Was trying to solve this problem but have hit a wall.
A psychiatrist believes that more than 50% of the users of sleeping pills sleep better simply because of the psychological effect of taking the pills. To substantiate this hypothesis with data, he selects a random sample of 20 insomniacs and gives each of them a box of pills to use. They are actually sugar pills but are otherwise identical to a popular brand of sleeping pills currently on the market. Subsequently, 15 of these patients report that the pills have been effective in inducing sleep. Assuming a level of significance close to 0.05, does this observation support the psychiatrist's conjecture.
Ok, so here is my interpretation of the problem:
- H0: The number of patients who sleep better with the help of sleeping pills is > 50%
- H1: The number of patients who sleep better with the help of sleeping pill is <= 50%
- Rejection region is X >= 15
- alpha <= 0.05
- Distribution is binomial b(20,0.5)
Now to solve this I look up the CPD tables for binomial distribution where:
The tables are P[X <= c] so P[X>=15]=1-[X<=14]=1-0.979=0.021 since 0.001 << 0.05 H0 is false and hence the conjecture of the psychiatrist true
However when I look up the answer key at the back of the book is says:
Rejection region X >=14 has an alpha of 0.058, claim is supported
Now I can't understand how he arrived at the rejection region being X >= 14