Let's suppose that heights of Americans are normally distributed with a mean of 67.1 inches and a standard deviation of 5.3 inches. If a particular American is shorter than only \( 3 \% \) of all Americans, how tall is that person in inches? 76.1 inches 77.1 inches 57.1 inches 78.1 inches

To find the height of an American who is shorter than only \( 3\% \) of all Americans, we need to determine the height corresponding to the \( 3\% \) percentile of the normal distribution with a mean (\( \mu \)) of \( 67.1 \) inches and a standard deviation (\( \sigma \)) of \( 5.3 \) inches. ### Step 1: Find the z-score for the \( 3\% \) percentile The z-score corresponding to the \( 3\% \) percentile can be found using the standard normal distribution table or a calculator. The z-score represents how many standard deviations an element is from the mean. ### Step 2: Calculate the height using the z-score formula The formula to convert a z-score to a height in a normal distribution is: \[ X = \mu + z \cdot \sigma \] Where: - \( X \) is the height we want to find, - \( \mu = 67.1 \) inches (mean), - \( z \) is the z-score for the \( 3\% \) percentile, - \( \sigma = 5.3 \) inches (standard deviation). ### Step 3: Find the z-score for the \( 3\% \) percentile Let's find the z-score for the \( 3\% \) percentile. Using a standard normal distribution table or calculator, the z-score for \( 0.03 \) (or \( 3\% \)) is approximately \( -1.88079 \). ### Step 4: Calculate the height Now we can substitute the values into the formula: \[ X = 67.1 + (-1.88079) \cdot 5.3 \] Let's calculate this. \[ X = 67.1 - 9.973 \] \[ X \approx 57.127 \] Thus, the height of the person who is shorter than only \( 3\% \) of all Americans is approximately \( 57.1 \) inches. ### Conclusion The correct answer is \( 57.1 \) inches.