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
(which simplifies to 
)
(which simplifies to 
)