average toss, based on 50 throws, was 47.5 yards, and we assumed the population standard
deviation was the same as the sample standard deviation, 8 yards. A test of the hypothesis that
Todd can throw the ball 50 yards on average against that alternative that he can’t throw that far
might look something like the following (we will fill in many of the details, e specially those in
the third part of the process, in the following chapters):• Let μ be the true average distance Todd can throw a football. H 0 : μ = 50 (or H 0 : μ ≥ 50, since the
alternative is one sided) and H (^) A : μ < 50
• Since we know σ , we will use a z -test. We assume the 50 throws is an SRS of all his throws, and the
central limit theorem tells us that the sampling distribution of is approximately normal. We will use a
significance level of α = 0.05.
• In the previous section, we determined that the P- value for this situation (the probability of getting an
average as far away from our expected value as we got) is 0.014.
• Since the P- value < α (0.014 < 0.05), we can reject H 0 . We have good evidence that the true mean
distance Todd can throw a football is actually less than 50 yards (note that we aren’t claiming anything
about how far Todd can actually throw the ball on average, just that it’s likely to be less than 50 yards).
Type I and Type II Errors and the Power of a Test
When we do a hypothesis test as described in the previous section, we never really know if we have
made the correct decision or not. We can try to minimize our chances of being wrong, but there are trade-
offs involved. If we are given a hypothesis, it may be true or it may be false. We can decide to reject the
hypothesis or not to reject it. This leads to four possible outcomes:
Two of the cells in the table are errors and two are not. Filling those in, we have