Confidence Intervals and Sample Size

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332 Chapter 7 Confidence Intervals and Sample Size
Figure 7–4
Finding �/2 for a 98%
Confidence Interval
Figure 7–5
Finding z �/2 for a 98%
Confidence Interval
7–8
Since other confidence intervals (besides 90%, 95%, and 99%) are sometimes used in
statistics, an explanation of how to find the values for z a/2 is necessary. As stated
previously, the Greek letter a represents the total of the areas in both tails of the normal
distribution. The value for a is found by subtracting the decimal equivalent for the desired
confidence level from 1. For example, if one wanted to find the 98% confidence interval,
one would change 98% to 0.98 and find a � 1 � 0.98, or 0.02. Then a/2 is obtained by
dividing a by 2. So a/2 is 0.02/2, or 0.01. Finally, z 0.01 is the z value that will give an area
of 0.01 in the right tail of the standard normal distribution curve. See Figure 7–4.
α
2
= 0.01
α = 0.02
α = 0.01
2
–z α /2
0
z α /2
Once a/2 is determined, the corresponding z a/2 value can be found by using the procedure
shown in Chapter 6 (see Example 6–17), which is reviewed here. To get the z a/2
value for a 98% confidence interval, subtract 0.01 from 0.5000 to get 0.4900. Next, locate
the area that is closest to 0.4900 (in this case, 0.4901) in Table E, and then find the
corresponding z value. In this example, it is 2.33. See Figure 7–5.
z
0.0
0.1
...
2.3
For confidence intervals, only the positive z value is used in the formula.
When the original variable is normally distributed and a is known, the standard
normal distribution can be used to find confidence intervals regardless of the size of the
sample. When n � 30, the distribution of means will be approximately normal even if
the original distribution of the variable departs from normality. Also, if n � 30 (some
authors use n � 30), s can be substituted for s in the formula for confidence intervals;
and the standard normal distribution can be used to find confidence intervals for means,
as shown in Example 7–3.
0.98
Table E
The Standard Normal Distribution
.00 .01 .02 .03 ... .09
.4901