8 Probability intervals - general normal distribution

We saw in subsection 3 that 95 % of the area under the standard normal curve lay between z 1 = 1.96 and z 2 = 1.96 . Using the formula Z = X μ σ in the re-arrangement X = μ + Z σ . We can see that 95 % of the area under the general normal curve lies between x 1 = μ 1.96 σ and x 2 = μ + 1.96 σ .

Figure 14

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Example 14

Suppose that the internal diameters of mass-produced pipes are normally distributed with mean 50 mm and standard deviation 2 mm. What are the 95 % probability limits on the internal diameter of a single pipe?

Solution

Here μ = 50 σ = 2 so that the 95 % probability limits are

50 ± 1.96 × 2 = 50 ± 3.92 mm

i.e. 46.08 mm and 53.92 mm.

The probability interval is (46.08, 53.92).

Task!

What is the 99 % probability interval for the lifetime of a bulb when the lifetimes of such bulbs are normally distributed with a mean of 2000 hours and standard deviation of 40 hours?

First sketch the standard normal curve marking the values z 1 , z 2 between which 99 % of the area under the curve is located:

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Now deduce the corresponding values x 1 , x 2 for the general normal distribution:

x 1 = μ 2.58 σ , x 2 = μ + 2.58 σ

Next, find the values for x 1 and x 2 for the given problem:

x 1 = 2000 2.58 × 40 = 1896.8 hours

x 2 = 2000 + 2.58 × 40 = 2103.2 hours

Finally, write down the 99 % probability interval for the lifetimes:

(1896.8 hours, 2103.2 hours).