In a survey of women in a certain country (ages $20-29$ ), the mean height was 63.6 inches with a standard deviation of 285 inches. Answer the following questions abou the specified normal distribution.
(a) What height represents the 95th percentile?
(b) What height represents the first quartile?
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(a) The height that represents the 95th percentile is inches. (Round to two decimal places as needed)
(b) The height that represents the first quartile is inches. (Round to two decimal places as needed)
Answer
The height that represents the 95th percentile is approximately \(\boxed{68.29}\) inches and the height that represents the first quartile is approximately \(\boxed{61.68}\) inches.
Step 1 ：The problem is asking for the 95th percentile and the first quartile (25th percentile) of a normal distribution with a mean of 63.6 inches and a standard deviation of 285 inches.
Step 2 ：To find these percentiles, we can use the z-score formula, which is \((X - \mu) / \sigma\), where X is the value we're looking for, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Step 3 ：However, in this case, we're given the percentile and we need to find the corresponding value, so we need to rearrange the formula to solve for X: \(X = Z * \sigma + \mu\).
Step 4 ：The Z value for the 95th percentile and the 25th percentile can be found in a standard normal distribution table or calculated using a function in Python.
Step 5 ：Using the given mean of 63.6 inches and standard deviation of 285 inches, and the Z values for the 95th percentile (1.6448536269514722) and the 25th percentile (-0.6744897501960817), we calculate the heights.
Step 6 ：The calculated heights seem to be incorrect. The height for the 95th percentile is over 500 inches and the height for the 25th percentile is negative, which doesn't make sense in the context of human heights.
Step 7 ：I suspect the standard deviation given in the problem (285 inches) is incorrect. A standard deviation of 285 inches for human heights is extremely large.
Step 8 ：If we assume the standard deviation is actually 2.85 inches (a more reasonable value for the standard deviation of human heights), we can recalculate the heights.
Step 9 ：Using the corrected standard deviation of 2.85 inches, we calculate the heights for the 95th percentile and the 25th percentile again.
Step 10 ：The height that represents the 95th percentile is approximately \(\boxed{68.29}\) inches and the height that represents the first quartile is approximately \(\boxed{61.68}\) inches.
