Respuesta :
Answer:
a) 84th percentile.
b) The 25th percentile is of 67 inches.
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.
The heights of the men age 18 and over in HANES5 averaged 69 inches; the SD was 3 inches.
This means that [tex]\mu = 69, \sigma = 3[/tex]
Such a man that is 6 feet tall is at the percentile of this height distribution.
6 feet = 6*12 inches = 72 inches. So this percentile is the p-value of Z when X = 72. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{72 - 69}{3}[/tex]
[tex]Z = 1[/tex]
[tex]Z = 1[/tex] has a p-value of 0.84, so 84th percentile.
What is the 25th percentile, to the nearest inch
X when Z has a p-value of 0.25, so X when Z = -0.675.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.675 = \frac{X - 69}{3}[/tex]
[tex]X - 69 = -0.675*3[/tex]
[tex]X = 67[/tex]
The 25th percentile is of 67 inches.
