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Part IV: Guesstimating and Hypothesizing with Confidence
ice cream (10 ounces each). You want to estimate the average weight of the
cones they make over a one-day period, including a margin of error. Instead
of weighing every single cone made, you ask each of your new employees to
randomly spot check the weights of a random sample of the large cones they
make and record those weights on a notepad. For n = 50 cones sampled, the
sample mean was found to be 10.3 ounces. Suppose the population standard
deviation of σ = 0.6 ounces is known.
What’s the margin of error? (Assume you want a 95% level of confidence.) It’s
calculated this way:
So to report these results, you say that based on the sample of 50 cones, you
estimate that the average weight of all large cones made by the new employees
over a one-day period is 10.3 ounces, with a margin of error of plus or minus
0.17 ounces. In other words, the range of likely values for the average weight
of all large cones made for the day is estimated (with 95% confidence) to be
between 10.30 – 0.17 = 10.13 ounces and 10.30 + 0.17 = 10.47 ounces. The new
employees appear to be giving out too much ice cream (but I have a feeling the
customers aren’t offended).
Notice in the ice-cream-cone example, the units are ounces, not percentages!
When working with and reporting results about data, always remember what
the units are. Also, be sure that statistics are reported with their correct units
of measure, and if they’re not, ask what the units are.
In cases where n is too small (in general, less than 30) for the Central Limit
Theorem to be used, but you still think the data came from a normal dis-
tribution, you can use a t*-value instead of a z*-value in your formulas. A
t*-value is one that comes from a t-distribution with n – 1 degrees of free-
dom. (Chapter 10 gives you all the in-depth details on the t-distribution.)
In fact, many statisticians go ahead and use t*-values instead of z*-values
consistently, because if the sample size is large, t*-values and z*-values are
approximately equal anyway. In addition, for cases where you don’t know
the population standard deviation, σ, you can substitute it with s, the sample
standard deviation; from there you use a t*-value instead of a z*-value in your
formulas as well.
Being confident you’re right
If you want to be more than 95% confident about your results, you need to
add and subtract more than 1.96 standard errors (see Table 12-1). For exam-
ple, to be 99% confident, you add and subtract 2.58 standard errors to obtain
your margin of error. More confidence means a larger margin of error, though
(assuming the sample size stays the same); so you have to ask yourself if it’s
worth it. When going from 95% to 99% confidence, the z*-value increases by

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