One year Perry had the lowest ERA (earned-run average, mean number of runs yielded per nine innings pitched) of any male pitcher at his school, with an ERA of 2.83. Also, Karla had the lowest ERA of any female pitcher at the school with an ERA of 3.37. For the males, the mean ERA was 3.771 and the standard deviation was 0.861. For the females, the mean ERA was 4.979 and the standard deviation was 0.508. Find their respective z-scores. Which player had the better year relative to their peers, Perry or Karla? (Note: In general, the lower the ERA, the better the pitcher.) Perry had an ERA with a z-score of Karla had an ERA with a z-score of (Round to two decimal places as needed.)
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
Given:
For males, the mean ERA is 3.771 and standard deviation is 0.861.
The mean ERA of Perry is 2.83
The Z score for perry is obtained as below:
Thus, the Z score for perry is -1.09.
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