The sizes of CDs are normally distributed with a standard deviation of 1 mm. The CDs are expected to be 120 mm in diameter. We need to find the percentage of the CDs that are wider than that. To begin, let's draw a normal curve for this distribution.
The normal curve is always symmetric with respect to the mean μ= 120 of the data set. In this case, this means that the number of CDs that are less than 120 mm in diameter is the same as the number of CDs wider than 120 mm.
Consequently, we expect 50 % of CDs to be greater than 120 mm.
b
The company manufactures 1000 CDs per hour. We are asked to find the number of CDs between 119 and 122 mm made in one hour. For that we can use the Empirical Rule.
The rule tells us that that some percentage of observed values lie within an integral estimate, as shown in the graph. The estimate depends on the standard deviation of the data set, in this case σ = 1.
The area under the curve representing the values within the given range of 119 mm and 122 mm can be represented as the sum of three areas.
If we add these areas, we get that 34 % + 34 % + 13.5 % = 81.5 % of the CDs manufactured is between 119 and 122 mm in diameter. Now, if 1000 CDs are made every hour, this gives us the number of CDs within the given range.
1000 * 81.5 % = 8150
c
In this exercise, we need to find the number of CDs made per hour that are too wide to fit the drives. We were told that any CD wider than 122 mm will not fit the drive. We can show that portion of the CDs in the graph.
Through summing the corresponding areas we get that 2.35 %+ 0.15 %=2.5 % of the CDs will be too wide. This gives us the number of such CDs made every hour.
1000 * 2.5 % = 25
