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Chapter 2: Problem 15

Find the mean, the median, and the mode for the number of vehicles owned in a survey of 52 households. \begin{tabular}{c|cccccccc} \(x\) & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline\(f\) & 2 & 12 & 15 & 11 & 6 & 3 & 1 & 2 \end{tabular}

Short Answer

Expert verified
Mean: 2.12, Median: 2, Mode: 2.

Step by step solution

01

Understanding the Data

First, observe the data given in the frequency distribution table. The table shows the number of vehicles owned (\(x\)) and their corresponding frequency (\(f\)) in 52 households.
02

Calculating the Mean

The mean is calculated using the formula: \[ \bar{x} = \frac{\sum (x \, f)}{\sum f} \].1. Calculate the total frequency: \(\sum f = 2 + 12 + 15 + 11 + 6 + 3 + 1 + 2 = 52\).2. Calculate the sum of products of each value and its frequency: \(\sum (x \, f) = (0 \times 2) + (1 \times 12) + (2 \times 15) + (3 \times 11) + (4 \times 6) + (5 \times 3) + (6 \times 1) + (7 \times 2) = 110\).3. Compute the mean: \(\bar{x} = \frac{110}{52} \approx 2.12\).
03

Finding the Median

The median is the middle value when data is arranged in order. For frequency data:1. Arrange the data in an increasing frequency cumulative format.2. Find the middle position: \(\frac{N+1}{2} \) where \( N = 52\), which gives \(\frac{53}{2} = 26.5\).3. Determine the median category: Cumulative Frequency: - 0: 2 - 1: 14 - 2: 29 (this includes the middle position)4. Thus, the median is the class of 2 vehicles.
04

Determining the Mode

The mode is the data value with the highest frequency in the dataset.- From the frequency table, the highest frequency, \(f = 15\), corresponds to \(x = 2\).- Thus, the mode is 2 vehicles.

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Key Concepts

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

Understanding Mean Calculation
To understand mean calculation in a simple way, think of it as the "average" of a set of numbers. It provides a central value for a dataset. Calculating the mean involves a couple of straightforward steps:

  • Add all the values together: This means you need to multiply each value of the dataset by its frequency, then sum these products. For example, if you have 5 occurrences of the number 2, you'll multiply 2 by 5.
  • Divide the total sum by the number of occurrences: Once you have the total from the first step, divide it by the number of all entries (total of all frequencies).

For our vehicle example, multiplying the number of vehicles by the number of households and summing them up gives us 110. Dividing this by the total number of households (52) results in a mean of approximately 2.12 vehicles. By using this method, any dataset's average, or central value, can be quickly determined.
Exploring Median Determination
The median is essentially the middle number of a dataset. To find the median, the data must be organized in sequence, often from smallest to largest. For frequency tables, like the one in our exercise, there is a slightly different approach.

Start by understanding the cumulative frequency:
  • Create a cumulative frequency table by adding up the frequencies successively. This will help to identify where the center of the dataset is.
  • Find the middle position: Use the formula \( \frac{N + 1}{2} \), where \( N \) is the total number of observations. In our scenario, the calculation \( \frac{53}{2} \) equals 26.5. This means the median lies between the 26th and 27th positions.
  • Locate the median: Using the cumulative frequencies, you'll find the median in the data category that includes this middle value.
In our vehicle count dataset, the median is the 2 vehicles category, as the middle position lies within its cumulative frequency.
Clarifying Mode
The mode is the simplest among the three central measures, as it represents the value that appears most frequently in a dataset. Identifying the mode in a frequency table involves finding the value with the highest frequency count.

To determine the mode:
  • Look at the frequency column: Identify the highest number. This is key because the mode is the number that occurs the most.
  • Match with the corresponding category: Once you have spotted the highest frequency, link it to the value it represents.

From the household vehicle survey, the highest frequency is 15, and this is associated with 2 vehicles. Thus, the mode for this dataset is 2. It's a simple reflection of the most popular or common value within the data.

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Most popular questions from this chapter

The IQ scores of ten students randomly selected from an elementary school are given. $$ \begin{array}{ccccc} 108 & 100 & 33 & 125 & 87 \\ 105 & 107 & 105 & 113 & 118 \end{array} $$ Grouping the measures in the 80s, the 90 s, and so on, construct a stem and leaf diagram, a frequency histogram, and a relative frequency histogram.

Thirty-six students took an exam on which the average was 80 and the standard deviation was \(6 .\) A rumor says that five students had scores 61 or below. Can the rumor be true? Why or why not?

For the population \(\begin{array}{llll}0 & 0 & 2 & 2\end{array}\) compute each of the following. a. The population mean \(\mu\). b. The population variance \(\sigma^{2}\). c. The population standard deviation \(\sigma\). d. The z-score for every value in the population data set.

A sample data set with a bell-shaped distribution has mean \(x=2\) and standard deviation \(s=1.1 .\) Find the approximate proportion of observations in the data set that lie: a. below-0.2; b. below \(3.1 ;\) c. between -1.3 and 0.9 .

Random samples, each of size \(n=10,\) were taken of the lengths in centimeters of three kinds of commercial fish, with the following results: $$ \begin{array}{lllllr} \text { Sample 1: } & 108 & 100 & 93 & 125 & 87 \\ & 105 & 107 & 105 & 119 & 118 \\ \text { Sample 2: } & 133 & 140 & 152 & 142 & 137 \\ & 145 & 160 & 138 & 139 & 138 \\ \text { Sample3: } & 82 & 60 & 83 & 82 & 82 \\ & 74 & 73 & 82 & 80 & 80 \end{array} $$ Grouping the measures by their common hundreds and tens digits, construct a stem and leaf diagram, a frequency histogram, and a relative frequency histogram for each of the samples. Compare the histograms and describe any patterns they exhibit.
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