Question

The heights of Russian men are Normally distributed with a mean of 65.2 inches and a standard deviation of 4.5 inches. What percentage of Russian men are: (a) Under 56.2 inches tall? (b) Over 51.7 inches tall? (c) Berween 62.185 and 68.215 inches tall?

Answer 1

Using the Z-score formula, we find that approximately 2.28% of Russian men are under 56.2 inches, about 99.73% are over 51.7 inches, and approximately 50% are between 62.185 and 68.215 inches tall.

Step-by-step explanation:

To find these percentages, we will use the Z-score formula which is Z = (X - μ) /σ, where X is the height, μ is the mean, and σ is the standard deviation. For Russian men, the mean (μ) is 65.2 inches and the standard deviation (σ) is 4.5 inches.

Part (a)

For a man under 56.2 inches:

Z = (56.2 - 65.2) / 4.5 = -2.0

Using a standard normal distribution table or a calculator, the percentage of men under this Z-score is approximately 2.28%.

Part (b)

For a man over 51.7 inches:

Z = (51.7 - 65.2) / 4.5 = -3.0

Since almost all values in a normal distribution lie within 3 standard deviations from the mean, the percentage over 51.7 inches is approximately 99.73%.

Part (c)

For men between 62.185 and 68.215 inches:

Z for 62.185 = (62.185 - 65.2) / 4.5 = -0.67

Z for 68.215 = (68.215 - 65.2) / 4.5 = 0.67

The percentage between these two Z-scores is approximately 50%.