Answer:
the probability that a randomly chosen refill will fit correctly is 0.9780 (rounded to three decimal places).
Explanation:
We can use the standard normal distribution to solve this problem. We first need to calculate the z-scores for the given diameters.
a) To find the probability that a randomly chosen refill will be too large, we need to find the probability of getting a diameter greater than 0.520 mm.
z-score = (0.520 - 0.5) / 0.007 = 2.857
Using a standard normal table or calculator, we find that the probability of getting a z-score of 2.857 or greater is 0.0021.
Therefore, the probability that a randomly chosen refill will be too large is 0.0021 (rounded to three decimal places).
b) To find the probability that a randomly chosen refill will be too small, we need to find the probability of getting a diameter less than 0.485 mm.
z-score = (0.485 - 0.5) / 0.007 = -2.143
Using a standard normal table or calculator, we find that the probability of getting a z-score of -2.143 or less is 0.0164.
Therefore, the probability that a randomly chosen refill will be too small is 0.0164 (rounded to three decimal places).
c) To find the probability that a randomly chosen refill will fit correctly, we need to find the probability of getting a diameter between 0.485 mm and 0.520 mm.
First, we find the z-scores for each diameter:
z-score for 0.485 mm = (0.485 - 0.5) / 0.007 = -2.143
z-score for 0.520 mm = (0.520 - 0.5) / 0.007 = 2.857
Using a standard normal table or calculator, we find that the probability of getting a z-score between -2.143 and 2.857 is 0.9780.
Therefore, the probability that a randomly chosen refill will fit correctly is 0.9780 (rounded to three decimal places).