Hypothesis tests with a normal approximation, Type I and Type II errors

Last time

Before we start: last time, in Hypothesis tests, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Is it chance, or is it real?

A ball lands in slot A fifteen times in a hundred throws. Is the slot really less likely than a quarter, or is that just chance? A hypothesis test answers by asking how rare the result would be if nothing special were going on. And a test can be wrong in two ways. Those two errors have names, and Cambridge asks for both.

The language of a test

The null hypothesis H zero is the claim being tested, written as a statement about the parameter, such as p equals one quarter. The alternative H one says how it might differ: smaller, larger, or just different. Different means a two-tailed test. The significance level is the probability of rejecting H zero when it is true. At 2 percent, a two-tailed test puts 1 percent in each tail.

A normal approximation

Under H zero, X is binomial with n equal to 100 and p equal to one quarter. Mean 25, variance 18.75. Use a normal approximation. The observed value is 15. Apply the continuity correction: P of X at most 15 becomes P of the normal below 15.5. z equals 15.5 minus 25, over root 18.75, which is minus 2.194. The probability is 0.0141. 0.0141 is bigger than 0.01, the tail for a 2 percent two-tailed test. So there is no evidence to reject H zero.