Suppose the military has designed a new missile-guidance system. With the old sy

Question

Suppose the military has designed a new missile-guidance system. With the old system, 80% of missiles landed within thirty meters of the target at a range of five hundred miles. The military claims that the new system is better (that more than 80% of missiles will land within thirty meters at five hundred miles). Ten randomly-chosen, identical missiles will be launched using the new guidance system in order to test the null hypothesis that the accuracy rate of the new system is no better than that of the old system against the alternative hypothesis that the accuracy rate of the new system is better.

a. State the null and alternative hypotheses in terms of the parameter of a probability model for the sample observations.

b. What is the form of the rejection region?

c. Is a sample of size ten sufficiently large for the maximum probability of Type I error to be controlled at 10%? Show why or why not. What is the minimum sample size that permits the maximum probability of Type I error to be controlled at 10%?

d. The military decides to increase the sample size to fifteen. What critical value should be used to control the maximum probability of Type I error at 10%?

e. What is the power of this test (the hypothesis test using the rejection region you just calculated) if the true accuracy rate is 90%?

Explanation / Answer

(A) H0: p = 0.80

Ha : p > 0.80

where p is the proportion of successful target achieved by missile.

(b) Here rejection region would be the number of successful missile launched as per the guidance above which we will consider that the new method is successful in increasing the accuracy rate.

(c) Here sample size = 10

So if type I error is assumed. Given the binomial distributioon where n = 10 and p = 0.8

Pr(X >= C ; 10 ; 0.8) < 0.10

so for Pr(X = 10) = 0.8 ^ 10 = 0.1074

so that is greater than 0.10 so no, the sample size 10 is not sufficient. As there is no rejection region here.

Let say minimum sample size is n that permits the maximum probability of Type I error to be controlled at 10%.

so that means at least we have one value such that we can have a rejection region.

Pr(X >= C ; n ; 0.8) <= 0.10

0.8n < 0.10

n ln (0.8) < ln (0.10)

n < 10.32

so n shall be minimum 11 to have atleast one value of C where we can have a rejection region.

(d) Sample size = 15

Now type I error = 10%

so let say critical value c above which we shall reject the null hypothesis,

Pr(X >= C ; 15 ; 0.8) <= 0.10

so BIN (X < C ; 15 ; 0.8) > = 0.90

Binomial table the value of C = 15

that means that if 15 out of 15 is successful here then we reject the null hypothesis.

(e) Now, if true accuracy rate is 90% then

Power of the test would be = Pr(X = 15 ; 15; 0.90) = 0.9015 = 0.2051

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