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9.3 Describing Dispersion

9.3 Describing Dispersion
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  • 7. Distribution of Years Worked by Cereal Factory Employees

    The histogram shows the distribution of the years worked by the 1820 employees of a cereal factory. Does this data set have a normal distribution? Explain your reasoning. (See Example 3.)

    • Worked-Out Solution

      The distribution of the data is approximately bell-shaped, so it appears to have a normal distribution.

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     _    _      ___      _____     ______   __   __  
    | |  | ||   / _ \\   |__  //   /_   _//  \ \\/ // 
    | |/\| ||  / //\ \\    / //     -| ||-    \   //  
    |  /\  || |  ___  ||  / //__    _| ||_    / . \\  
    |_// \_|| |_||  |_|| /_____||  /_____//  /_//\_\\ 
    `-`   `-` `-`   `-`  `-----`   `-----`   `-`  --` 
                                                      
    
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  • 8. Percent of Years Worked within 1 Standard Deviation

    The histogram shows the distribution of the years worked by the 1820 employees of a cereal factory. What percent of the employees lie within 1 standard deviation of the mean? (See Example 3.)

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     __   _    _    _     _____      ___      _  __  
    | || | || | || | ||  /  ___||   / _ \\   | |/ // 
    | '--' || | || | || | // __    | / \ ||  | ' //  
    | .--. || | \\_/ || | \\_\ ||  | \_/ ||  | . \\  
    |_|| |_||  \____//   \____//    \___//   |_|\_\\ 
    `-`  `-`    `---`     `---`     `---`    `-` --` 
                                                     
    
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  • 9. Percent of Years Worked within 2 Standard Deviations

    The histogram shows the distribution of the years worked by the 1820 employees of a cereal factory. What percent of the employees lie within 2 standard deviations of the mean? (See Example 3.)

    • Worked-Out Solution

      The mean is about 10 years and the standard deviation is about 4 years. Data values that lie within 2 standard deviations lie within the range 10 - 2(4) to 10 + 2(4), or from 2 years to 18 years.

      Add the number of employees who have years within the range from 2 years to 18 years. The number of employees with years worked within 2 standard deviations of the mean is

      Divide this number by the total number of employees.

      So, about 97% of the employees have years that lie within 2 standard deviations of the mean.

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     _    _    _    _    __   _     _____     ______  
    | |  | || | || | || | || | ||  |  ___||  /_   _// 
    | |/\| || | || | || | '--' ||  | ||__      | ||   
    |  /\  || | \\_/ || | .--. ||  | ||__     _| ||   
    |_// \_||  \____//  |_|| |_||  |_____||  /__//    
    `-`   `-`   `---`   `-`  `-`   `-----`   `--`     
                                                      
    
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  • 10. Comparing Distributions with the Normal Distribution

    The histogram shows the distribution of the years worked by the 1820 employees of a cereal factory. Compare the percents in Exercises 8 and 9 with the percents given by the normal distribution. (See Example 3.)

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                 ___      ______     ___     _    _   
      ____      / _ \\   /_____//   / _ \\  | |  | || 
     |    \\   / //\ \\  `____ `   / //\ \\ | |/\| || 
     | [] ||  |  ___  || /___//   |  ___  |||  /\  || 
     |  __//  |_||  |_|| `__ `    |_||  |_|||_// \_|| 
     |_|`-`   `-`   `-`  /_//     `-`   `-` `-`   `-` 
     `-`                 `-`                          
    
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  • 11. Comparing Distributions of Cereal Box Weights

    The histograms show the distributions of samples of weights of boxes of cereal filled by two machines. What is the significance of the smaller standard deviation for machine 2? (See Example 4.)

    • Worked-Out Solution

      The smaller standard deviation for machine 2 means that the distribution of data values for machine 2 has less dispersion than the distribution of the data values for machine 1. In other words, the weights in machine 2's distribution are more tightly clustered about the mean than the weights in machine 1's distribution.

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     __   _    _    _    _    _      ___              
    | || | || | || | || | \  / ||   / _ \\      ___   
    | '--' || | || | || |  \/  ||  / //\ \\    /   || 
    | .--. || | \\_/ || | .  . || |  ___  ||  | [] || 
    |_|| |_||  \____//  |_|\/|_|| |_||  |_||   \__ || 
    `-`  `-`    `---`   `-`  `-`  `-`   `-`     -|_|| 
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  • 12. Estimating the Percent of Cereal Boxes that Pass the Weight Requirement

    The histograms show the distributions of samples of weights of boxes of cereal filled by two machines. The boxes must weigh at least 12.2 ounces. Estimate the percent of boxes from each machine that pass the weight requirement. (See Example 4.)

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     __   __   __   __               ___      ____    
     \ \\/ //  \ \\/ //     ___     / _ \\   |  _ \\  
      \   //    \ ` //     /   ||  | / \ ||  | |_| || 
      / . \\     | ||     | [] ||  | \_/ ||  | .  //  
     /_//\_\\    |_||      \__ ||   \___//   |_|\_\\  
     `-`  --`    `-`'       -|_||   `---`    `-` --`  
                             `-`                      
    
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