Question

Heights of Men in the US Heights of adult males in the US are approximately normally distributed with mean 70 inches \((5\) ft 10 in \()\) and standard deviation 3 inches. (a) What proportion of US men are between \(5 \mathrm{ft}\) 8 in and \(6 \mathrm{ft}\) tall \((68\) and 72 inches, respectively)? (b) If a man is at the 10 th percentile in height, how tall is he?

Short answer

(a) Approximately 49.72% of US men are between 68 and 72 inches tall. (b) A man at the 10th percentile in height is about 5 ft 6.16 inches tall.

Step-by-step solution

Calculate Z-Scores for Part (a)

To determine the proportion of US men between \(68\) inches and \(72\) inches, convert these measurements into Z-scores. Z-score formula is \(Z = \frac{x - \mu}{\sigma}\). Using this formula, calculate the Z-scores for \(68\) inches and \(72\) inches, where \(\mu = 70\) inches (mean height) and \(\sigma = 3\) inches (standard deviation). The Z-score for \(68\) inches is \(Z1 = \frac{68 - 70}{3} = -\frac{2}{3} = -0.67\) and the Z-score for \(72\) inches is \(Z2 = \frac{72-70}{3} = \frac{2}{3} = 0.67\).

Find Probabilities for Part (a)

Use the Z-scores to find the corresponding cumulative probabilities from the standard normal distribution table. The probability for \(Z1 = -0.67\) is \(0.2514\) and the probability for \(Z2 = 0.67\) is \(0.7486\).

Find Proportion for Part (a)

Subtract the probability for \(Z1\) from the probability for \(Z2\) to find the proportion of men between \(68\) inches and \(72\) inches. The proportion is \(0.7486 - 0.2514 = 0.4972\) which corresponds to approximately \(49.72\%\) of men.

Calculate Z-Score for Part (b)

For the 10th percentile height, first find the corresponding Z-score from the standard normal distribution table. The Z-score for the 10th percentile (\(0.10\) or \(10\%\)) is approximately \(-1.28\).

Find Height for Part (b)

Use the Z-score to find the corresponding height. Use the formula \(x = Z*\sigma + \mu\), where \(Z = -1.28\), \(\sigma = 3\) inches and \(\mu = 70\) inches. The height is \(x = -1.28 * 3 + 70 = 66.16\) inches or approximately \(5\) ft \(6.16\) inches.