Further Statistics 1 Quality of Tests

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K.C.
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Could someone explain how to do part b of this question?

The random variable S has the distribution B(14, p). A significance test is carried out of the null hypothesis H0 : p = 0.3 against the alternative hypothesis H1 : p > 0.3. The critical region for the test is S ≥ 8.

(a) (2 marks) Find the significance level of the test, correct to 3 significant figures.

(b) (5 marks) It is given that, on each occasion that the test is carried out, the true value of p is equally likely to be 0.3, 0.5 or 0.7, independently of any other test. Four independent tests are carried out. Find the probability that at least one of the tests results in a Type II error.
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ghostwalker
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(Original post by K.C.)
Could someone explain how to do part b of this question?

The random variable S has the distribution B(14, p). A significance test is carried out of the null hypothesis H0 : p = 0.3 against the alternative hypothesis H1 : p > 0.3. The critical region for the test is S ≥ 8.

(a) (2 marks) Find the significance level of the test, correct to 3 significant figures.

(b) (5 marks) It is given that, on each occasion that the test is carried out, the true value of p is equally likely to be 0.3, 0.5 or 0.7, independently of any other test. Four independent tests are carried out. Find the probability that at least one of the tests results in a Type II error.
P(At least one results in a Type II error) = 1-P(None give a type II error)

Since all the tests are independent, this equals

1-(P(Not getting a type II error on a single test))^4

=1-(1-P(type II error))^4

Since p is equally likely to be 0.3, 0.5, or 0.7 in each test independently of other test, we have:

P(type II error) = P("type II error" I p=0.3) P(p=0.3) +P("type II error" I p=0.5) P(p=0.5)+P("type II error" I p=0.7) P(p=0.7)

And put the two equations together.

Part (a) will give you the "type II error | p=0.3", and you'll need to repeat it for 0.5, and 0.7
Last edited by ghostwalker; 2 years ago
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