Final Answer:
Bob's z-score for running a 540-second mile is 1.33.
Step-by-step explanation:
The z-score is a statistical measure that indicates how many standard deviations a data point is from the mean of a data set. In this case, we are given the mean time for 10th graders running a mile as 420 seconds with a standard deviation of 60 seconds. The formula to calculate the z-score is (X - μ) / σ, where X is the individual data point, μ is the mean, and σ is the standard deviation.
For Bob, who ran a 540-second mile, the calculation would be (540 - 420) / 60 = 1.33. This positive z-score indicates that Bob's mile time is 1.33 standard deviations above the mean for 10th graders. In other words, his performance is above average compared to his peers.
Standardized scores, like z-scores, play a crucial role in statistical analysis by providing a common scale for comparing different sets of data. They help identify outliers and assess the relative position of individual data points within a distribution. Understanding z-scores is fundamental in interpreting and making meaningful comparisons in various fields such as education, sports, and research.