Question

the army reports that the distribution of head circumference 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 is the smallest head size before a soldier needs to get a custom - made helmet? round your answer to 2 decimal places inches what is the largest head size before a soldier needs to get a custom - made helmet? round your answer to 2 decimal places inches

Explanation:

Step1: Find the z - score for the lower 5%

We use the standard normal distribution table (z - table). The area to the left of the z - score for the lower 5% (or 0.05) is 0.05. Looking up in the z - table, the z - score $z_1\approx - 1.645$.

Step2: Find the z - score for the upper 5%

The area to the left of the z - score for the upper 5% is $1 - 0.05=0.95$. Looking up in the z - table, the z - score $z_2\approx1.645$.

Step3: Use the z - score formula to find the head - size values

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the original normal distribution, $\mu$ is the mean, and $\sigma$ is the standard deviation. We know that $\mu = 22.8$ inches and $\sigma = 1.1$ inches.
For the lower value:
\[

$$\begin{align*} -1.645&=\frac{x_1 - 22.8}{1.1}\\ x_1-22.8&=-1.645\times1.1\\ x_1&=22.8-1.645\times1.1\\ x_1&=22.8 - 1.8095\\ x_1&\approx20.99 \end{align*}$$

\]
For the upper value:
\[

$$\begin{align*} 1.645&=\frac{x_2 - 22.8}{1.1}\\ x_2-22.8&=1.645\times1.1\\ x_2&=22.8+1.645\times1.1\\ x_2&=22.8 + 1.8095\\ x_2&\approx24.61 \end{align*}$$

\]

Answer:

The smallest head size: 20.99 inches
The largest head size: 24.61 inches