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Chapter 2: Problem 14

Consider the following data. \(8.9 \quad 10.2 \quad 11.5 \quad 7.8 \quad 10.0 \quad 12.2 \quad 13.5 \quad 14.1 \quad 10.0\) \(\begin{array}{llllllll}6.8 & 9.5 & 11.5 & 11.2 & 14.9 & 7.5 & 10.0 & 6.0 & 15.8 & 11.5\end{array}\) a. \(\quad\) Construct a dot plot. b. Construct a frequency distribution. c. Construct a percent frequency distribution.

Short Answer

Expert verified
a. Create a dot plot using the data values. b. Organize data into frequency distribution. c. Calculate percent frequencies, considering total counts.

Step by step solution

01

Construct the Dot Plot

First, list the unique values from the data set in ascending order, then count how many times each value appears. Place a dot above the corresponding number on a number line for each occurrence. These are the unique values in order: 6.0, 6.8, 7.5, 7.8, 8.9, 9.5, 10.0, 10.2, 11.2, 11.5, 12.2, 13.5, 14.1, 14.9, 15.8. Plot each value with the corresponding number of dots: 6.0 (1 dot), 6.8 (1 dot), 7.5 (1 dot), 7.8 (1 dot), 8.9 (1 dot), 9.5 (1 dot), 10.0 (3 dots), 10.2 (1 dot), 11.2 (1 dot), 11.5 (3 dots), 12.2 (1 dot), 13.5 (1 dot), 14.1 (1 dot), 14.9 (1 dot), 15.8 (1 dot).
02

Construct the Frequency Distribution

Organize the data into a table with two columns, one for the unique data values and another for their corresponding frequencies. Using the unique values identified earlier, list them in the first column and their counts as frequencies: \(6.0: 1\), \(6.8: 1\), \(7.5: 1\), \(7.8: 1\), \(8.9: 1\), \(9.5: 1\), \(10.0: 3\), \(10.2: 1\), \(11.2: 1\), \(11.5: 3\), \(12.2: 1\), \(13.5: 1\), \(14.1: 1\), \(14.9: 1\), \(15.8: 1\).
03

Construct the Percent Frequency Distribution

Calculate the percent frequency of each data value by dividing the frequency of each value by the total number of data points and multiplying by 100. The total number of data points is 19. Use the formula \(\frac{\text{Frequency of Value}}{19} \times 100\). For example, for the value 10.0, \(\frac{3}{19} \times 100 = 15.79\%\). Repeat this for each unique value: 6.0 (5.26%), 6.8 (5.26%), 7.5 (5.26%), 7.8 (5.26%), 8.9 (5.26%), 9.5 (5.26%), 10.0 (15.79%), 10.2 (5.26%), 11.2 (5.26%), 11.5 (15.79%), 12.2 (5.26%), 13.5 (5.26%), 14.1 (5.26%), 14.9 (5.26%), 15.8 (5.26%).

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Key Concepts

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

Dot Plot
A dot plot is an intuitive way to visualize the frequency of data points along a number line. This simple graphical display is particularly useful for small datasets. Each unique data point is represented along the horizontal axis, and the number of dots above each point corresponds to the number of times that value appears in the dataset.
To construct a dot plot, follow these steps:
  • Identify unique values: List all distinct values in the dataset in ascending order.
  • Count frequency: Count how many times each value appears in the dataset.
  • Place dots: Plot a dot for each occurrence of a value directly above its position on the number line.
For example, if your data set has a value of 10.0 appearing three times, you will place three dots directly above 10.0 on the number line. A dot plot is a straightforward tool that helps identify clusters, gaps, or outliers at a glance.
Frequency Distribution
A frequency distribution organizes data into distinct categories and shows the number of occurrences (or frequency) of each category. This method efficiently summarizes data, making it easier to discern patterns and trends.
To create a frequency distribution:
  • List unique values: Use the unique data values extracted from the dataset.
  • Count occurrences: Create a table with one column for each unique value and another for the frequency of occurrences for each value.
  • Tabulate the data: Fill in the table with the number of times each unique value appears.
The frequency distribution provides a clear picture of how data points are spaced out across their range. For example, a value that appears three times in a dataset will have a frequency of 3 in the frequency table. By structuring data this way, it becomes easier to detect data concentration and distribution.
Percent Frequency Distribution
The percent frequency distribution gives a relative understanding of how often each category occurs compared to the entire dataset. This distribution is useful because it shows the data percentages rather than raw counts, making comparisons between different data sets more intuitive.
Here's how to calculate percent frequency:
  • Determine total number: Count the total number of data points in the dataset.
  • Calculate percentage: For each unique value, divide its frequency by the total number of data points, then multiply by 100 to convert it to a percentage.
  • Present in table: Record each value with its corresponding percent frequency in an organized manner.
For example, if a value with a frequency of 3 is divided by a total of 19 data points, its percent frequency is \( \left( \frac{3}{19} \right) \times 100 \approx 15.79\% \). This method allows for easy visualization of how data compares proportionally within the dataset.

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Most popular questions from this chapter

The 2004 Naples, Florida, mini marathon \((13.1 \text { miles) had } 1228\) registrants (Naples Daily News, January 17,2004 ). Competition was held in six age groups. The following data show the ages for a sample of 40 individuals who participated in the marathon. \\[ \begin{array}{lcccc} 49 & 33 & 40 & 37 & 56 \\ 44 & 46 & 57 & 55 & 32 \\ 50 & 52 & 43 & 64 & 40 \\ 46 & 24 & 30 & 37 & 43 \\ 31 & 43 & 50 & 36 & 61 \\ 27 & 44 & 35 & 31 & 43 \\ 52 & 43 & 66 & 31 & 50 \\ 72 & 26 & 59 & 21 & 47 \end{array} \\] a. Show a stretched stem-and-leaf display. b. What age group had the largest number of runners? c. What age occurred most frequently? d. A Naples Daily News feature article emphasized the number of runners who were "20something." What percentage of the runners were in the 20 -something age group? What do you suppose was the focus of the article?

Sorting through unsolicited e-mail and spam affects the productivity of office workers. An InsightExpress survey monitored office workers to determine the unproductive time per day devoted to unsolicited e-mail and spam (USA Today, November 13,2003 ). The following data show a sample of time in minutes devoted to this task. \\[ \begin{array}{rrrr} 2 & 4 & 8 & 4 \\ 8 & 1 & 2 & 32 \\ 12 & 1 & 5 & 7 \\ 5 & 5 & 3 & 4 \\ 24 & 19 & 4 & 14 \end{array} \\] Summarize the data by constructing the following: a. A frequency distribution (classes \(1-5,6-10,11-15,16-20, \text { and so on })\) b. A relative frequency distribution c. A cumulative frequency distribution d. A cumulative relative frequency distribution e. An ogive f. What percentage of office workers spend 5 minutes or less on unsolicited e-mail and spam? What percentage of office workers spend more than 10 minutes a day on this task?

Approximately 1.5 million high school students take the Scholastic Aptitude Test (SAT) each year and nearly \(80 \%\) of the college and universities without open admissions policies use SAT scores in making admission decisions (College Board, March 2006). A sample of SAT scores for the combined math and verbal portions of the test are as follows: \\[ \begin{array}{rrrrr} 1025 & 1042 & 1195 & 880 & 945 \\ 1102 & 845 & 1095 & 936 & 790 \\ 1097 & 913 & 1245 & 1040 & 998 \\ 998 & 940 & 1043 & 1048 & 1130 \\ 1017 & 1140 & 1030 & 1171 & 1035 \end{array} \\] a. Show a frequency distribution and histogram for the SAT scores. Begin the first class with an SAT score of 750 and use a class width of 100 . b. Comment on the shape of the distribution. c. What other observations can be made about SAT scores based on the tabular and graphical summaries?

The Nielsen Media Research television rating measures the percentage of television owners who are watching a particular television program. The highest-rated television program in television history was the \(M^{*} A^{*} S^{*} H\) Last Episode Special shown on February 28,1983 A 60.2 rating indicated that \(60.2 \%\) of all television owners were watching this program. Nielsen Media Research provided the list of the 50 top-rated single shows in television history (The New York Times Almanac, 2006). The following data show the television net- work that produced each of these 50 top-rated shows. \(\begin{array}{lllll}\mathrm{ABC} & \mathrm{ABC} & \mathrm{ABC} & \mathrm{NBC} & \mathrm{CBS} \\ \mathrm{ABC} & \mathrm{CBS} & \mathrm{ABC} & \mathrm{ABC} & \mathrm{NBC} \\ \mathrm{NBC} & \mathrm{NBC} & \mathrm{CBS} & \mathrm{ABC} & \mathrm{NBC} \\ \mathrm{CBS} & \mathrm{ABC} & \mathrm{CBS} & \mathrm{NBC} & \mathrm{ABC} \\ \mathrm{CBS} & \mathrm{NBC} & \mathrm{NBC} & \mathrm{CBS} & \mathrm{NBC} \\ \mathrm{CBS} & \mathrm{CBS} & \mathrm{CBS} & \mathrm{NBC} & \mathrm{NBC} \\ \mathrm{FOX} & \mathrm{CBS} & \mathrm{CBS} & \mathrm{ABC} & \mathrm{NBC} \\ \mathrm{ABC} & \mathrm{ABC} & \mathrm{CBS} & \mathrm{NBC} & \mathrm{NBC} \\ \mathrm{NBC} & \mathrm{CBS} & \mathrm{NBC} & \mathrm{CBS} & \mathrm{CBS} \\ \mathrm{ABC} & \mathrm{CBS} & \mathrm{ABC} & \mathrm{NBC} & \mathrm{ABC}\end{array}\) a. Construct a frequency distribution, percent frequency distribution, and bar graph for the data. b. Which network or networks have done the best in terms of presenting top- rated television shows? Compare the performance of \(\mathrm{ABC}, \mathrm{CBS},\) and \(\mathrm{NBC}\)

The Higher Education Research Institute at UCLA provides statistics on the most popular majors among incoming college freshmen. The five most popular majors are Arts and Humanities (A), Business Administration (B), Engineering (E), Professional (P), and Social Science (S) (The New York Times Almanac, 2006). A broad range of other (O) majors, including biological science, physical science, computer science, and education, are grouped together. The majors selected for a sample of 64 college freshmen follow. \(\begin{array}{llllllllllllllll}\mathrm{S} & \mathrm{P} & \mathrm{P} & \mathrm{O} & \mathrm{B} & \mathrm{E} & \mathrm{O} & \mathrm{E} & \mathrm{P} & \mathrm{O} & \mathrm{O} & \mathrm{B} & \mathrm{O} & \mathrm{O} & \mathrm{O} & \mathrm{A} \\ \mathrm{O} & \mathrm{E} & \mathrm{E} & \mathrm{B} & \mathrm{S} & \mathrm{O} & \mathrm{B} & \mathrm{O} & \mathrm{A} & \mathrm{O} & \mathrm{E} & \mathrm{O} & \mathrm{E} & \mathrm{O} & \mathrm{B} & \mathrm{P}\end{array}\) \(\begin{array}{llllllllllllllll}\mathrm{B} & \mathrm{A} & \mathrm{S} & \mathrm{O} & \mathrm{E} & \mathrm{A} & \mathrm{B} & \mathrm{O} & \mathrm{S} & \mathrm{S} & \mathrm{O} & \mathrm{O} & \mathrm{E} & \mathrm{B} & \mathrm{O} & \mathrm{B} \\ \mathrm{A} & \mathrm{E} & \mathrm{B} & \mathrm{E} & \mathrm{A} & \mathrm{A} & \mathrm{P} & \mathrm{O} & \mathrm{O} & \mathrm{E} & \mathrm{O} & \mathrm{B} & \mathrm{B} & \mathrm{O} & \mathrm{P} & \mathrm{B}\end{array}\) a. Show a frequency distribution and percent frequency distribution. b. Show a bar graph. c. What percentage of freshmen selects one of the five most popular majors? d. What is the most popular major for incoming freshmen? What percentage of freshmen select this major?
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