(g) Customers buying computer parts supplied by a certain manufacturer require that the parts be between 9.9cm and 10.5 cm, inclusive. The manufacturing process is such that the actual part lengths are well approximated by a normal distribution with mean 10.1 cm and standard deviation 0.20 cm. Determine the percentage of the manufacturer's production that is acceptable to the customer. (5 marks)
Question
(g) Customers buying computer parts supplied by a certain manufacturer require that the parts be between 9.9cm and 10.5 cm, inclusive. The manufacturing process is such that the actual part lengths are well approximated by a normal distribution with mean 10.1 cm and standard deviation 0.20 cm. Determine the percentage of the manufacturer's production that is acceptable to the customer. (5 marks)
Solution
To solve this problem, we need to use the properties of the normal distribution.
Step 1: Standardize the values We first convert the given lengths into z-scores, which are standard deviations from the mean. The formula for z-score is (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
For the lower limit of 9.9 cm: Z1 = (9.9 - 10.1) / 0.20 = -1.0
For the upper limit of 10.5 cm: Z2 = (10.5 - 10.1) / 0.20 = 2.0
Step 2: Find the probabilities Next, we look up these z-scores in the standard normal distribution table or use a calculator with a normal distribution function to find the probabilities.
The probability that a part is less than 9.9 cm is P(Z < -1.0) = 0.1587 (or 15.87%).
The probability that a part is less than 10.5 cm is P(Z < 2.0) = 0.9772 (or 97.72%).
Step 3: Find the probability that the part is acceptable The probability that a part is between 9.9 cm and 10.5 cm is the difference between these two probabilities.
P(9.9 < X < 10.5) = P(Z < 2.0) - P(Z < -1.0) = 0.9772 - 0.1587 = 0.8185 (or 81.85%).
So, 81.85% of the manufacturer's production is acceptable to the customer.
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