Why is type II error (Beta risk) more likely if the actual process mean falls wi

Question

Why is type II error (Beta risk) more likely if the actual process mean falls within the control limits of a variable control chart as opposed to outside the control limits of such a control chart?  Defend your explanation by discussing Equation (6.19) and assume L = 3.

My thinking is that the probability of not detecing a shift when the mean is within the control limits (L=3) is that even when the mean is within the control limits, there is still a risk of 0.0708 that a type II error would occur. If the mean were to go out of the control limits, the likelihood of detecting that shift would increase, but I'm not sure how this is shown in the equation.

6.2.6 The Operating Characteristic Function (6.19) To illustrate the use of equation 6.19, suppose that we are using an x chart with 3 (the usual three-sigma limits), the sample size n = 5, and we wish to determine the probabil- ity of detecting a shift to ,-H0 + 20 on the first sample following the shift. Then, since L = 3, k = 2, and n 5, we have (-1.47)-0(-7.37) 0.0708 This is the -risk, or the probability of not detecting such a shift. The probability that such a shift will be detected on the first subsequent sample is 1- = 1-0.0708 = 0.9292.

Explanation / Answer

Answer to the question)

The type II error is denoted by Beta and the value of Beta in this situation is 0.0708

Which is very less

This equation basically aims to tell us that fi the mean is within the control limits , than the power of the test is good

the power of the test here is given by 1- beta

So this entire example tells us that if the mean value is within the control limits, then the power of detecting the same is strong , as given by the value 0.9292

This implies there is 92.92% chance of detecting the mean within the control limits.

[It seems you are considering beta 0.0708 to be higher as compared to the significance level 0.05. Significance level 0.05 denotes Type I error , so we do not compare beta to it. A value of beta 0.0708 is still considered to be less. it depends on the researcher which risk he is willing to take : type I or type II. In any case either of the two has to be more]

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