Final answer:
The mean diameter of the screws is 0.15 cm and the standard deviation is 0.0145 cm. The probability that 50 randomly selected screws will be within the stated tolerance levels is very small, indicating that the company's diameter claim may not be plausible.
Step-by-step explanation:
To find the mean diameter, we can use the formula:
Mean = (Lower Limit + Upper Limit) / 2
Given the range of 0.10 cm to 0.20 cm, the lower limit is 0.10 cm and the upper limit is 0.20 cm. Plugging these values into the formula, we get:
Mean = (0.10 + 0.20) / 2 = 0.15 cm
To find the standard deviation, we can use the formula:
Standard Deviation = (Upper Limit - Lower Limit) / (2 * √3)
Plugging in the values, we get:
Standard Deviation = (0.20 - 0.10) / (2 * √3) = 0.05 / (2 * √3) ≈ 0.0145 cm
For part b of the question, we need to calculate the probability that 50 randomly selected screws will be within the stated tolerance levels. Since the distribution is uniform within the range, the probability is calculated as:
Probability = (Number of screws in tolerance range) / (Total number of screws)
Given that the range is 0.10 cm to 0.20 cm, the number of screws in the tolerance range is 0.20 cm - 0.10 cm = 0.10 cm. The total number of screws is 20 screws per box, so for 50 randomly selected screws, the total number of screws is 50 * 20 = 1000 screws. Plugging in these values, we get:
Probability = 0.10 cm / 1000 screws ≈ 0.0001
Since the probability is very small, it is unlikely that 50 randomly selected screws will be within the stated tolerance levels. Therefore, the company's diameter claim may not be plausible.