Statistics for Dummies (33 page)

When a characteristic being measured is categorical (for example, opinion on an issue [support, oppose, or are neutral], gender, political party, or type of behavior [do/don't wear a seatbelt while driving]), most people want to report the proportion (or percentage) of people in the population that fall into a certain category of interest. For example, the percentage of people in favor of a four day work week, the percentage of Republicans who voted in the last election, or the proportion of drivers who don't wear seat belts. In each of these cases, the object is to estimate a population proportion using a sample proportion plus or minus a margin of error. The result is called a
confidence interval for the population proportion.

The formula for a CI for a population proportion is
where
is the sample proportion,
n
is the sample size, and Z is the appropriate value from the standard normal distribution for your desired confidence level. (See
Chapter 3
for formulas and calculations for
; see
Chapter 10
[
Table 10-1
] for values of Z for certain confidence levels.)

To calculate a CI for the population proportion:

1. Determine the confidence level and find the appropriate Z-value.
   

   See
   Chapter 10
   (
   Table 10-1
   ).
   

2. Find the sample proportion (
   ) by taking the number of people in the sample having the characteristic of interest, divided by the sample size
   (n)
   .
   

   Note:
   should be a decimal value between 0 and 1.
   

3. Multiply
   times (1 –
   ), and then divide that amount by
   n
   .
   

4. Take the square root of the result from Step 3.
   

5. Multiply your answer by Z.
   

   This is the margin of error.
   

6. Take
   plus or minus the margin of error to obtain the CI. The lower end of the CI is
   minus the margin of error and the upper end of the CI is
   plus the margin of error.
   

For example, suppose you want to estimate the percentage of the times you get a red light at a certain intersection.

Because you want a 95% confidence interval, your Z-value is 1.96.

You take a random sample of 100 different trips through this intersection, and you find that you hit a red light 53 times, so
= 53 ÷ 100 = 0.53.

Take 0.53 times (1
−
0.53) and divide by 100 to get 0.2491 ÷ 100 = 0.002491.

Take the square root to get 0.0499.

The margin of error is, therefore, plus or minus 1.96 × (0.0499) = 0.0978.

Your 95% confidence interval for the percentage of times you will ever hit a red light at that particular intersection is 0.53 (or 53%) plus or minus 0.0978 (rounded to 0.10 or 10%). (The lower end of the interval is 0.53
−
0.10 = 0.43 or 43%; the upper end is 0.53 + 0.10 = 0.63 or 63%.) In other words, you can say that with 95% confidence, the percentage of the times you should expect to hit a red light at this intersection is somewhere between 43% and 63%, based on your sample.