adult males have normally distributed heights with mean of 69.0 in and a standard deviation of 2.8 in. if a clothing manufacture decides to produce goods that exclude the shortest 12% and the tallest 12% of adult males, find the minimum and maximum heights that they will fit.
To find the minimum and maximum heights that would fit within the range excluding the shortest 12% and the tallest 12% of adult males, we need to find the z-scores corresponding to these percentiles.
First, let's find the z-score for the shortest 12% of adult males:
To find the z-score corresponding to a given percentile, we can use the z-score formula:
z = (x - μ) / σ
where:
- z is the z-score
- x is the value we want to convert into a z-score
- μ is the mean of the distribution
- σ is the standard deviation
For the shortest 12% of adult males, we want to find the z-score that corresponds to the 12th percentile. Since this is a lower tail percentile, we need to find the z-score associated with the cumulative probability of 0.12.
Using a standard normal distribution table or a calculator, we find that the z-score for a cumulative probability of 0.12 is approximately -1.18.
Now we can calculate the minimum height as follows:
Minimum Height = μ + (z * σ)
Minimum Height = 69.0 + (-1.18 * 2.8)
Minimum Height ≈ 65.12 inches
Similarly, we can find the z-score for the tallest 12% of adult males. Since this is an upper tail percentile, we need to find the z-score associated with the cumulative probability of (1 - 0.12) = 0.88.
Using the same method described above, we find that the z-score for a cumulative probability of 0.88 is approximately 1.17.
We can now calculate the maximum height:
Maximum Height = μ + (z * σ)
Maximum Height = 69.0 + (1.17 * 2.8)
Maximum Height ≈ 73.27 inches
Therefore, the clothing manufacturer should produce goods that fit adult males with heights between approximately 65.12 inches and 73.27 inches.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.12) and its Z score. Insert Z score in above equation to find heights.