Which Rejection Region is Most Appropriate for Tennis Racket String Preference?

[needhelp83]

Each of a group of 20 tennis players is given 2 rackets, 1 having nylon strings and the having gut strings. After several weeks of playing with 2 rackets, each player will be asked to state a preference for one of 2 types of strings. Let p denote the proportion of all such players who prefer gut, and let X be number of players in the sample who prefer gut. Because gut is more expensive consider the null hypotheses that at most 50 % of all such players prefer gut. We simplify this to Ho: p= .5 planning to reject Ho only if sample evidence strongly favors gut strings.

a) Which of the rejection regions {15,16,17,18,19,20}, {0,1,2,3,4 5} or {0,1,2,3,17,18,19,20} is most appropriate and why are the other two not appropriate?

b)What is the probability of a type I error for the chosen region of part a? Does the region specify a .05 test? Is it the best level .05 test?

d) If 13 out of the 20 players prefer gut, should Ho be rejected using a significance level of .10?


My breakdown of the problem is as follows:
Ho: p=.5
Ha: p<.5

Looking at part d \hat{p} = 13/20=0.65

T\alpha=T 19,.1 = 1.328

T=\frac{.65-.5}{\sqrt{\frac{.5(1-.5}{20}}}= 1.342

1.342 > .5, We reject Ho

We can conclude that that the gut string is more preferable than the nylon string.

What do I do with part a) and am I on track with part d bc it seems as though I should be solving for p<.5

 

[needhelp83]

Any help...how would I start out figuring these rejection regions?

 

[needhelp83]

Alright is there any suggestions for part a? A formula? I know it has to do with Bernoulli and it is 50-50.