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Chapter 2: Problem 22

Unclaimed Expired Prizes The number of unclaimed expired prizes (in millions of dollars) for lottery tickets bought in a sample of states is shown. Construct a frequency distribution for the data, using 5 classes. $$ \begin{array}{crcc} 28.5 & 51.7 & 19 & 5 \\ 2 & 1.2 & 14 & 14.6 \\ 0.8 & 11.6 & 3.5 & 30.1 \\ 1.7 & 1.3 & 13 & 14 \end{array} $$

Short Answer

Expert verified
The frequency distribution is already created with 5 classes, showing frequencies for each class range.

Step by step solution

01

Determine the Range

First, identify the range of the data by subtracting the smallest value from the largest value. The data set has a minimum value of 0.8 and a maximum value of 51.7. Thus, the range is: \[ \text{Range} = 51.7 - 0.8 = 50.9 \]
02

Calculate the Class Width

To determine the class width, divide the range by the number of classes. You need 5 classes, so:\[ \text{Class Width} = \frac{50.9}{5} \approx 10.18 \]Round up to the next whole number to ensure all data points are covered, giving a class width of 11.
03

Define the Classes

Start with the smallest value and create classes using the class width. The classes are: 1. 0.8 to 11.8 2. 11.8 to 22.8 3. 22.8 to 33.8 4. 33.8 to 44.8 5. 44.8 to 55.8
04

Tally the Frequencies

Go through the data and count how many values fall into each class: - 0.8 to 11.8: 5 values (0.8, 5, 2, 1.2, 1.3) - 11.8 to 22.8: 6 values (19, 14, 11.6, 14.6, 13, 14) - 22.8 to 33.8: 3 values (28.5, 30.1) - 33.8 to 44.8: 0 values - 44.8 to 55.8: 1 value (51.7)
05

Construct the Frequency Distribution

Present the frequency distribution: - 0.8 to 11.8: 5 - 11.8 to 22.8: 6 - 22.8 to 33.8: 3 - 33.8 to 44.8: 0 - 44.8 to 55.8: 1

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Key Concepts

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

Class Width
In statistics, class width is the difference between the upper and lower boundaries of any class interval in a frequency distribution. It helps divide the entire data range into logical segments, making analysis simpler.
To determine the class width, use the formula:
  • First, find the range by subtracting the minimum data value from the maximum data value.
  • Then, divide the range by the desired number of classes.

For example, in the given exercise, the range was calculated as 50.9, and with a requirement of 5 classes, the class width was computed as \[ \frac{50.9}{5} = 10.18 \approx 11\]. Always remember to round up - this ensures all values are covered. Whether you choose classes by years, dollars, or other units, the class width ensures a clear partition of your data so that each class shows consistent intervals.
Range in Statistics
The statistical range provides a clear, bounded measure of where all your data lies. It is a crucial step in creating a frequency distribution as it determines the span of the entire dataset. To find the range:
  • Look for the highest value in your data, called the maximum.
  • Identify the lowest value in your data, called the minimum.
  • The range is simply: \( \text{Range} = \text{Maximum} - \text{Minimum} \).

In the provided data, the smallest value was 0.8 and the largest was 51.7, resulting in a range of 50.9. By simply knowing the range, you can grasp how wide the data spread is, which informs how to classify the data in subsequent steps. The range gives you the full picture of variability within the dataset.
Tally Frequencies
Tally frequencies is a method used to count occurrences of data points within intervals of classes in a frequency distribution. It's helpful in organizing data and reveals patterns about how often data points appear within defined boundaries.
Here's how tallying works:
  • List your classes alongside your data.
  • For each piece of data, mark which class range it belongs to.
  • Count the total number of data points falling within each class range.

This exercise demonstrated tallying each point to its class, revealing that:
  • The class 0.8 to 11.8 held 5 values: (0.8, 5, 2, 1.2, 1.3).
  • The class 11.8 to 22.8 contained 6 values: (19, 14, 11.6, 14.6, 13, 14).
By these means, tallying converts raw data into meaningful information, showing how distribution varies across the set. Thus, it facilitates visualizing and understanding the data more coherently.

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Most popular questions from this chapter

The data show the number of years of experience the players on the Pittsburgh Steelers football team have at the beginning of the season. Draw and analyze a dot plot for the data $$ \begin{array}{rrrrrrrrr} 4 & 4 & 2 & 9 & 7 & 3 & 7 & 12 & 6 \\ 5 & 1 & 4 & 5 & 2 & 7 & 6 & 12 & 3 \\ 12 & 4 & 0 & 4 & 0 & 0 & 0 & 2 & 9 \\ 2 & 6 & 7 & 13 & 4 & 2 & 6 & 9 & 4 \\ 4 & 0 & 3 & 5 & 4 & 2 & 6 & 9 & 4 \\ 4 & 0 & 3 & 5 & 3 & 11 & 1 & 4 & 2 \\ 3 & 15 & 1 & 6 & 0 & 11 & 3 & 10 & 3 \end{array} $$

Find the class boundaries, midpoints, and widths for each class. $$ 58-62 $$

The U.S. health dollar is spent as indicated below. Construct two different types of graphs to represent the data. $$ \begin{array}{lr} \text { Government administration } & 9.7 \% \\ \text { Nursing home care } & 5.5 \\ \text { Prescription drugs } & 10.1 \\ \text { Physician and clinical services } & 20.3 \\ \text { Hospital care } & 30.5 \\ \text { Other (OTC drugs, dental, etc.) } & 23.9 \end{array} $$

The frequency distribution shows the blood glucose levels (in milligrams per deciliter) for 50 patients at a medical facility. Construct a histogram, frequency polygon, and ogive for the data. Comment on the shape of the distribution. What range of glucose levels did most patients fall into? $$ \begin{array}{lr} \text { Class limits } & \text { Frequency } \\ \hline 60-64 & 2 \\ 65-69 & 1 \\ 70-74 & 5 \\ 75-79 & 12 \\ 80-84 & 18 \\ 85-89 & 6 \\ 90-94 & 5 \\ 95-99 & \frac{1}{50} \end{array} $$

The data show the most number of home runs hit by a batter in the American League over the last 30 seasons. Construct a frequency distribution using 5 classes. Draw a histogram, a frequency polygon, and an ogive for the date, using relative frequencies. Describe the shape of the histogram. $$ \begin{array}{lll} 40 & 43 & 40 \\ 53 & 47 & 46 \\ 44 & 57 & 43 \\ 43 & 52 & 44 \\ 54 & 47 & 51 \\ 39 & 48 & 36 \\ 37 & 56 & 42 \\ 54 & 56 & 49 \\ 54 & 52 & 40 \\ 48 & 50 & 40 \end{array} $$
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