Page 63 - Statistics II for Dummies

Chapter 3: Reviewing Confidence Intervals and Hypothesis Tests
Suppose someone claims that the mean time to deliver packages for a company 47
is 3.0 days on average (so Ho is µ = 3.0), but you believe it’s not equal to
that (so Ha is µ ≠ 3.0). Your α level is 0.05, and because you have a two-sided
test, you have 0.025 on each side. Your sample of 100 packages has a mean
of 3.5 days with a standard deviation of 1.5 days. The test statistic equals
, which is greater than 1.96 (the value on the last row and the

0.025 column of the t-distribution table — see Table A-1 in the appendix). So
3.0 is not a likely value for the mean time of delivery for all packages, and you
reject Ho.

But suppose that just by chance, your sample contained some longer than
normal delivery times and that, in reality, the company’s claim is right. You
just made a Type I error. You made a false alarm about the company’s claim.

To reduce the chance of a Type I error, reduce your value of α. However I
don’t recommend reducing it too far. On the positive side, this reduction
makes it harder to reject Ho because you need more evidence in your data to
do so. On the negative side, by reducing your chance of a false alarm (Type I
error) you increase the chance of a missed opportunity (a Type II error.)

Missing an opportunity with a Type II error
A Type II error is the conditional probability of not rejecting Ho, given that Ho
is false. I call it a missed opportunity because you were supposed to be able
to find a problem with Ho and reject it, but you didn’t. You didn’t blow the
whistle when you should have.

The chance of making a Type II error depends on a couple of things:

✓ Sample size: If you have more data, you’re less likely to miss something
that’s going on. For example, if a coin actually is unfair, flipping the coin
only ten times may not reveal the problem. But if you flip the coin 1,000
times, you have a good chance of seeing a pattern that favors heads
over tails, or vice versa.
✓ Actual value of the parameter: A Type II error is also related to how big
the problem is that you’re trying to uncover. For example, suppose a
company claims that the average delivery time for packages is 3.5 days.
If the actual average delivery time is 5.0 days, you won’t have a very
hard time detecting that with your sample (even a small sample). But if
the actual average delivery time is 4.0 days, you have to do more work
to actually detect the problem.

To reduce the chance of a Type II error, take a larger sample size. A greater
sample size makes it easier to reject Ho but increases the chance of a Type I
error.

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