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Chapter 1: Problem 13

Create a data set that fits the information. (Th) a. Ten students were asked the number of times they had flown in an airplane. The range of data values was 7 . The minimum was 0 and the mode was 2 . b. Eight students each measured the length of their right foot. The range of data values was \(8.2 \mathrm{~cm}\), and the maximum value was \(30.4 \mathrm{~cm}\). There was no mode.

Short Answer

Expert verified
Data sets: a) 0, 1, 2, 2, 2, 2, 3, 5, 6, 7; b) 22.2 cm, 23.0 cm, 24.5 cm, 25.3 cm, 26.6 cm, 27.8 cm, 29.1 cm, 30.4 cm.

Step by step solution

01

Analyze the Information for Section 'a'

The problem states there are 10 students. The data about flights is: - Range is 7 - Minimum is 0 - Mode is 2 The range of 7 means the difference between the maximum and minimum values is 7. So the maximum value is 0 + 7 = 7.
02

Create Data Points for Section 'a'

To meet the condition of the mode being 2, it should appear more frequently than any other number. Proposal: 0, 1, 2, 2, 2, 2, 3, 5, 6, 7 - Mode condition met with four occurrences of 2. - Range condition met as the difference between 0 and 7 is 7.
03

Analyze the Information for Section 'b'

For the measurement, the problem states: - Range is 8.2 cm - Maximum is 30.4 cm - No mode exists. The range of 8.2 cm means the minimum is 30.4 - 8.2 = 22.2 cm.
04

Create Data Points for Section 'b'

To ensure there is no mode, make sure each number appears only once. Proposal: 22.2 cm, 23.0 cm, 24.5 cm, 25.3 cm, 26.6 cm, 27.8 cm, 29.1 cm, 30.4 cm - No mode as all values are unique. - Range condition is met as the difference between 22.2 cm and 30.4 cm is 8.2 cm.

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Key Concepts

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

Statistics
Statistics is a branch of mathematics that deals with collecting, analyzing, interpreting, and presenting data. It helps us understand the world by providing tools to make inferences about a population based on a sample.
  • Data Description: This involves summarizing data using measures like the mean, median, mode, and range. In our exercise, we focused on the mode and range.
  • Mode: The mode is the number that appears most frequently in a data set. It's useful for categorical data where numbers repeat. In Section 'a', the mode was 2, meaning this number appeared more than any other in the data set.
  • Range: The range gives us the spread of the data and is calculated as the difference between the maximum and minimum values. For example, in Section 'a', a range of 7 indicates the data spans from 0 to 7. In Section 'b', a range of 8.2 cm suggests a more varied distribution from 22.2 cm to 30.4 cm.
By understanding these terms, students can better analyze data and make informed decisions based on statistical evidence.
Data Sets
Data sets encompass a collection of data points or values relating to a particular subject. Each data point within a data set provides insight into the overall trend or pattern. In the context of the exercise, students dealt with two distinct data sets:
  • Defining Features: Each data set comes with particular properties that need to be satisfied, such as having a specific range or repeated values to establish a mode.
  • Creating Data Sets: Students were asked to create data sets meeting specified criteria. This requires a thoughtful balance, ensuring conditions like the range and mode are strictly adhered to.
    • In Section 'a', students explored flights with numbers designed to have a mode of 2 and a range of 7.
    • In Section 'b', they created foot lengths with a unique single appearance for each value, establishing no mode but a range which met the constraints.
Understanding how to construct and manipulate data sets is crucial for many types of data analysis work, allowing students to visualize and interpret data trends.
Mathematical Problem Solving
Mathematical problem solving is about using logic and understanding to find a solution to a problem. This process can vary widely depending on the problem but often involves a few key steps:
  • Analyze Information: Before starting to solve a problem, it's important to understand what's being asked. Here, understanding the properties of the range, mode, and maximum/minimum values was crucial.
  • Solution Planning: Once the details are clear, plan how to meet these criteria. This often means deciding on a strategy to achieve specific numerical features within the data set.
    • For example, in the exercise, students needed to ensure the number of flights and foot lengths met all outlined conditions.
  • Execution and Testing: Finally, execute the plan and verify your solution by checking if all conditions are satisfied, such as the correct mode and range.
    • Ensuring compliance with no mode in Section 'b' required attention to detail so that no repetition occurred.
By mastering problem-solving skills, students enhance their ability to tackle various mathematical challenges, making them more adept at thinking critically and logically.

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Most popular questions from this chapter

Thirty students participated in a 20 -problem mathematics competition. Here are the numbers of problems they got correct: $$ \\{12,7,8,3,5,7,10,13,7,10,2,1,11,12,17,4,11,7,6,18,14,17,11,9,1,12,10,12,2 \text {, } $$ a. Construct two histograms for the data. Use different bin widths for each. b. What patterns do you notice in the data? What do the histograms tell you about the number of problems that students tend to get correct? c. Give the five-number summary for the data and construct a box plot. d. Give the mode(s) for the data.

Give the five-number summary and create a box plot for the listed values. $$ \\{2,6,4,9,1,6,4,7,2,8,5,6,9,3,6,7,5,4,8\\} $$

Is \([A]+[B]\) equal to \([B]+[A]\) ? Do you think this result is always true for matrix addition? Explain.

Use matrices \([A]\) and \([B]\) to write the new matrices asked for in \(9 \mathrm{a}-\mathrm{d}\). \([A]=\left[\begin{array}{rrr}3 & -4 & 2.5 \\ -2 & 6 & 4\end{array}\right]\) $$ [B]=\left[\begin{array}{rrr} 5 & -1 & 2 \\ -4 & 3.5 & 1 \end{array}\right] $$ a. \([A]+[B]\) b. \(-[A]\) (a) c. \(3 \cdot[\mathrm{B}]\) d. the average of \([A]\) and \([B]\) (?)

Lucia and Malcolm each estimated the weights of five different items from a grocery store. Each of Lucia's estimates was too low. Each of Malcolm's was too high. The scatter plot at right shows the (actual weight, estimated weight) data collected. The line drawn shows when an estimate is the same as the actual measurement. a. Which points represent Lucia's estimates? b. Which points represent Malcolm's estimates?
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