How and why is B= 0.1 in this question? Example 9-3 Propellant Burning Rate Type
ID: 3320974 • Letter: H
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How and why is B= 0.1 in this question?
Example 9-3 Propellant Burning Rate Type Il Error Consider the rocket propellant problem of Example 9-2. Suppose that the true burning rate is 49 centimeters per second. What is for the two-sided test with = 0.05, = 2, and n = 25? Section 9-2/Tests on the Mean of a Normal Distribution, Variance Known 327 Here = 1 and z = 1.96. From Equation 9-20, 25 25 = | 1 .96 -o(-0.54)-(-4.46) = 0.295 The probability is about 0.3 that this difference from 50 centimeters per second will not be detected. That is, the probabil- ity is about 0.3 that the test will fail to reject the null hypothesis when the true burning rate is 49 centimeters per second. Practical Interpretation: A sample size of n - 25 results in reasonable, but not great, power-1--1-0.3 0.70 Suppose that the analyst wishes to design the test so that if the true mean burning rate differs from 50 centimeters per second by as much as 1 centimeter per second, the test will detect this (i.e., reject Ho: = 50) with a high prob- ability, say, 0.90. Now we note that = 2, = 51-50 = 1, = 0.05, and = 0.10. Because z/,-z002,-1.96 and = z0.10 = 1.28, the sample size required to detect this departure from Ho: = 50 is found by Equation 9-22 as n » Ka/? +zs)o(1.96+1.28)2 - 42 2 The approximation is good here, because (-an-5vn )-0(-1.96-(1) V42 2) = (-5.20)-0, which is small relative to Practical Interpretation: To achieve a much higher power of 0.90, you will need a considerably large sample size, n = 42 instead of n = 25Explanation / Answer
You should have given page number 326 also to make you understand better.However,"will be detected" means H_0 is declared false,"will not be detected means" H_0 is declared true,now power means probability of declaring H_0 false when H_) is actally false in nature. "The test will reject H_0:mu=50 when the true difference is shifted even by 1 cm per second from 50 cm per second" is basically the power of the test". So,beta(B) is basically the type II error which means "not detecing H_0 false when H_0 is actually false in nature".In this case,B is probability of the event that "even if the true mean differs from 50 cm by 1 cm,it will not be detected from the test" So, when H_0 is false(because true mean is 50-1cm=49cm,not 50cm),there are two events,"Declaring H_0 false" and "declaring H_0 true" and these events are mutually exclusive and exhaustive.Now, power=P("The test will reject H_0:mu=50 when the true difference is shifted even by 1 cm per second from 50 cm per second")=0.9(given) So,being mutually exclusive and exhaustive, B=P("The test will not reject H_0:mu=50 when the true difference is shifted even by 1 cm per second from 50 cm per second")1-power=1-0.9=0.1
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