Automobile repair costs continue to rise with the average cost now at $367 per repair (U.S. News & World Report website, January 5, 2015). 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. Use Table 1 in Appendix B.
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To answer the questions about the cost of automobile repairs, we can use the normal distribution and the given information regarding the average cost and standard deviation.
1. What is the probability that an automobile repair cost is less than $250?
To find the probability that an automobile repair cost is less than $250, we need to find the z-score corresponding to this value and then look up the corresponding probability in the standard normal distribution table (Table 1 in Appendix B).
The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
x = given value ($250 in this case)
μ = mean average cost ($367)
σ = standard deviation ($88)
Plugging in the values:
z = (250 - 367) / 88
z ≈ -1.33
Now, we can look up the z-score of -1.33 in the standard normal distribution table to find the corresponding probability. The table will give us the area under the curve to the left of the z-score.
Looking up -1.33 in the z-score column, we find that the corresponding probability is approximately 0.0918 or 9.18%.
Therefore, the probability that an automobile repair cost is less than $250 is approximately 0.0918 or 9.18%.
2. What is the probability that an automobile repair cost is between $400 and $500?
To find the probability that an automobile repair cost is between $400 and $500, we need to find the z-scores corresponding to these values and calculate the area between them.
First, we calculate the z-score for $400:
z1 = (400 - 367) / 88
z1 ≈ 0.375
Next, we calculate the z-score for $500:
z2 = (500 - 367) / 88
z2 ≈ 1.511
Now, we need to find the area between these two z-scores in the standard normal distribution table.
Looking up z1 = 0.375 in the z-score column, we find that the corresponding cumulative probability is approximately 0.6480 or 64.80%.
Looking up z2 = 1.511 in the z-score column, we find that the corresponding cumulative probability is approximately 0.9340 or 93.40%.
To find the area between these two probabilities, we subtract the smaller probability from the larger probability:
Area between = 0.9340 - 0.6480
Area between ≈ 0.2860 or 28.60%
Therefore, the probability that an automobile repair cost is between $400 and $500 is approximately 0.2860 or 28.60%.
Please note that the values obtained may be approximations due to the rounded z-scores used in the calculations. For more precise values, a more accurate z-score may be needed.