Question

Suppose a company manufactures plastic lids for disposable coffee cups. When the manufacturing process is working correctly, the diameters of the lids are approximately Normally distributed with a mean diameter of 4 inches and a standard deviation of 0.02 inches. You can assume the diameter of each lid is independent from the diameter of other lids produced. Show your work for each part.

a) What percentage of lids off the production line have a diameter less than 4.01 inches?

b) To make sure the machine is working correctly, each hour a random sample of 10 lids are selected and the sample mean is calculated. What is the probability that the mean of the 10 lids in a sample is greater than 4.01 inches, assuming the manufacturing process is working correctly?

c) Calculate the difference in diameter between the lid with the median diameter and the lid with the diameter at the 95th percentile.

Answer 1

We have the distibution of the lid as normally distributed.

z-score = = (x - 4) / 0.02

Then we will use the normal distrbution tables to calculate the probabilities.

1. P(X < 4.01) = P( Z < 0.5)

= 0.6915

Ans: 69.15%

2.

Here we have to find the probability of the average so we will need the distribution of the mean. We will use the central limit theorem which states that if population is normal then the distribution of the means of its sampls is also normal. The parameters are as follows.

Where n = 10

Therefore the z-score = ( - 4) / 0.0063

P( > 4.01) = 1 - P( < 4.01)

= 1 - P(Z <1.58)

=1 - 0.94308

= 0.0569

Ans: 5.69%

3.

Median is the value under and above which 50% of the data lies. So if 'x' is the median then

P( X < x) = P(X > x) = 50%

Using the normal percentage tables we have

P( X > x) = 50%

P( Z > (x - 4)/0.02) = 50% ..........next we will use normal percentage tables

(x - 4) / 0.02 = 0 ............................very small value so rounded to zero.

x = 4

The median = 4.

95th percentile means that a value 'x' below which 95% of the data lies

P( X <x) = 95%

Subtracting from 1 on both sides

P( X > x) = 0.05

P( Z > (x-4)/0.02) = 0.05

(x- 4)/0.02 = 1.6449

95th percentile x = 4.0329

Difference = 4.0329 - 4

Ans: 0.0329

If we see the median = mean this is because normal distribution is symmetrical which is why it is very likely of the valeus being equal. It is a property of the symmetrical distribution.

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