8. What \(\alpha\) levels are possible with a decision rule of the form \"Reject \(H_0\) if \(k \ge k*\)\" when \(H_0: p = 0.5\) is to be tested against \(H_1: p > 0.5\) using a random sample of size \(n = 7\)?
9. The following is a printout of the binomial pdf
\(P_X(k) = \binom{9}{k} (0.6)^k (0.4)^{9-k},\) \(k = 0, 1, \dots, 9.\)
Suppose \(H_0: p = 0.6\) is to be tested against \(H_1: p > 0.6\) and we wish the level of significance to be exactly 0.05. Use Theorem 2.4.1 to combine two different critical regions into a single randomized decision rule for which \(\alpha = 0.05.\)
options (scipen=999)
k<-0:9; binomial<-dbinom(k,9,0.6); cbind(k,binomial)
k
binomial
[1,] 0 0.000262144
[2,] 1 0.003538944
[3,] 2 0.021233664
[4,] 3 0.074317824
[5,] 4 0.167215104
[6,] 5 0.250822656
[7,] 6 0.250822656
[8,] 7 0.161243136
[9,] 8 0.060466176
[10,] 9 0.010077696
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