- Use the \(z\)-score formula: \(z = \frac{x - \bar{x}}{s}\).
- Substitute the given values: \(-1.5 = \frac{x - 29}{3.6}\).
- Solve for \(x\): \(x = 29 + (-1.5 \times 3.6) = 23.6\).
- Select the option closest to the result, which is \(\boxed{24}\) miles.
Explanation
- Analyze the problem and given data
We are given the following data from a supervisor's study of employee commute distances:
- Mean number of miles (\(\bar{x}\)): \(29\)
- Standard deviation of miles (\(s\)): \(3.6\)
- Target \(z\)-score (\(z\)): \(-1.5\)
The \(z\)-score describes how many standard deviations a value is from the mean. It is calculated using the formula:
\[z = \frac{x - \bar{x}}{s}\]
Our objective is to find the mileage value \(x\) that corresponds to a \(z\)-score of \(-1.5\).
- Solve for the mileage \(x\)
We use the \(z\)-score formula to solve for \(x\):
\[-1.5 = \frac{x - 29}{3.6}\]
To isolate \(x\), we first multiply both sides of the equation by \(3.6\):
\[-1.5 \times 3.6 = x - 29\]
\[-5.4 = x - 29\]
Now, we add \(29\) to both sides:
\[x = 29 - 5.4\]
\[x = 23.6\]
The calculated mileage is \(23.6\) miles.
- Compare with the provided options
Comparing our calculated result of \(23.6\) miles to the given options:
- \(21\) miles
- \(24\) miles
- \(36\) miles
- \(41\) miles
The value \(23.6\) miles is closest to \(24\) miles. Given the choices, \(24\) is the best fit for the intended result of the calculation.
Examples
Calculating the \(z\)-score is a fundamental tool in statistics used to identify outliers and normalize data. For example, if a teacher wants to identify students who are performing exceptionally low or high compared to the class average, they can use \(z\)-scores to determine the exact threshold values, allowing for better identification of students who may need additional support or enrichment. This helps in making data-driven decisions in various fields like education, quality control, and finance.