OK, folks. O-6 will challenge you. Here's the answers:
a) Ho: p = 0.5; Ha: p=/ 0.5
b) Conditions for a 1-prop z-test are NOT satisfied, because 8*0.5 = 4 which is a lot less than 10. In other words, the expected number of successes is too small.
c) To work around this problem, we use binomial pdf. We use p = 0.5, n = 8 and fill in the table:
0 0.00391
1 0.03125
2 0.10937
3 0.21875
4 0.27344
5 0.21875
6 0.10937
7 0.03125
8 0.00391
d) Since the test is two-sided, we see this this:
If x = 0 or 8, p = 2*0.0391 which is less than 5%
If x <= 1, then we add up the lower 2 and upper 2 and get 0.07312, which is greater than 5%
So 5% is not possible.
e) For the provided data, x = 2. So the p-value is 2*P(x <= 2) = 2*(0.003906 + 0.03125 + 0.109375) = 28.9% We would fail to reject and fail to find evidence of any difference between the two brands.
f) Increase sample size! It lowers the expected sd and enables us to more clearly see any differences that might exist between the two brands. sqrt(p*1-p/n) gets smaller!
That's a doozy of a problem, but hopefully this helps!
