Questions: Engineers must consider the breadths of male heads when designing helmets. The company researchers have determined that the population of potential clientele have head breadths that are normally distributed with a mean of 5.9 inches and a standard deviation of 0.8 inches. X be the head breadth of a potential clientele. a. What is the distribution of X ? X N 0 ( , ) b. If a client is randomly chosen, find the probability that his head breadth is at least 6.3 inches. c. If a client is randomly chosen, find the probability that his head breadth is between 5.5 and 5.9 inches. d. Due to financial constraints, the helmets will be designed to fit all men except those with head breadths that are in the smallest 3.9 % or largest 3.9 %. What is the minimum and maximum head breadths that will fit the clientele? Minimum and maximum = and inches.

Engineers must consider the breadths of male heads when designing helmets. The company researchers have determined that the population of potential clientele have head breadths that are normally distributed with a mean of 5.9 inches and a standard deviation of 0.8 inches. X be the head breadth of a potential clientele.

a. What is the distribution of X ? X

N 0 ( , )

b. If a client is randomly chosen, find the probability that his head breadth is at least 6.3 inches.

c. If a client is randomly chosen, find the probability that his head breadth is between 5.5 and 5.9 inches.

d. Due to financial constraints, the helmets will be designed to fit all men except those with head breadths that are in the smallest 3.9 % or largest 3.9 %. What is the minimum and maximum head breadths that will fit the clientele? Minimum and maximum = and inches.

Transcript text: Engineers must consider the breadths of male heads when designing helmets. The company researchers have determined that the population of potential clientele have head breadths that are normally distributed with a mean of 5.9 inches and a standard deviation of 0.8 inches. $X$ be the head breadth of a potential clientele. a. What is the distribution of $X$ ? $X$ N 0 ( , ) b. If a client is randomly chosen, find the probability that his head breadth is at least 6.3 inches. c. If a client is randomly chosen, find the probability that his head breadth is between 5.5 and 5.9 inches. d. Due to financial constraints, the helmets will be designed to fit all men except those with head breadths that are in the smallest $3.9 \%$ or largest $3.9 \%$. What is the minimum and maximum head breadths that will fit the clientele? Minimum and maximum = and inches.

Solution

Solution Steps

Step 1: Distribution of $ X $

The head breadth $ X $ of potential clientele is normally distributed with a mean $ \mu = 5.9 $ inches and a variance $ \sigma^2 = 0.64 $ inches. Thus, we can express the distribution as: \[ X \sim N(5.9, 0.64) \]

Step 2: Probability of Head Breadth at Least 6.3 Inches

To find the probability that a randomly chosen client's head breadth is at least 6.3 inches, we calculate: \[ P(X \geq 6.3) = 1 - P(X < 6.3) = 1 - \Phi(Z_{end}) = 1 - \Phi(0.5) = 0.6915 \] where $ Z_{end} $ corresponds to the z-score for 6.3 inches. Therefore, the probability that a client has a head breadth of at least 6.3 inches is: \[ P(X \geq 6.3) = 0.6915 \]

Step 3: Probability of Head Breadth Between 5.5 and 5.9 Inches

Next, we find the probability that a randomly chosen client's head breadth is between 5.5 and 5.9 inches: \[ P(5.5 < X < 5.9) = P(X < 5.9) - P(X < 5.5) = \Phi(0.0) - \Phi(-0.5) = 0.1915 \] Thus, the probability that a client has a head breadth between 5.5 and 5.9 inches is: \[ P(5.5 < X < 5.9) = 0.1915 \]