A researcher performs a sequence of independent experiments, up to and including

Question

A researcher performs a sequence of independent experiments, up to and including the first 'success', after which the researcher stops. Each experiment has the same probability p of success. Let T be the number of experiments performed (including the first observed success). The researcher wishes to test the null hypothesis Ho: p = 0.5. against the alternative hypothesis Hi: p 3. (a) What is the significance level a of the test proposed by the re- searcher? Hint: it may be easier to first find 1-a alternative hypothesis that p = 0.25. in terms of the likelihood ratio: p( p = 0.5)/p( p = 0.25) (b) What is the power of the researcher's test against the specific (c) Re-express the researcher's rule for rejecting the null hypothesis

Explanation / Answer

H0 : p = 0.5

Ha : p < 0.5

if T > 3

(a) So first we will find that he wil get success in T 3 events if null hypothesis is correct.

Pr(T 3) = Pr(T = 1) + Pr(T =2) + Pr(T =3) = 0.5 + 0.52 + 0.53 = 0.875

so significance level alpha = 1 - 0.875 = 0.125

(b) Here if p = 0.25

then power = Pr(T > 3; 0.25) = 1 - [Pr(T = 1) + Pr(T =2) + Pr(T =3)]

= 1 - [0.25 + 0.75 * 0.25 + 0.752 * 0.25]

=1 - 0.578125 = 0.4219

(c) p(t l p = 0.5)/ p(t l p = 0.25) = (1-p0)3/(1-p1)3 = (1-0.5)3 /(1-0.25)3 = (0.5/0.75)3 = 0.2963

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.