Certainly! We can use the normal cumulative distribution function (normalcdf) on a calculator or software to directly find the required percentages.

Part (a): Percentage of lids too small to fit

We need to find the percentage of lids with a diameter less than 3.95 inches.

Using the normalcdf function: $\text{normalcdf}(-\infty, 3.95, 3.98, 0.02)$

• Lower bound: $-\infty$ (or a very large negative number, but in practice, we use a low number like -1000)
• Upper bound: 3.95
• Mean ($\mu$): 3.98
• Standard deviation ($\sigma$): 0.02

Using this in a calculator: $\text{normalcdf}(-1000, 3.95, 3.98, 0.02) \approx 0.0668$

So, about 6.68% of the lids are too small to fit.

Part (b): Percentage of lids too big to fit

We need to find the percentage of lids with a diameter greater than 4.05 inches.

Using the normalcdf function: $\text{normalcdf}(4.05, \infty, 3.98, 0.02)$

• Lower bound: 4.05
• Upper bound: $\infty$ (or a very large positive number, but in practice, we use a high number like 1000)
• Mean ($\mu$): 3.98
• Standard deviation ($\sigma$): 0.02

Using this in a calculator: $\text{normalcdf}(4.05, 1000, 3.98, 0.02) \approx 0.0002$

So, about 0.02% of the lids are too big to fit.

Part (c): Comparison and Analysis

The calculations confirm that 6.68% of the lids are too small, while only 0.02% are too large. This large difference suggests that the manufacturing process might slightly favor producing lids that are closer to the lower bound of the acceptable range.

Do you want more details or have any questions?

Follow-up Questions:

1. How would the percentages change if the mean were changed to 4.00 inches?
2. What is the percentage of lids that are within the range of 3.96 to 4.04 inches?
3. How does changing the standard deviation to 0.01 inches affect the results?
4. What would the normalcdf output be for the entire range of acceptable diameters (3.95 to 4.05 inches)?
5. How would these percentages influence the design choices for the lid manufacturing process?

Tip: When using normalcdf in a calculator, ensure you input the lower and upper bounds carefully. Misinputting them can lead to incorrect results.