### 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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```  ______   __   __    _____    _    _    _    _
/_____//  \ \\/ //  / ____|| | || | || | || | ||
`____ `    \ ` //  / //---`' | || | || | || | ||
/___//      | ||   \ \\___   | \\_/ || | \\_/ ||
`__ `       |_||    \_____||  \____//   \____//
/_//        `-`'     `----`    `---`     `---`
`-`
```
• 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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``` _    _    _    _      _____   __   __   _____
| |  | || | || | ||   / ___//  \ \\/ // |  __ \\
| |/\| || | || | ||   \___ \\   \ ` //  | |  \ ||
|  /\  || | \\_/ ||   /    //    | ||   | |__/ ||
|_// \_||  \____//   /____//     |_||   |_____//
`-`   `-`   `---`   `-----`      `-`'    -----`

```
• 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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```  ____     _    _     ______    _____    _____
|  _ \\  | || | ||  /_   _//  |  ___|| |  __ \\
| |_| || | || | ||    | ||    | ||__   | |  \ ||
| .  //  | \\_/ ||   _| ||    | ||__   | |__/ ||
|_|\_\\   \____//   /__//     |_____|| |_____//
`-` --`    `---`    `--`      `-----`   -----`

```
• 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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```  ____      ______    _  _    __   __    _  _
|  _ \\   /_   _//  | \| ||  \ \\/ //  | \| ||
| |_| ||   -| ||-   |  ' ||   \ ` //   |  ' ||
| .  //    _| ||_   | .  ||    | ||    | .  ||
|_|\_\\   /_____//  |_|\_||    |_||    |_|\_||
`-` --`   `-----`   `-` -`     `-`'    `-` -`

```
• 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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``` _    _      ___                ______   _____
| |  | ||   / _ \\    ____     /_   _// |  __ \\
| |/\| ||  / //\ \\  |    \\    -| ||-  | |  \ ||
|  /\  || |  ___  || | [] ||    _| ||_  | |__/ ||
|_// \_|| |_||  |_|| |  __//   /_____// |_____//
`-`   `-` `-`   `-`  |_|`-`    `-----`   -----`
`-`
```
• 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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``` __   _     ______    ______     ___      ______