The diameters of bolts produced by a certain machine are normally distributed with a mean of 0.25 inches and a standard deviation of 0.01 inches. What percentage of bolts will have a diameter greater than 0.24 inches?
A. 79.13%
B. 82.46%
C. 84.13%
D. 84.46%
answer:C
correct,
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To find the percentage of bolts that will have a diameter greater than 0.24 inches, we need to calculate the area under the normal distribution curve to the right of 0.24 inches.
1. We first calculate the z-score for the value 0.24 inches using the formula: z = (x - μ) / σ.
- x is the value (0.24 inches),
- μ is the mean (0.25 inches), and
- σ is the standard deviation (0.01 inches).
Plugging in the values, we get: z = (0.24 - 0.25) / 0.01 = -1.
2. Once we have the z-score, we can use a z-table or a calculator to find the area under the curve to the right of the z-score.
Looking up the z-score of -1 in a standard normal distribution table or using a calculator, we find that the area to the left of -1 is 0.1587.
3. The total area under the curve is 1, so to find the area to the right of -1, we subtract the area to the left of -1 from 1:
Area to the right = 1 - 0.1587 = 0.8413.
4. Finally, we convert the decimal to a percentage by multiplying by 100:
Percentage = 0.8413 × 100 = 84.13%.
Therefore, the percentage of bolts that will have a diameter greater than 0.24 inches is 84.13%.
Hence, the correct answer is C. 84.13%.