Example 6.33

EXAMPLE 6.33

Photo by The Photo Works

Outer diameter of a skateboard bearing. The mean outer diameter of a skateboard bearing is supposed to be 22.000 millimeters (mm). The outer diameters vary Normally with standard deviation σ = 0.010 mm. When a lot of the bearings arrives, the skateboard manufacturer takes an SRS of five bearings from the lot and measures their outer diameters. The manufacturer rejects the bearings if the sample mean diameter is significantly different from 22 mm at the 5% significance level.

This is a test of the hypotheses

H₀: μ = 22

H_a: μ ≠ 22

398

To carry out the test, the manufacturer computes the z statistic:

and rejects H₀ if

z < −1.96 or z > 1.96

A Type I error is to reject H₀ when in fact μ = 22.

What about Type II errors? Because there are many values of μ in H_a, we will concentrate on one value. The producer and the manufacturer agree that a lot of bearings with mean 0.015 mm away from the desired mean 22.000 should be rejected. So a particular Type II error is to accept H₀ when in fact μ = 22.015.

Figure 6.21 shows how the two probabilities of error are obtained from the two sampling distributions of , for μ = 22 and for μ = 22.015. When μ = 22, H₀ is true and to reject H₀ is a Type I error. When μ = 22.015, accepting H₀ is a Type II error. We will now calculate these error probabilities.

Figure 6.21 The two error probabilities, Example 6.33. The probability of a Type I error (yellow area) is the probability of rejecting H₀: μ = 22 when, in fact, μ = 22. The probability of a Type II error (blue area) is the probability of accepting H₀ when, in fact, μ = 22.015.