Answer:
a. Z = 1.99
b. 97.67%
c. Z = 2.56
d. 0.52%
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean and standard deviation
, the z-score of a measure X is given by:
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.
a. If a man is 6 feet 3 inches tall, what is his z-score (to two decimal places)?
The heights of adult men in America are normally distributed, with a mean of 69.7 inches and a standard deviation of 2.66 inches, and thus, we have
6 feet 3 inches = 6*12 + 3 = 75 inches, which means that we have to find z when X = 75. So
b. What percentage of men are SHORTER than 6 feet 3 inches?
The proportion is the p-value of Z = 1.99.
Looking at the z-table, Z = 1.99 has a p-value of 0.9767.
0.9767*100% = 97.67%, which is the answer.
c. If a woman is 5 feet 11 inches tall, what is her z-score (to two decimal places)?
The heights of adult women in America are also normally distributed, but with a mean of 64.5 inches and a standard deviation of 2.54 inches, and thus, we have . We have to find Z when X = 5*12 + 11 = 71. So
d. What percentage of women are TALLER than 5 feet 11 inches?
The proportion is 1 subtracted by the p-value of Z = 2.56.
Looking at the z-table, Z = 2.56 has a p-value of 0.9948.
1 - 0.9948 = 0.0052
0.0052*100% = 0.52%, which is the answer.