The army reports that the distribution of head circumfererence among soldiers is approximately Normal with mean 22.8 inches and standard deviation 1.1 inches. Helmets are mass produced for all except the smallest 5 % and the largest 5 % of head sizes. soldiers in the smallest or largest 5% get custom made helmets. What head sizes get custom-made helmets?
Ok I don't see what I'm doing wrong here
.0500 = 1.1^-1 (x - 22.8)
x = .0500 (1.1) + 22.8
x = 22.855
To determine the head sizes that require custom-made helmets, we can use the concept of Z-scores and the standard normal distribution.
Given that the head circumference follows a normal distribution with a mean of 22.8 inches and a standard deviation of 1.1 inches, we need to find the head sizes corresponding to the smallest and largest 5% of the distribution.
To find the Z-scores associated with these percentiles, we can use the standard normal distribution table or a calculator:
For the smallest 5%:
Z-score = -1.645 (approximately)
For the largest 5%:
Z-score = 1.645 (approximately)
Now, we can calculate the head sizes corresponding to these Z-scores:
For the smallest 5%:
x = mean + (Z-score * standard deviation)
x = 22.8 + (-1.645 * 1.1)
x ≈ 20.853 inches
For the largest 5%:
x = mean + (Z-score * standard deviation)
x = 22.8 + (1.645 * 1.1)
x ≈ 24.747 inches
Therefore, soldiers with head circumferences less than approximately 20.853 inches or greater than approximately 24.747 inches would require custom-made helmets.