Answer:
a) 1.93
b) 97.32% of men are SHORTER than 6 feet 3 inches
c) 2.71
d) 0.34% of women are TALLER than 5 feet 11 inches
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.
a. If a man is 6 feet 3 inches tall, what is his z-score (to two decimal places)?
For man, ![\mu = 69.8, \sigma = 2.69](https://tex.z-dn.net/?f=%5Cmu%20%3D%2069.8%2C%20%5Csigma%20%3D%202.69)
A feet has 12 inches, so this is Z when X = 6*12 + 3 = 75. 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{75 - 69.8}{2.69}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B75%20-%2069.8%7D%7B2.69%7D)
![Z = 1.93](https://tex.z-dn.net/?f=Z%20%3D%201.93)
b. What percentage of men are SHORTER than 6 feet 3 inches?
Z = 1.93 has a pvalue of 0.9732
97.32% of men are SHORTER than 6 feet 3 inches
c. If a woman is 5 feet 11 inches tall, what is her z-score (to two decimal places)?
For woman, ![\mu = 64.1, \sigma = 2.55](https://tex.z-dn.net/?f=%5Cmu%20%3D%2064.1%2C%20%5Csigma%20%3D%202.55)
Here we have X = 5*12 + 11 = 71.
![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{71 - 64.1}{2.55}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B71%20-%2064.1%7D%7B2.55%7D)
![Z = 2.71](https://tex.z-dn.net/?f=Z%20%3D%202.71)
d. What percentage of women are TALLER than 5 feet 11 inches?
Z = 2.71 has a pvalue of 0.9966
1 - 0.9966 = 0.0034
0.34% of women are TALLER than 5 feet 11 inches