https://file.aiquickdraw.com/mathbot/file/1729131174863ptmwoo7w.jpg

To solve this problem, we need to calculate the probability that a randomly selected pilot has a weight between 150 lb and 191 lb, given that the weights follow a normal distribution with a mean of 158 lb and a standard deviation of 30.5 lb.

The formula for the standard normal variable \( Z \) is:
\[
Z = \frac{X - \mu}{\sigma}
\]
where:
- \( X \) is the value of interest,
- \( \mu \) is the mean,
- \( \sigma \) is the standard deviation.

We calculate the \( Z \)-scores for 150 lb and 191 lb.

### Step 1: Calculate the \( Z \)-score for 150 lb
\[
Z_{150} = \frac{150 - 158}{30.5} = \frac{-8}{30.5} \approx -0.2623
\]

### Step 2: Calculate the \( Z \)-score for 191 lb
\[
Z_{191} = \frac{191 - 158}{30.5} = \frac{33}{30.5} \approx 1.081967
\]

### Step 3: Use the standard normal distribution table or a calculator
- For \( Z = -0.2623 \), the cumulative probability is approximately 0.3966.
- For \( Z = 1.081967 \), the cumulative probability is approximately 0.8599.

### Step 4: Calculate the probability
The probability that the pilot's weight is between 150 lb and 191 lb is the difference between the two cumulative probabilities:
\[
P(150 \leq X \leq 191) = 0.8599 - 0.3966 = 0.4633
\]

Thus, the probability is approximately **0.4633**.

Do you have any questions or need further details?

Here are 5 related questions to expand on this:
1. How do we use the standard normal table to find probabilities?
2. What if the weight range was different, like 140 lb to 180 lb?
3. How would we solve this problem using a graphing calculator or software like Excel?
4. What is the significance of the standard deviation in this context?
5. How do changes in the mean affect the probability for different weight ranges?

**Tip:** When working with normal distributions, always convert raw values into \( Z \)-scores before consulting the standard normal distribution table.

An engineer is redesigning an ejection seat for an airplane. The seat was designed for pilots weighing between 150 lb and 191 lb. The new population of pilots has a normally distributed weight with a mean of 158 lb and a standard deviation of 30.5 lb. What is the probability that a randomly selected pilot weighs between 150 lb and 191 lb?

Learn how to calculate the probability that a pilot's weight falls between 150 lb and 191 lb. This example uses a normal distribution with a mean of 158 lb and a standard deviation of 30.5 lb, applying Z-scores to find the solution. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation