Travel Time Use the results of Problem 22 in Section 3.1 and Problem 22 in Section 3.2 to compute the z-scores for all the students. Compute the mean and standard deviation of these z-scores.

Verified step by step guidance

1. Identify the raw scores for each student from the results given in Problem 22 of Section 3.1 and Problem 22 of Section 3.2. These raw scores represent the travel times or related measurements for each student.
2. Recall the formula for computing a z-score for each student: \[ z = \frac{X - \mu}{\sigma} \] where \(X\) is the student's raw score, \(\mu\) is the mean of the distribution, and \(\sigma\) is the standard deviation of the distribution. Use the mean and standard deviation values provided or calculated in the referenced problems.
3. Calculate the z-score for each student by substituting their raw score and the known mean and standard deviation into the formula above. This standardizes each student's score relative to the group.
4. Once all z-scores are computed, find the mean of these z-scores by summing all the z-scores and dividing by the number of students: \[ \bar{z} = \frac{1}{n} \sum_{i=1}^n z_i \] where \(n\) is the number of students.
5. Calculate the standard deviation of the z-scores using the formula: \[ s_z = \sqrt{ \frac{1}{n-1} \sum_{i=1}^n (z_i - \bar{z})^2 } \] This measures the spread of the standardized scores around their mean.

Key Concepts

Z-Score

A z-score measures how many standard deviations a data point is from the mean of its distribution. It standardizes different data points, allowing comparison across different scales. Calculating z-scores involves subtracting the mean and dividing by the standard deviation.

Mean of Z-Scores

The mean of z-scores in a dataset is typically zero because z-scores are centered around the mean of the original data. This property helps verify correct standardization and ensures the transformed data has a balanced distribution.

Standard Deviation of Z-Scores

The standard deviation of z-scores is usually one, reflecting the spread of the standardized data. This normalization allows for consistent comparison of variability across different datasets or variables.