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 $440? (Round your answer to four decimal places.) (b) What is the probability that the cost will be less than $270? (Round your answer to four decimal places.) (c) What is the probability that the cost will be between $270 and $440? (Round your answer to four decimal places.) (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.)

Answer 1

The probability that the cost of an automobile repair will be more than $440 is 0.2266. The probability that the cost will be less than $270 is 0.1546. The probability that the cost will be between $270 and $440 is 0.8620. The maximum possible cost in the lower 5% of automobile repair charges is $270.98.

Step-by-step explanation:

To answer these questions about the cost of automobile repairs, we will use the normal distribution. The average cost is $367, with a standard deviation of $88.

(a) To find the probability that the cost will be more than $440, we need to find the area to the right of $440 in the normal distribution curve. We can use the z-score formula to standardize the value of $440 using the formula: (440 - average) / standard deviation. Once we have the z-score, we can look up the corresponding probability using a standard normal distribution table. The probability turns out to be 0.2266.

(b) To find the probability that the cost will be less than $270, we need to find the area to the left of $270 in the normal distribution curve. Again, we can use the z-score formula to standardize the value of $270. The resulting z-score is -1.0227, and the corresponding probability is 0.1546.

(c) To find the probability that the cost will be between $270 and $440, we need to find the area between these two values in the normal distribution curve. We can use the z-score formula to standardize the values of $270 and $440, and then subtract the probability of being less than $270 from the probability of being less than $440. The resulting probability is 0.8620.

(d) To find the maximum possible cost in the lower 5% of automobile repair charges, we need to find the z-score that corresponds to the 5th percentile of the normal distribution. We can use a standard normal distribution table to look up the z-score, which turns out to be -1.645. Then, we can use the z-score formula to find the corresponding value of the cost: (z-score * standard deviation) + average. The maximum possible cost is $270.98.

Answer 2

The probability that the cost of automobile repair will be more than $440 is approximately 20.31%. The probability that the cost will be less than $270 is approximately 13.57%. The probability that the cost will be between $270 and $440 is approximately 6.74%. The maximum possible cost in the lower 5% of automobile repair charges is approximately $244.18.

To answer these probability questions, we can use the z-score formula. The z-score is calculated by subtracting the mean from the given value and dividing by the standard deviation.

(a) To find the probability that the cost will be more than $440, we need to find the area under the curve to the right of $440. Using the z-score formula, the z-score would be (440 - 367) / 88 = 0.83. We can then find this probability using a standard normal distribution table or calculator, which is approximately 0.2031 or 20.31%.

(b) To find the probability that the cost will be less than $270, we need to find the area under the curve to the left of $270. Using the z-score formula, the z-score would be (270 - 367) / 88 = -1.10. We can then find this probability using a standard normal distribution table or calculator, which is approximately 0.1357 or 13.57%.

(c) To find the probability that the cost will be between $270 and $440, we can subtract the probability from part (b) from the probability from part (a). This gives us approximately 0.0674 or 6.74%.

(d) To find the maximum possible cost in the lower 5% of automobile repair charges, we need to find the z-score corresponding to the 5th percentile. The 5th percentile is -1.645. Using the z-score formula, we can solve for the cost: (x - 367) / 88 = -1.645. Solving for x, we find that the maximum possible cost is approximately $244.18.