Respuesta :
Answer:
1. The probability that x will be more than 25 is 0.6305.
2. The 55th percentile is 26.38.
Step-by-step explanation:
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score 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 p-value, we get the probability that the value of the measure is greater than X.
Mean 26 and standard deviation 3.
This means that [tex]\mu = 26, \sigma = 3[/tex]
1 Find the probability that x will be more than 25.
This is 1 subtracted by the p-value of Z when X = 25. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{25 - 26}{3}[/tex]
[tex]Z = -0.333[/tex]
[tex]Z = -0.333[/tex] has a p-value of 0.3695.
1 - 0.3695 = 0.6305.
The probability that x will be more than 25 is 0.6305.
2 Find the 55th percentile of x.
This is X when Z has a p-value of 0.55, so X when Z = 0.125.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.125 = \frac{X - 26}{3}[/tex]
[tex]X - 26 = 0.125*3[/tex]
[tex]Z = 26.38[/tex]
The 55th percentile is 26.38.
