Hi there I am struggling to understand the binomial distribution. How does X get determined? why is it 3 and 4 and then 13 and 14? Kind regards
Flip to page 188 of the Tables and we have the CDF F(x) = P(X <= x) given for n=20 and p=0.4. For a 95% confidence interval we're trying to find the x value such that P(X<=x) = 2.5% and a second x value such that P(X>=x) = 2.5%. The first value is fine - looking down the column you can see that P(X<=3) = 0.0160 and P(X<=4) = 0.0510 and so the x such that P(X<=x) = 0.025 will lie between 3 and 4. For the other end we have to be careful. For example P(X>=5) = 1 - P(X<5). So P(X<5) = 97.5%. However P(X<5) = P(X<=4) since X can only take values 0, 1, 2, 3, 4, 5, 6, 7, .... So we see that P(X<=11) = 0.9435 and P(X<=12) = 0.9790. Hence P(X>11) = P(X>=12) = 1 - 0.9435 = 0.0565 and P(X>12) = P(X>=13) = 1 - 0.9790 = 0.0210.
