Question 8

Engineers must consider the breadths of male heads when designing heimets. The company researchers have determined that the population of potential clientele have head breadths that are Normally distributed with a mean of 6.4 inches and a standard deviation of 0.9 inches.

According to the 68-95:99.7 rule we expect 95%of head breadths to be between and inches.

Step 1 ：The problem involves designing helmets considering the breadths of male heads. The company researchers have determined that the population of potential clientele have head breadths that are Normally distributed with a mean of 6.4 inches and a standard deviation of 0.9 inches.

Step 2 ：We are asked to find the range of head breadths for 95% of the population. According to the 68-95-99.7 rule, also known as the empirical rule, 95% of the data falls within two standard deviations of the mean in a normal distribution.

Step 3 ：Let's denote the mean as \( \mu \) and the standard deviation as \( \sigma \). In our case, \( \mu = 6.4 \) inches and \( \sigma = 0.9 \) inches.

Step 4 ：To find the range, we need to calculate \( \mu - 2\sigma \) and \( \mu + 2\sigma \).

Step 5 ：Calculating \( \mu - 2\sigma \), we get \( 6.4 - 2 \times 0.9 = 4.6 \) inches.

Step 6 ：Calculating \( \mu + 2\sigma \), we get \( 6.4 + 2 \times 0.9 = 8.2 \) inches.

Step 7 ：Final Answer: We expect 95% of head breadths to be between \( \boxed{4.6} \) inches and \( \boxed{8.2} \) inches.