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Chapter 2: Problem 20

Salaries of Governors Here are the salaries (in dollars) of the governors of 25 randomly selected states. Construct a grouped frequency distribution with 6 classes. $$ \begin{aligned} &\begin{array}{rrrrr} 112,895 & 117,312 & 140,533 & 110,000 & 115,331 \\ 95,000 & 177,500 & 120,303 & 139,590 & 150,000 \\ 173,987 & 130,000 & 133,821 & 144,269 & 142,542 \\ 150,000 & 145,885 & 105,000 & 93,600 & 166,891 \\ 130,273 & 70,000 & 113,834 & 117,817 & 137,092 \end{array}\\\ &\text { Source: World Almanac } \end{aligned} $$

Short Answer

Expert verified
The frequency distribution consists of 6 classes, with frequencies 1, 1, 6, 7, 6, and 4 respectively.

Step by step solution

01

Determine Range of Salaries

To construct a frequency distribution, start by finding the range of the data. The range is the difference between the highest and lowest salary. The highest salary is \(177,500 and the lowest is \)70,000. Thus, the range is:\[177,500 - 70,000 = 107,500\]
02

Determine Class Width

Divide the range by the number of classes to determine the class width. Round up to the nearest whole number. For 6 classes, the class width is:\[\text{Class Width} = \frac{107,500}{6} \approx 17,917\]Rounding up, we choose a class width of 18,000.
03

Define Class Limits

Start the first class below the smallest data point or at a convenient number near it. Start at $70,000 and use the class width to define the class limits as follows: - Class 1: $70,000 to $(70,000 + 18,000 - 1) = $87,999 - Class 2: $88,000 to $105,999 - Class 3: $106,000 to $123,999 - Class 4: $124,000 to $141,999 - Class 5: $142,000 to $159,999 - Class 6: $160,000 to $177,999.
04

Tally Frequencies

Go through each salary and tally the number of salaries that fall into each class range. This gives you the frequency distribution: - Class 1 (70,000 - 87,999): 1 - Class 2 (88,000 - 105,999): 1 - Class 3 (106,000 - 123,999): 6 - Class 4 (124,000 - 141,999): 7 - Class 5 (142,000 - 159,999): 6 - Class 6 (160,000 - 177,999): 4.
05

Record the Frequency Distribution

Using the tallies from Step 4, the grouped frequency distribution is:\[\begin{array}{|c|c|}\hline\text{Class Range} & \text{Frequency} \\hline70,000 - 87,999 & 1 \88,000 - 105,999 & 1 \106,000 - 123,999 & 6 \124,000 - 141,999 & 7 \142,000 - 159,999 & 6 \160,000 - 177,999 & 4 \\hline\end{array}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Class Width
Understanding class width is essential for creating an accurate frequency distribution. Class width refers to the range of values covered by each class in a grouped frequency distribution. The formula to calculate the class width is:
  • Subtract the lowest data point from the highest data point to get the range.
  • Divide the range by the desired number of classes.
  • Round up to the nearest whole number to find the class width, ensuring all data points are included within the specified classes.
For example, if you have a salary range of 107,500 dollars and six classes, you divide 107,500 by 6, which gives you a class width approximation of 17,917. Rounding this up results in a class width of 18,000 dollars, which ensures the distribution encompasses all values accurately.
Range of Data
When constructing a frequency distribution, calculating the range of the data is the first crucial step. The range provides the span of the data set by identifying the difference between the highest and lowest values. This calculation helps determine appropriate class intervals.

Here's how you calculate the range:
  • Identify the highest data point in the set. In our case, it's 177,500 dollars.
  • Identify the lowest data point, which is 70,000 dollars.
  • Subtract the lowest value from the highest value: 177,500 - 70,000 = 107,500 dollars.
With this complete range of salaries, you have the essential span needed to determine class intervals, ensuring all data points have a place in the distribution.
Frequency Distribution
A frequency distribution provides a snapshot of how data is dispersed across different intervals or classes. It's a powerful tool to understand data distribution at a glance by tallying how many data points fall within each specified range.

To create a frequency distribution:
  • Set the class limits based on the class width, ensuring continuous coverage of the entire data range.
  • Tally each data point within its respective class range.
  • Record the frequency, or count of data points, in each class.
This categorization into various classes helps identify patterns, trends, and deviations in data. For the governor salaries, this exercise defines six classes covering ranges from 70,000 to 177,999 dollars, with the number of salaries in each class recorded to visualize the distribution effectively.
Class Limits
Class limits are boundaries that separate classes in a frequency distribution, defining where a class starts and ends. They ensure every data point fits within a class without overlapping or leaving gaps between classes.

To set class limits:
  • Start the first class limit at or just below the smallest data point. For example, at 70,000 dollars in the case of governor salaries.
  • Use the calculated class width to establish the subsequent limits. For instance, add the class width (18,000 dollars) to the lower limit of the previous class to set the upper limit.
  • Adjust for overlap by subtracting 1 from the upper result to prevent data falling into two classes simultaneously.
This setup allows for a clear, organized grouping of data points, making the information accessible and understandable. The smooth transition between class limits ensures that data is categorized efficiently.

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Most popular questions from this chapter

A listing of calories per 1 ounce of selected salad dressings (not fat-free) is given below. Construct a stem and leaf plot for the data. $$ \begin{array}{llllllllll} 100 & 130 & 130 & 130 & 110 & 110 & 120 & 130 & 140 & 100 \\ 140 & 170 & 160 & 130 & 160 & 120 & 150 & 100 & 145 & 145 \\ 145 & 115 & 120 & 100 & 120 & 160 & 140 & 120 & 180 & 100 \\ 160 & 120 & 140 & 150 & 190 & 150 & 180 & 160 & & \end{array} $$

Show frequency distributions that are incorrectly constructed. State the reasons why they are wrong. $$ \begin{array}{cc} \text { Class } & \text { Frequency } \\ \hline 5-9 & 1 \\ 9-13 & 2 \\ 13-17 & 5 \\ 17-20 & 6 \\ 20-24 & 3 \end{array} $$

The data show the lengths (in hundreds of miles) of major rivers in South America and Europe. Construct a back-to-back stem and leaf plot, and compare the distributions. $$ \begin{array}{llrl|rrrr} &&&{\text { South America }} && {\text { Europe }} \\ \hline 39 & 21 & 10 & 10 & 5 & 12 & 7 & 6 \\ 11 & 10 & 2 & 10 & 5 & 5 & 4 & 6 \\ 10 & 14 & 10 & 12 & 18 & 5 & 13 & 9 & \\ 17 & 15 & 10 & & 14 & 6 & 6 & 11 & \\ 15 & 25 & 16 & & 8 & 6 & 3 & 4 & \end{array} $$

The following data show where children obtain guns for committing crimes. Draw and analyze a pie graph for the data. $$ \begin{array}{lcccc} \text { Source Friend } & \text { Family } & \text { Street } & \text { Gun or Pawn Shop } & \text { Other } \\ \hline \text { Number } & 24 & 15 & 9 & 9 & 6 \end{array} $$

Ages of Declaration of Independence Signers The ages of the signers of the Declaration of Independence are shown. (Age is approximate since only the birth year appeared in the source, and one has been omitted since his birth year is unknown.) Construct a grouped frequency distribution and a cumulative frequency distribution for the data, using 7 classes. \(\begin{array}{llllllllllll}41 & 54 & 47 & 40 & 39 & 35 & 50 & 37 & 49 & 42 & 70 & 32 \\ 44 & 52 & 39 & 50 & 40 & 30 & 34 & 69 & 39 & 45 & 33 & 42 \\ 44 & 63 & 60 & 27 & 42 & 34 & 50 & 42 & 52 & 38 & 36 & 45 \\ 35 & 43 & 48 & 46 & 31 & 27 & 55 & 63 & 46 & 33 & 60 & 62 \\ 35 & 46 & 45 & 34 & 53 & 50 & 50 & & & & & \end{array}\) Source: The Universal Almanac.
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