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Chapter 3: Building Confidence and Testing Models
Every margin of error is interpreted as plus or minus a certain number of
standard errors. The number of standard errors added and subtracted is
determined by the confidence level. If you need more confidence, you add
and subtract more standard errors. If you need less confidence, you add and
subtract fewer standard errors. The number that represents how many stan-
dard errors to add and subtract is different from situation to situation. For
one population mean, you use a value on the t-distribution, represented by
t n – 1 , where n is the sample size. See Table A-1 in the Appendix.
Here’s an example. Suppose you have a sample size of 20, and you want to
estimate the mean of a population. The number of standard errors you add
and subtract is represented by t n – 1 , which in this case is t 19 . Suppose your
confidence level is 90 percent. To find the value of t, you look at row 19 in the
t-distribution table (Table A-1 in the Appendix). The table uses the area to the
right, so that area in this case is 0.05. (You get this value because 90 percent
is within the confidence interval, so 10 percent is outside of it. Half of that 10
percent lies above the confidence interval, and the other half lies below it.)
So look at row 19 and the column headed by the value 0.05. You get the value 55
of t = 1.73. So to be 90 percent confident with a sample size of 20, you need to
add and subtract 1.73 standard errors.
Now suppose you want to be 95 percent confident in your results, with the
same sample size of n = 20. The area above the interval is now half of 5 percent,
which is 2.5 percent or 0.025. Row 19 and column 0.025 in Table A-1 gives you
the value of t 19 = 2.09. Notice that this value of t is larger than the value of t for
90 percent confidence, because in order to be more confident, you need to go
out more standard deviations on the t-distribution table to cover more possible
results.
Large confidence, narrow intervals — just the right size
A narrow confidence interval is much more desirable than a wide one. For
example, if you said that you think the average cost of a new home is $150,000
plus or minus $100,000, that wouldn’t be helpful at all because this makes
your estimate anywhere between $50,000 and $250,000. (Who has an extra
hundred grand to throw around?) But you have to be 99 percent confident, so
your statistician has to add and subtract more standard errors to get there,
which makes the interval that much wider (a downer). She tells you to be
happy with 95 percent confidence, but no!
Wait, don’t panic — you can have your cake and eat it too! If you know you
want to have a high level of confidence, but you don’t want a wide confidence
interval, just increase your sample size to meet that level of confidence. The
effect of sample size and the effect of confidence level cancel each other out,
so you can have a precise (narrow) confidence interval and a high level of
confidence at the same time. It all depends on sample size, something you
can control (up to the size of your pocketbook of course).