Respuesta :
Answer:
For the 99th percentile, we have X = 206 seconds.
Step-by-step explanation:
Problems of normally distributed samples can be solved using 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 problem, we have that:
[tex]\mu = 138.5, \sigma = 29[/tex]
99th percentile:
Value of X when Z has a pvalue of 0.99. So we use [tex]Z = 2.325[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]2.325 = \frac{X - 138.5}{29}[/tex]
[tex]X - 138.5 = 29*2.325[/tex]
[tex]X = 205.92 = 206[/tex]
For the 99th percentile, we have X = 206 seconds.
