You may need to use the appropriate appendix table to answer this question.
Automobile repair costs continue to rise with the average cost now at $367 per repair.† Assume that the cost for an automobile repair is normally distributed with a standard deviation of $88. Answer the following questions about the cost of automobile repairs.
(a)
What is the probability that the cost will be more than $460? (Round your answer to four decimal places.)
0.15876
 
Incorrect: Your answer is incorrect.
(b)
What is the probability that the cost will be less than $250? (Round your answer to four decimal places.)
0.1587
 
Incorrect: Your answer is incorrect.
(c)
What is the probability that the cost will be between $250 and $460? (Round your answer to four decimal places.)
0.6826
 
Incorrect: Your answer is incorrect.
(d)
If the cost for your car repair is in the lower 5% of automobile repair charges, what is your maximum possible cost in dollars? (Round your answer to the nearest cent.)
$ 
224.54
 
Incorrect: Your answer is incorrect.

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.

What is the probability that the cost of automobile repairs will be more than $460, less than $250, between $250 and $460, and what is the maximum cost for repairs in the lower 5%?

Explore the probabilities related to automobile repair costs, including the chances of costs exceeding $460, falling below $250, and being between $250 and $460. Learn about the maximum cost for repairs in the lower 5%, based on a normal distribution with a mean of $367 and a standard deviation of $88.

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