Questions: Engineers must consider the diameters of heads when designing helmets. The company researchers have determined that the population of potential clientele have head diameters that are normally distributed with a mean of 5.7-in and a standard deviation of 1.1-in. Due to financial constraints, the helmets will be designed to fit all men except those with head diameters that are in the smallest 0.8% or largest 0.8%. What is the minimum head diameter that will fit the clientele? min = What is the maximum head diameter that will fit the clientele? max = Enter your answer as a number accurate to 1 decimal place. Answers obtained using exact z-scores or z scores rounded to 3 decimal places are accepted.

Engineers must consider the diameters of heads when designing helmets. The company researchers have determined that the population of potential clientele have head diameters that are normally distributed with a mean of 5.7-in and a standard deviation of 1.1-in. Due to financial constraints, the helmets will be designed to fit all men except those with head diameters that are in the smallest 0.8% or largest 0.8%.

What is the minimum head diameter that will fit the clientele?
min =

What is the maximum head diameter that will fit the clientele?

max =

Enter your answer as a number accurate to 1 decimal place. Answers obtained using exact z-scores or z scores rounded to 3 decimal places are accepted.

Transcript text: Engineers must consider the diameters of heads when designing helmets. The company researchers have determined that the population of potential clientele have head diameters that are normally distributed with a mean of $5.7-\mathrm{in}$ and a standard deviation of $1.1-\mathrm{in}$. Due to financial constraints, the helmets will be designed to fit all men except those with head diameters that are in the smallest $0.8 \%$ or largest $0.8 \%$. What is the minimum head diameter that will fit the clientele? $\min =$ $\square$ What is the maximum head diameter that will fit the clientele? \[ \max = \] $\square$ Enter your answer as a number accurate to 1 decimal place. Answers obtained using exact $\mathbf{z}$-scores or $z$ scores rounded to 3 decimal places are accepted.

Solution

Solution Steps

To determine the minimum and maximum head diameters that will fit the clientele, we need to find the head diameters corresponding to the 0.8th percentile and the 99.2th percentile of a normal distribution with a mean of 5.7 inches and a standard deviation of 1.1 inches. We can use the inverse cumulative distribution function (percent point function) to find these values.

Step 1: Identify the Given Parameters

The head diameters are normally distributed with:

Mean ($\mu$) = 5.7 inches
Standard deviation ($\sigma$) = 1.1 inches

Step 2: Determine the Percentiles

We need to find the head diameters corresponding to the 0.8th percentile and the 99.2th percentile of the normal distribution.

Step 3: Calculate the Z-Scores

Using the inverse cumulative distribution function (percent point function), we find:

$ z_{\text{lower}} = \text{ppf}(0.008) $
$ z_{\text{upper}} = \text{ppf}(0.992) $

Step 4: Convert Z-Scores to Head Diameters

The head diameters corresponding to these percentiles are calculated as:

$ \text{min\_diameter} = \mu + z_{\text{lower}} \cdot \sigma $
$ \text{max\_diameter} = \mu + z_{\text{upper}} \cdot \sigma $

Step 5: Substitute the Values

Substituting the values, we get:

$ \text{min\_diameter} \approx 5.7 + (-2.4098) \cdot 1.1 \approx 3.0502 $ inches
$ \text{max\_diameter} \approx 5.7 + 2.4098 \cdot 1.1 \approx 8.3498 $ inches

Step 6: Round to One Decimal Place

Rounding the results to one decimal place: