A coin is tossed 20 times and x = 6 ears are observed. Let p = P(head). A test of Ho: p > or = to .5 versus Ha: p < .5 of size at most .10 is desired.

a) Theorem 12.4.1
Let S ~ BIN (n,p). For large n, an approximate size alpha test of Ho: p < or equal to po against Ha: p > po is to reject Ho if zo = [s-n*po] / sort[n*po(1 - po)] > z(1-alpha)

Perform a test using theorem 12.4.1

b) Theorem 12.4.2
Suppose that S ~ BIN(n,p), and B(s;n,p) denotes a binomial CDF. Denote by s an observed value of S.

1. A conservative size alpha test of Ho: p < or equal to po against Ha: p > po is to reject Ho: if
1 - B(s-1; n, po) < or equal to alpha.

2. A conservative size alpha test of Ho: p > or equal to po against Ha: p < po is to reject Ho if
B(s; n, po) < or equal to alpha

3. A conservative two-sided test of Ho: p = po against Ha: p not equal to po is to reject Ho if
B(s; n, po) < or equal to alpha/2 or 1 - B(s-1; n, po) < or equal to alpha/2

Perform a test using theorem 12.4.2

c) What is the power of a size alpha = .0577 test of Ho: p > or equal to .5 for the alternative p = .2?