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The problem provided involves automobile repair costs which are normally distributed with a mean of $367 and a standard deviation of $88. Several probability-related questions are asked, which we can solve using properties of the normal distribution.

### Given information:
- Mean (μ) = $367
- Standard deviation (σ) = $88
- Z-score formula: $ Z = \frac{X - \mu}{\sigma} $, where X is the cost of the repair.

Let's break down each part:

### (a) What is the probability that the cost will be more than $450?

To solve this, first, we calculate the Z-score for $450.

\[
Z = \frac{450 - 367}{88} = \frac{83}{88} \approx 0.9432
\]

We can now look up this Z-score in the standard normal distribution table (or use a calculator). The probability for a Z-score of 0.9432 is approximately 0.8264. Thus, the probability that the cost is more than $450 is:

\[
P(X > 450) = 1 - 0.8264 = 0.1736
\]

### (b) What is the probability that the cost will be less than $250?

Next, we calculate the Z-score for $250.

\[
Z = \frac{250 - 367}{88} = \frac{-117}{88} \approx -1.3295
\]

Looking up this Z-score in the normal distribution table, the cumulative probability is approximately 0.0918. So the probability that the cost is less than $250 is:

\[
P(X < 250) = 0.0918
\]

### (c) What is the probability that the cost will be between $250 and $450?

This requires finding the probability between two Z-scores: one for $250 and one for $450, which we already calculated.

For $250, the cumulative probability is 0.0918, and for $450, it's 0.8264. So the probability that the cost lies between these two values is:

\[
P(250 < X < 450) = 0.8264 - 0.0918 = 0.7346
\]

### (d) If the cost for your car repair is in the lower 5% of charges, what is your cost?

For the lower 5%, we need to find the Z-score corresponding to a cumulative probability of 0.05. From the Z-table, this Z-score is approximately -1.645.

Now, convert the Z-score back to the actual cost using the formula $ X = Z \cdot \sigma + \mu $:

\[
X = -1.645 \cdot 88 + 367 = -144.68 + 367 = 222.32
\]

So, if your car repair cost is in the lower 5%, it is approximately **$222.24** (rounded to two decimals).

---

Would you like further details on any part of the solution or have additional questions? 

Here are 5 related questions for deeper understanding:
1. What is the probability that the cost will exceed $500?
2. What would be the cost if it lies in the upper 10%?
3. How do we calculate the Z-score for a given value?
4. Can the normal distribution be used for repair costs under $100?
5. How would the probability change if the standard deviation were higher?

**Tip:** The Z-score helps normalize any value by comparing it to the mean in terms of standard deviations.

Automobile repair costs continue to rise with the average cost now at $367 per repair (U.S. News & World Report website). 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:

1. What is the probability that the cost will be more than $450 (to four decimals)?
2. What is the probability that the cost will be less than $250 (to four decimals)?
3. What is the probability that the cost will be between $250 and $450 (to four decimals)?
4. If the cost for your car repair is in the lower 5% of automobile repair charges, what is your cost (to two decimals)?

Solve probability problems related to automobile repair costs using normal distribution. Learn to calculate Z-scores and probabilities for costs exceeding $450, below $250, and within a specific range. Includes a calculation for the lower 5% of repair charges. Suitable for advanced high school or college-level statistics students.

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