Question

In Exercises 1–6, refer to the data below, which are total home game playing times (hours) for all Major League Baseball teams in a recent year (based on data from Baseball Prospectus).

236 237 238 239 241 241 242 245 245 245 246 247 247 248 248 249 250 250 250 251 252 252 253 253 258 258 258 260 262 264

Frequency Distribution For the frequency distribution from Exercise 1, find the following.

a. Class limits of the first class

b. Class boundaries of the first class

c. Class midpoint of the first class

Short answer

a. Class limits of the first class are 235 hours and 239 hours.

b. Class boundaries of the first class are 234.5 hours and 239.5 hours.

c. Class midpoint of the class is 237 hours.

Step-by-step solution

Given information

Data are given on playing times for Major League Baseball Teams.

Refer to Exercise 1, the frequency distribution for the data on playing times is given as follows:

Playing Times (in hours)	Frequency
235-239	4
240-244	3
245-249	9
250-254	8
255-259	3
260-264	3

Determine the class limits of the first class

a.

For an inclusive type frequency distribution, the class limits of a class interval are the minimum and the maximum values included in that interval.

The first class interval is 235-239.

The minimum value included in this interval is equal to 235 hours, and the maximum value included in this interval is 239 hours.

Therefore, 235 hours and 239 hours are the class limits of the first class interval.

Determine the class boundaries of the first class

b.

The value of the gap between each successive interval divided by two is subtracted from the lower limit and added to the upper limit of a class interval to obtain the class boundaries.

The gap is calculated as shown below:

$$\begin{gathered} \text{Gap}=2^{\text{nd}}\text{lower}\text{class}\text{limit}-1^{\text{st}}\text{upper}\text{class}\text{limit} \\ =240-239 \\ =1 \end{gathered}$$

The value equal to $\frac{1}{2}=0.5$is subtracted from the lower class limit of the first class interval:

$235-0.5=234.5$

The value equal to 0.5 is added to the upper-class limit of the first class interval:

$239+0.5=239.5$

Thus, the class boundaries of the first class interval are 234.5 hours and 239.5 hours.

Determine the class midpoint of the first class 

c.

The midpoint of the class interval is calculated using the given formula:

$\text{Midpoint}=\frac{\text{Lower}\text{limit}+\text{Upper}\text{limit}}{2}$

The first class interval is 235-239.

The midpoint is calculated as follows:

$$\begin{gathered} \text{Midpoint}=\frac{\text{Lower}\text{limit}+\text{Upper}\text{limit}}{2} \\ =\frac{235+239}{2} \\ =237 \end{gathered}$$

Thus, the class midpoint of the first class interval is equal to 237 hours.