The heights of adult men in America are normally distributed, with a mean of 69.1 inches and a standard deviation of 2.62 inches. The heights of adult women in America are also normally distributed, but with a mean of 64.1 inches and a standard deviation of 2.59 inches.
a) If a man is 6 feet 3 inches tall, what is his z-score?
z =
b) What percentage of men are SHORTER than 6 feet 3 inches?
c) If a woman is 5 feet 6 inches tall, what is her z-score?
z =
d) What percentage of women are TALLER than 5 feet 6 inches?
To find the z-score, we can use the formula:
z = (x - μ) / σ
where z is the z-score, x is the individual's height, μ is the mean height, and σ is the standard deviation.
a) If a man is 6 feet 3 inches tall, we need to convert this height into inches, which is 6 * 12 + 3 = 75 inches. Then we can calculate the z-score.
z = (75 - 69.1) / 2.62 = 2.26
So, the z-score for a man who is 6 feet 3 inches tall is 2.26.
b) To find the percentage of men who are shorter than 6 feet 3 inches, we need to find the area under the normal distribution curve to the left of the z-score. We can use a standard normal distribution table or a statistical calculator to find this value.
Using a standard normal distribution table or a calculator, we find that the area to the left of z = 2.26 is approximately 0.9872, or 98.72%. This means that approximately 98.72% of men are shorter than 6 feet 3 inches.
c) If a woman is 5 feet 6 inches tall, we need to convert this height into inches, which is 5 * 12 + 6 = 66 inches. Then we can calculate the z-score.
z = (66 - 64.1) / 2.59 = 0.73
So, the z-score for a woman who is 5 feet 6 inches tall is 0.73.
d) To find the percentage of women who are taller than 5 feet 6 inches, we need to find the area under the normal distribution curve to the right of the z-score. Again, we can use a standard normal distribution table or a statistical calculator to find this value.
Using a standard normal distribution table or a calculator, we find that the area to the right of z = 0.73 is approximately 0.7673, or 76.73%. This means that approximately 76.73% of women are taller than 5 feet 6 inches.