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Chapter 3: Problem 20

Thirty automobiles were tested for fuel efficiency (in miles per gallon). This frequency distribution was obtained. Find the variance and standard deviation for the data. $$ \begin{array}{lr} \text { Class boundaries } & \text { Frequency } \\ \hline 7.5-12.5 & 3 \\ 12.5-17.5 & 5 \\ 17.5-22.5 & 15 \\ 22.5-27.5 & 5 \\ 27.5-32.5 & 2 \end{array} $$

Short Answer

Expert verified
The variance is 25 and the standard deviation is 5.

Step by step solution

01

Calculate the Midpoints

For each class interval, calculate the midpoint. The midpoint is the average of the lower and upper class boundaries: \[\text{Midpoint} = \frac{\text{Lower boundary} + \text{Upper boundary}}{2}\]- 7.5-12.5: Midpoint = \(\frac{7.5 + 12.5}{2} = 10\)- 12.5-17.5: Midpoint = \(\frac{12.5 + 17.5}{2} = 15\)- 17.5-22.5: Midpoint = \(\frac{17.5 + 22.5}{2} = 20\)- 22.5-27.5: Midpoint = \(\frac{22.5 + 27.5}{2} = 25\)- 27.5-32.5: Midpoint = \(\frac{27.5 + 32.5}{2} = 30\)
02

Calculate the Weighted Mean

Find the weighted average of midpoints using frequencies to find the mean (\(\bar{x}\)):\[\bar{x} = \frac{\sum f_i \cdot x_i}{\sum f_i}\]\[\bar{x} = \frac{3 \cdot 10 + 5 \cdot 15 + 15 \cdot 20 + 5 \cdot 25 + 2 \cdot 30}{30} = \frac{600}{30} = 20\]This 20 is the mean of the fuel efficiency.
03

Find Each Class's Contribution to Variance

Calculate \((x_i - \bar{x})^2\) for each midpoint and multiply it by its frequency \(f_i\):- \(f_1 (x_1 - \bar{x})^2 = 3 \cdot (10 - 20)^2 = 300\)- \(f_2 (x_2 - \bar{x})^2 = 5 \cdot (15 - 20)^2 = 125\)- \(f_3 (x_3 - \bar{x})^2 = 15 \cdot (20 - 20)^2 = 0\)- \(f_4 (x_4 - \bar{x})^2 = 5 \cdot (25 - 20)^2 = 125\)- \(f_5 (x_5 - \bar{x})^2 = 2 \cdot (30 - 20)^2 = 200\)
04

Calculate Total Variance

Sum all contributions and divide by total number of data points to find variance:\[\text{Variance} = \frac{\sum f_i \cdot (x_i - \bar{x})^2}{N}\]\[Variance = \frac{300 + 125 + 0 + 125 + 200}{30} = 25\]
05

Calculate the Standard Deviation

Find the square root of the variance to determine the standard deviation:\[\text{Standard Deviation} = \sqrt{25} = 5\]

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Key Concepts

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

Frequency Distribution
Understanding frequency distribution is key in statistics as it gives us insights into how data is spread across different intervals or classes. Imagine you have a collection of data, like in our exercise, where we measure the fuel efficiency of cars in miles per gallon. Rather than listing each data point, we group them into ranges or intervals called class boundaries. Each class, like 7.5-12.5, has a certain count of data points, which we call its frequency.
  • Frequency distribution table: Lists classes (ranges) and their corresponding frequencies.
  • It provides a compact numerical summary of the data.
  • Essential for visual representation and further calculations.
By understanding how many data points are within each class, we get a clearer picture of the distribution of values.
Mean Calculation
Calculating the mean, or average, is foundational in analyzing datasets. The mean provides a central value that represents the entire set of data. To find the mean in a frequency distribution, we focus on the midpoints of each class interval. Instead of just summing values, we calculate the weighted mean based on these midpoints and their frequencies.
  • Midpoint calculation: Average of the lower and upper boundary of each class.
  • Weighted mean: Uses frequencies as weights to balance out the contribution of each class to the overall mean.
  • Formula: \( \bar{x} = \frac{\sum f_i \cdot x_i}{\sum f_i} \) where \( f_i \) is frequency and \( x_i \) is the midpoint.
This approach ensures that each data point's influence on the mean reflects how commonly it appears in the dataset.
Class Midpoints
The concept of class midpoints is essential for working with grouped data like frequency distributions. A midpoint represents the center of a class interval. In our example, we have intervals such as 7.5-12.5. For these intervals, midpoints are calculated as the average of the lower and upper boundaries.
  • Midpoint: \( \frac{\text{Lower boundary} + \text{Upper boundary}}{2} \)
  • Used in weighted mean calculations and variance analysis.
  • Midpoints make it possible to perform arithmetic operations on classes representing multiple data points.
By calculating midpoints, you're essentially "finding the average" of each interval, making further analysis manageable.
Weighted Mean
The weighted mean is a central statistical measure, especially when dealing with frequency distributions. Unlike a simple average, a weighted mean considers how often each value appears. This makes it especially useful for datasets where different values carry different importance or frequency.
  • Weighted Mean Formula: \( \bar{x} = \frac{\sum f_i \cdot x_i}{\sum f_i} \) where \( f_i \) is the frequency of each class, and \( x_i \) is the corresponding midpoint.
  • Ensures that classes with more data points have a greater influence on the final mean.
  • Provides a balanced representation of grouped data's central tendency.
Employing a weighted mean is essential when data is presented in groups, allowing for a better representation of the entire dataset.

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Most popular questions from this chapter

For this data set, find the mean and standard deviation of the variable. The data represent the ages of 30 customers who ordered a product advertised on television. Count the number of data values that fall within 2 standard deviations of the mean. Compare this with the number obtained from Chebyshev's theorem. Comment on the answer. \(\begin{array}{lllll}42 & 44 & 62 & 35 & 20 \\ 30 & 56 & 20 & 23 & 41 \\ 55 & 22 & 31 & 27 & 66 \\ 21 & 18 & 24 & 42 & 25 \\ 32 & 50 & 31 & 26 & 36 \\\ 39 & 40 & 18 & 36 & 22\end{array}\)

The average number of calories in a regular-size bagel is \(240 .\) If the standard deviation is 38 calories, find the range in which at least \(75 \%\) of the data will lie. Use Chebyshev's theorem.

Of the 25 brightest stars, the distances from earth (in light-years) for those with distances less than 100 light-years are found below. Find the mean, median, mode, and midrange for the data. $$\begin{array}{lllllll}8.6 & 36.7 & 42.2 & 16.8 & 33.7 & 77.5 & 87.9 \\\4.4 & 25.3 & 11.4 & 65.1 & 25.1 & 51.5 &\end{array}$$

The costs of three models of helicopters are shown here. Find the weighted mean of the costs of the models. $$\begin{array}{lrr}\text { Model } & \text { Number sold } & \text { Cost } \\\\\hline \text { Sunscraper } & 9 & \$ 427,000 \\\\\text { Skycoaster } & 6 & 365,000 \\ \text { High-flyer } & 12 & 725,000\end{array}$$

Using Chebyshev's theorem, solve these problems for a distribution with a mean of 80 and a standard deviation of \(10 .\) a. At least what percentage of values will fall between 60 and \(100 ?\) b. At least what percentage of values will fall between 65 and \(95 ?\)
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