To solve these questions, we'll use the properties of the normal distribution, specifically the Z-score formula and standard normal distribution tables (or a calculator).

Given:

• Mean (μ) = $367
• Standard Deviation (σ) = $88

(a) Probability that the cost will be more than $460

1. Calculate the Z-score: $Z = \frac{X - \mu}{\sigma} = \frac{460 - 367}{88} = \frac{93}{88} \approx 1.057$

2. Find the probability from the Z-table: Look up $Z = 1.057$. The cumulative probability for $Z = 1.057$ is approximately 0.8554.

3. Calculate the probability of being more than $460: $P(X > 460) = 1 - P(Z < 1.057) = 1 - 0.8554 \approx 0.1446$

(b) Probability that the cost will be less than $250

1. Calculate the Z-score: $Z = \frac{250 - 367}{88} = \frac{-117}{88} \approx -1.33$

2. Find the probability from the Z-table: Look up $Z = -1.33$. The cumulative probability for $Z = -1.33$ is approximately 0.0918.

(c) Probability that the cost will be between $250 and $460

1. Using previously found probabilities: $P(250 < X < 460) = P(X < 460) - P(X < 250) = P(Z < 1.057) - P(Z < -1.33)$ From earlier:

   • $P(X < 460) \approx 0.8554$
   • $P(X < 250) \approx 0.0918$

2. Calculate: $P(250 < X < 460) = 0.8554 - 0.0918 \approx 0.7636$

(d) Maximum cost in the lower 5% of automobile repair charges

1. Find the Z-score corresponding to the lower 5%: The Z-score for the lower 5% is approximately -1.645.

2. Calculate the maximum cost: $X = \mu + Z\sigma = 367 + (-1.645)(88) \approx 367 - 145.56 \approx 221.44$

Summary of Results:

• (a) $P(X > 460) \approx 0.1446$
• (b) $P(X < 250) \approx 0.0918$
• (c) $P(250 < X < 460) \approx 0.7636$
• (d) Maximum cost in the lower 5% is approximately $221.44.

If you would like further details or clarifications, feel free to ask! Here are some related questions you might consider:

1. How do you interpret the results of normal distribution probabilities?
2. What other factors could influence automobile repair costs?
3. How would changes in the mean or standard deviation affect the probabilities?
4. What is the significance of the lower 5% in practical terms?
5. Can you explain the concept of Z-scores in more detail?

Tip: Always ensure that you round correctly based on the context of the problem, as it can significantly affect your final answers.