Final answer:
To ensure 80,000 non-defective smartphone cases, the company should produce approximately 83,830 cases, based on the normal distribution of case lengths with a mean of 13 cm and a standard deviation of 0.125 cm.
Step-by-step explanation:
To calculate how many total cases the company should produce to get 80,000 non-defective cases, we must understand the distribution of the case lengths and the proportion that are expected to be within the specification limits. Since the company has a normal distribution of case lengths with a mean of 13 cm and a standard deviation of 0.125 cm, we can use the standard normal distribution to find the proportion of cases that fall within the spec limits of 12.75 cm and 13.25 cm.
Let's calculate the Z-scores for the lower and upper specification limits:
- Zlower = (12.75 - 13) / 0.125 = -2
- Zupper = (13.25 - 13) / 0.125 = 2
Looking up these Z-scores in a standard normal distribution table, we find that approximately 95.45% of cases will be within specs. To find the total number of cases the company should produce, we divide the desired number of 80,000 by the proportion within specs:
Total cases needed = 80,000 / 0.9545 ≈ 83,830 cases.
The company should therefore produce approximately 83,830 cases to ensure that at least 80,000 cases meet the specification limits.