Answer:
Find answers below
Step-by-step explanation:
H0: P <= 0.5
Ha: P > 0.5
, the number who prefer gut strings is <= a number or the test tends towards the  left-tailed.
{0,1,2,3,4,5} ;  
{15,16,17,18,19,20} is a right-tailed test and not appropriate for H0:
{ 0,1,2,3,17,18,19,20} is two-tailed and not appropriate for H0:
b)
Does the region specify a level .05 test? No
  
P = proportion who prefer gut strings to nylon
P = X /20
Assume alpha = 0.05
z(alpha) = -1.645
Reject if (x/20 - 0.5) / sqrt[ (0.5)(0.5)/20 ] < -1.645
Reject if (x/20 - 0.5) <  < (-1.645) sqrt ( (0.5)(0.5)/20 )
Reject if x/20   < (-1.645) sqrt ( (0.5)(0.5)/20 ) + 0.5
Reject if x/20   < 0.316
Reject if x   < (0.316)(20) = 6.32
{0,1,2,3,4,5,6} is the region for the best level 0.05 test
c)
According to (a),  reject H0 if x <= 5
P( Type II error) = P( do not reject H0/ when Ha is true)
P( Type II error) = P( x > 5/ P=0.6)
x ---p(x)
6  0.004854  0.998388  
7  0.014563    
8  0.035497  
9  0.070995  
10  0.117142  
11  0.159738  
12  0.179706  
13  0.165882  
14  0.124412  
15  0.074647  
16  0.034991  
17  0.012350  
18  0.003087  
19  0.000487  
20  0.000037  
add: 0.9984 --  proba bility of a type II error
Assuming P=0.8
P( Type II error) = P( x > 5/ P=0.8)
6  0.000002  1.000000  
7  0.000013  
8  0.000087  
9  0.000462  
10  0.002031  
11  0.007387  
12  0.022161  
13  0.054550  
14  0.109100  
15  0.174560  
16  0.218199  
17  0.205364  
18  0.136909  
19  0.057646  
20  0.011529  
add: 1.0000  probability of a type II error
d)
P( x <= 13) =  
0  0.000001  
1  0.000019  
2  0.000181  
3  0.001087  
4  0.004621  
5  0.014786  
6  0.036964  
7  0.073929  
8  0.120134  
9  0.160179  
10  0.176197  
11  0.160179  
12  0.120134  
13  0.073929  
add: 0.9423 < 0.10 ,  H0 cannot be rejected