Confidence Intervals

Read the section The Number Of Interviews Or Sample Size Required in the Gallup Organization's How Polls Are Conducted

1. Consider the interval 1.96. How often will this interval contain ? If the sampling distribution is normal, 95% of the time. This interval is a large sample 95% confidence interval.
    
2. Confidence Interval: If you repeatedly take random samples from a population, 95% of the 95% confidence intervals you calculate from those samples will contain .
    
   Many students when asked on a test will say that 95% of 95% confidence intervals will contain , the sample mean. This is not true! Every confidence interval contains the sample mean.
    
3. We use the large sample confidence interval when we assume that the central limit theorem holds because our sample so large. We pretend that the sample standard error is the population standard error in this case. For large samples you would expect these two values to be pretty close to each other anyway.
    
4. You can have confidence intervals with a confidence level other than 95%. You just need to replace 1.96 with the correct number in the formula 1.96.
    
   1. In general, the formula for a 100(1 - )% confidence interval is  .
       
   2. For a 95% confidence interval,  = 0.05. For a 90% confidence interval,  = 0.1. And so on.
       
   3. is a value from the standard normal distribution so that the area to the right of that number is /2. For instance,  = 1.96 because P(Z > 1.96) = 0.025.
       
5. Confidence intervals for a percentage ()
    
   1. For instance, you might want to estimate the population proportion (p) of people who will vote Republican. So you take a random sample of 1,000 voters (n = 1000) and calculate the proportion of the respondents who say they will vote Republican ().
       
   2. These confidence intervals are the same as the ones for any large sample confidence interval. You just use in place of and use the fact that =. The sample proportion () is in fact a sample mean and so the Central Limit Theorem is in effect and tells us that is normally distributed when n is large.
       
   3. Since we don't know p we use in it's place in the formula for .
       
   4. is at it's largest possible value when p = 0.5
       
   5. Look at the "Sampling Errors" listed in the recent CNN Poll results concerning the New Hampshire presidential primaries. How do they generate those margins of error?
       
6. Small sample confidence intervals
    
   1. These are the same as large sample confidence intervals except we use the T-distribution instead of the standard normal tables. So we use the formula:
        
      
       
      where can be found just like you find except use the t-table for n-1 df instead of the z-table. You can also use Table VI from the text to find for common values of .
       
   2. Since we are now dealing with small samples we can no longer use the Central Limit Theorem. So we need to assume that our sample was randomly selected from a normally distributed parent population.

E-mail Mr. Callahan at stat110@edcallahan.com with questions or comments about this web site or about the class itself.