry, skewness, unimodality or multimodality, and kurtosis.

Calculate the Proportions

To calculate the proportion of subdivisions with total length less than 2000, divide the number of suburbs with a total street length of less than 2000 by the total number of suburbs. Similarly calculate the proportion of suburbs with a total length between 2000 and 4000.