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Chapter 1: Problem 10

Use the information given to find a convenient class width. Then list the class boundaries that can be used to create a relative frequency histogram. 6 classes for \(n=20\) measurements; minimum value \(=25.5 ;\) maximum value \(=76.8\)

Short Answer

Expert verified
Answer: The class width is 9, and the class boundaries are (25.5, 34.5), (34.5, 43.5), (43.5, 52.5), (52.5, 61.5), (61.5, 70.5), (70.5, 79.5).

Step by step solution

01

Find the range

To find the range, subtract the minimum value from the maximum value: Range = Maximum value - Minimum value Range = \(76.8 - 25.5 = 51.3\)
02

Determine the class width

To find the class width, divide the range by the number of classes: Class width = Range / Number of classes Class width = \(51.3 / 6 \approx 8.55\) We can round up the class width to 9 for convenience.
03

Find the class boundaries

To find the class boundaries, we will start with the minimum value and add the class width to find the upper boundary of the first class. Then, we will continue to add the class width to find the remaining class boundaries. 1. Lower boundary = Minimum value, Upper boundary = Minimum value + Class width - Lower boundary = 25.5, Upper boundary = \(25.5 + 9 = 34.5\) 2. Lower boundary = Previous upper boundary, Upper boundary = Previous upper boundary + Class width - Lower boundary = 34.5, Upper boundary = \(34.5 + 9 = 43.5\) 3. Lower boundary = 43.5, Upper boundary = \(43.5 + 9 = 52.5\) 4. Lower boundary = 52.5, Upper boundary = \(52.5 + 9 = 61.5\) 5. Lower boundary = 61.5, Upper boundary = \(61.5 + 9 = 70.5\) 6. Lower boundary = 70.5, Upper boundary = \(70.5 + 9 = 79.5\) The class boundaries are: (25.5, 34.5), (34.5, 43.5), (43.5, 52.5), (52.5, 61.5), (61.5, 70.5), (70.5, 79.5).

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Key Concepts

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

Class Width Calculation
When organizing data into a frequency distribution, such as a histogram, 'class width' is of fundamental importance. Essentially, the class width is the interval size between the lower boundary of the first class and the lower boundary of the second class. Calculating it correctly ensures that the data is evenly distributed across the histogram without any gaps or overlaps between classes.

To calculate class width, you follow a simple formula: divide the range of the data set by the desired number of classes. The range is the difference between the largest and smallest values. In the given exercise, the range is found to be 51.3 and the number of classes is 6, resulting in a class width of approximately 8.55. For practical purposes, rounding up to the nearest whole number often makes for a more user-friendly histogram, so we adjust it to 9.
Range in Statistics
The range is a basic statistical measure that indicates the spread or dispersion of a data set. It provides a quick glimpse into the variability of the data by showing the difference between the highest (maximum) and the lowest (minimum) values. To calculate the range, you simply subtract the minimum value from the maximum value.

In our example, the range is calculated from the given data set where the smallest measurement is 25.5 and the largest is 76.8. Subtracting these figures, the range is determined to be 51.3, which directly informs us about the extent of variation in the measurements. While this number is simple to find, it's sensitive to outliers and doesn't give information about the distribution of data values between the extremes.
Class Boundaries
Class boundaries are the actual edges of the classes that contain the data values in a histogram. They help in distinguishing where one class ends and the next begins, thereby preventing any ambiguity about which class an individual data value belongs to. When setting up a histogram, it is essential to define these boundaries clearly to ensure that each data point is accounted for in a single, appropriate category.

In the context of this exercise, the boundaries are determined by starting at the minimum value and consistently adding the class width of 9 to obtain subsequent boundaries. For example, with a lower boundary of 25.5 for the first class, the upper boundary is calculated to be 34.5. This process repeats until all class boundaries are established, ensuring no overlap and that the entire range of data is covered.

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Most popular questions from this chapter

Construct a relative frequency histogram for these 50 measurements using classes starting at 1.6 with a class width of .5. Then answer the questions. $$\begin{array}{llllllllll}3.1 & 4.9 & 2.8 & 3.6 & 2.5 & 4.5 & 3.5 & 3.7 & 4.1 & 4.9 \\ 2.9 & 2.1 & 3.5 & 4.0 & 3.7 & 2.7 & 4.0 & 4.4 & 3.7 & 4.2 \\ 3.8 & 6.2 & 2.5 & 2.9 & 2.8 & 5.1 & 1.8 & 5.6 & 2.2 & 3.4 \\ 2.5 & 3.6 & 5.1 & 4.8 & 1.6 & 3.6 & 6.1 & 4.7 & 3.9 & 3.9 \\ 4.3 & 5.7 & 3.7 & 4.6 & 4.0 & 5.6 & 4.9 & 4.2 & 3.1 & 3.9\end{array}$$ How would you describe the shape of the distribution?

Determine whether the data collected represents a population or a sample. The income distribution of the top \(10 \%\) of wage earners in the United States is determined using data from the Internal Revenue Service.

Are the variables in Exercises 6-9 qualitative or quantitative? Number of students in a first-grade classroom.

The social networking site Facebook has grown rapidly in the last 10 years. The following table shows the average number of daily users (in millions) as it has grown from 2010 to 2017 in different regions in the world. \(^{3}\) $$\begin{array}{lrc}\hline \text { Region } & 2010 & 2017 \\\\\hline \text { United States/Canada } & 99 & 183 \\\\\text { Europe } & 107 & 271 \\\\\text { Asia } & 64 & 453 \\\\\text { Rest of the world } & 58 & 419 \\ \hline \text { Total } & 328 & 1,326 \\\\\hline\end{array}$$ How would you describe the changes in the distribution of average daily users during this 7 -year period?

The ages (in months) at which 50 children were first enrolled in a preschool are listed as follows. $$ \begin{array}{llllllllll} 38 & 40 & 30 & 35 & 39 & 40 & 48 & 36 & 31 & 36 \\ 47 & 35 & 34 & 43 & 41 & 36 & 41 & 43 & 48 & 40 \\ 32 & 34 & 41 & 30 & 46 & 35 & 40 & 30 & 46 & 37 \\ 55 & 39 & 33 & 32 & 32 & 45 & 42 & 41 & 36 & 50 \\ 42 & 50 & 37 & 39 & 33 & 45 & 38 & 46 & 36 & 31 \end{array} $$ a. Construct a relative frequency histogram for these data. Start the lower boundary of the first class at 30 and use a class width of 5 months. b. What proportion of the children were 35 months or older, but less than 45 months of age when first enrolled in preschool? c. If one child were selected at random from this group of children, what is the probability that the child was less than 50 months old when first enrolled in preschool?
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