Page 225 - Statistics for Dummies

Chapter 13: Confidence Intervals: Making Your Best Guesstimate
the mean (average) length of the cobs of two varieties of sweet corn (allow-
ing them to grow the same number of days under the same conditions). Call
the two varieties Corn-e-stats and Stats-o-sweet. Assume by prior research
that the population standard deviations for Corn-e-stats and Stats-o-sweet
are 0.35 inches and 0.45 inches, respectively.
1. Because you want a 95% confidence interval, your z* is 1.96.
Suppose you want to estimate with 95% confidence the difference between 209
2. Suppose your random sample of 100 cobs of the Corn-e-stats variety
averages 8.5 inches, and your random sample of 110 cobs of Stats-o-
sweet averages 7.5 inches. So the information you have is: ,
, n = 100, , , and n = 110.
1 2
3. The difference between the sample means, , from Step 3, is 8.5 – 7.5 =
+1 inch. This means the average for Corn-e-stats minus the average for
Stats-o-sweet is positive, making Corn-e-stats the larger of the two variet-
ies, in terms of this sample. Is that difference enough to generalize to the
entire population, though? That’s what this confidence interval is going
to help you decide.
4. Square (0.35) to get 0.1225; divide by 100 to get 0.0012. Square (0.45)
and divide by 110 to get 0.2025 ÷ 110 = 0.0018. The sum is 0.0012 + 0.0018 =
0.0030; the square root is 0.0554 inches (if no rounding was done).
5. Multiply 1.96 times 0.0554 to get 0.1085 inches, the margin of error.
6. Your 95% confidence interval for the difference between the average
lengths for these two varieties of sweet corn is 1 inch, plus or minus
0.1085 inches. (The lower end of the interval is 1 – 0.1085 = 0.8915
inches; the upper end is 1 + 0.1085 = 1.1085 inches.) Notice all the values
in this interval are positive. That means Corn-e-stats is estimated to be
longer than Stats-o-sweet, based on your data.
To interpret these results in the context of the problem, you can say
with 95% confidence that the Corn-e-stats variety is longer, on average,
than the Stats-o-sweet variety, by somewhere between 0.8915 and 1.1085
inches, based on your sample.

Notice that you could get a negative value for . For example, if you had
switched the two varieties of corn, you would have gotten –1 for this difference.
You would say that Stats-o-sweet averaged one inch shorter than Corn-e-stats
in the sample (the same conclusion stated differently).

If you want to avoid negative values for the difference in sample means,
always make the group with the larger sample mean your first group — all
your differences will be positive (that’s what I do).

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