A baseball pitcher, concerned about losing speed from his fastball, undertook a new training regimen during the offseason. His team's pitching coach measured the speed of 20 random fastballs (in miles per hour) thrown by the pitcher during spring training, and compared it with a sample of 20 random fastballs thrown during the pitcher's last five starts in the previous season. The results are shown in the following table. Assume that the pitcher's fastball speeds had a standard deviation of 2.9 miles per hour both before and after the training regimen and that the speeds for both time periods are normally distributed. Let the pitcher's fastball speeds in the previous season be the first sample, and let the pitcher's fastball speeds in spring training be the second sample. At the 0.05 level of significance, is there evidence that the pitcher is throwing fastballs at higher speeds? Find the test statistic, rounded to two decimal places, and the p-value, rounded to three decimal places. Last Season's Fastball This Season's Fastball 92 90 91 95 96 93 94 94 92 91 94 95 92 97 95 92 94 97 93 92 96 93 89 92 95 97 94 93 91 95 95 96 94 92 94 97 90 99 95 93
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.
A baseball pitcher, concerned about losing speed from his fastball, undertook a new training regimen during the offseason. His team's pitching coach measured the speed of 20 random fastballs (in miles per hour) thrown by the pitcher during spring training, and compared it with a sample of 20 random fastballs thrown during the pitcher's last five starts in the previous season. The results are shown in the following table.
Assume that the pitcher's fastball speeds had a standard deviation of 2.9 miles per hour both before and after the training regimen and that the speeds for both time periods are
At the 0.05 level of significance, is there evidence that the pitcher is throwing fastballs at higher speeds?
Find the test statistic, rounded to two decimal places, and the p-value, rounded to three decimal places.
| Last Season's Fastball | This Season's Fastball | ||
| 92 | 90 | ||
| 91 | 95 | ||
| 96 | 93 | ||
| 94 | 94 | ||
| 92 | 91 | ||
| 94 | 95 | ||
| 92 | 97 | ||
| 95 | 92 | ||
| 94 | 97 | ||
| 93 | 92 | ||
| 96 | 93 | ||
| 89 | 92 | ||
| 95 | 97 | ||
| 94 | 93 | ||
| 91 | 95 | ||
| 95 | 96 | ||
| 94 | 92 | ||
| 94 | 97 | ||
| 90 | 99 | ||
| 95 | 93 |
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