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Part IV: Guesstimating and Hypothesizing with Confidence
In the fish hatchery example from Case 1, suppose your sample size was 10
instead of 100, and everything else was the same. The t*-value in this case
comes from a t-distribution with 10 – 1 = 9 degrees of freedom. This t*-value is
found by looking at the t-table (in the appendix). Look in the last row where the
confidence levels are located, and find the confidence level of 95%; this marks
the column you need. Then find the row corresponding to df = 9. Intersect the
row and column, and you find t* = 2.262. This is the t*-value for a 95% confi-
dence interval for the mean with a sample size of 10. (Notice this is larger than
the z*-value of 1.96 found in Table 13-1.) Calculating the confidence
interval, you get , or 5.86 to 9.15 inches. (Chapter 10
gives you the full details on the t-distribution and how to use the t-table.)
Notice this confidence interval is wider than the one found when n = 100.
In addition to having a larger critical value (t* versus z*), the sample size
is much smaller, which increases the margin of error, because n is in its
denominator.
In a case where you need to use s because you don’t know , the confidence
interval will be wider as well. It is also often the case that is unknown and
the sample size is small, in which case the confidence interval is also wider.
Figuring Out What Sample

Size You Need

The margin of error of a confidence interval is affected by size (see the ear-
lier section “Factoring In the Sample Size”); as size increases, margin of error
decreases. Looking at this the other way around, if you want a smaller margin
of error (and doesn’t everyone?), you need a larger sample size. Suppose
you are getting ready to do your own survey to estimate a population mean;
wouldn’t it be nice to see ahead of time what sample size you need to get the
margin of error you want? Thinking ahead will save you money and time and
it will give you results you can live with in terms of the margin of error — you
won’t have any surprises later.

The formula for the sample size required to get a desired margin of error (MOE)
when you are doing a confidence interval for is ; always round up
the sample size no matter what decimal value you get. (For example, if your cal-
culations give you 126.2 people, you can’t just have 0.2 of a person — you need
the whole person, so include him by rounding up to 127.)

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