Answer:
97% of her laps are completed in less than 134 seconds.
The fastest 5% of her laps are under 125.96 seconds.
Step-by-step explanation:
When the distribution is normal, we use the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this question, we have that:
![\mu = 129.71, \sigma = 2.28](https://tex.z-dn.net/?f=%5Cmu%20%3D%20129.71%2C%20%5Csigma%20%3D%202.28)
Find the percent of her laps that are completed in less than 134 seconds.
We have to find the pvalue of Z when X = 134. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{134 - 129.71}{2.28}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B134%20-%20129.71%7D%7B2.28%7D)
![Z = 1.88](https://tex.z-dn.net/?f=Z%20%3D%201.88)
has a pvalue of 0.9699, so 97% of her laps are completed in less than 134 seconds.
The fastest 5% of her laps are under how many seconds?
This is the 5th percentile of times, which is X when Z has a pvalue of 0.05, that is, X when Z = -1.645. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![-1.645 = \frac{X - 129.71}{2.28}](https://tex.z-dn.net/?f=-1.645%20%3D%20%5Cfrac%7BX%20-%20129.71%7D%7B2.28%7D)
![X - 129.71 = -1.645*2.28](https://tex.z-dn.net/?f=X%20-%20129.71%20%3D%20-1.645%2A2.28)
![X = 125.96](https://tex.z-dn.net/?f=X%20%3D%20125.96)
The fastest 5% of her laps are under 125.96 seconds.