Respuesta :
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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:
[tex]\mu = 129.71, \sigma = 2.28[/tex]
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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{134 - 129.71}{2.28}[/tex]
[tex]Z = 1.88[/tex]
[tex]Z = 1.88[/tex] 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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.645 = \frac{X - 129.71}{2.28}[/tex]
[tex]X - 129.71 = -1.645*2.28[/tex]
[tex]X = 125.96[/tex]
The fastest 5% of her laps are under 125.96 seconds.
