To solve these questions, we'll use the normal distribution properties. Given information:

• Mean ($\mu$) = 244 feet
• Standard deviation ($\sigma$) = 42 feet

Let's calculate each probability.

Question (a)

We want the probability that a randomly chosen fly ball traveled fewer than 210 feet.

1. First, find the z-score for 210 feet:

   $z = \frac{X - \mu}{\sigma} = \frac{210 - 244}{42} = \frac{-34}{42} \approx -0.81$

2. Using a standard normal table or calculator, we find the cumulative probability for $z = -0.81$.

   $P(Z < -0.81) \approx 0.2090$

So, the probability is approximately 20.90%.

Question (b)

We want the probability that a randomly chosen fly ball traveled more than 228 feet.

1. First, find the z-score for 228 feet:

   $z = \frac{X - \mu}{\sigma} = \frac{228 - 244}{42} = \frac{-16}{42} \approx -0.38$

2. Using a standard normal table or calculator, find the cumulative probability for $z = -0.38$.

   $P(Z < -0.38) \approx 0.3520$

3. Since we want the probability of traveling more than 228 feet, we calculate $P(Z > -0.38)$:

   $P(Z > -0.38) = 1 - P(Z < -0.38) = 1 - 0.3520 = 0.6480$

So, the probability is approximately 64.80%.

Answers

• (a) $P(\text{fewer than 210 feet}) \approx 20.90\%$
• (b) $P(\text{more than 228 feet}) \approx 64.80\%$

Would you like more details on any step, or do you have any additional questions?

Here are some related questions for further practice:

1. What is the probability that a randomly chosen fly ball traveled exactly 244 feet?
2. What is the probability that a fly ball traveled between 210 and 228 feet?
3. How would the probabilities change if the standard deviation were smaller?
4. How does changing the mean affect the probability calculations?
5. What is the z-score for a fly ball that traveled 300 feet?

Tip: For normal distribution problems, calculating the z-score helps standardize the values, making it easier to find probabilities.