8. Confidence Intervals
With confidence intervals we will make our first statistical inference. Confidence intervals give us a direct inference about the population from a sample. The probability statement is one about hypotheses about the mean of the population based on the mean and standard deviation of the sample. This is a fine point. The frequentist definition of probability gives no way to assign a probability to a hypothesis. How do you count hypotheses? The central limit theorem makes a statement about the sample means on the basis of a hypothesis about a population, about its mean and standard deviation . If the population is fixed then the central limit theorem gives the results of counting sample means, frequentist probabilities. If we let represent a hypothesis about a population (i.e. that it is described by and ) and let represent data (with mean ) then the central limit theorem gives the probability . The confidence intervals that we’ll look at first give . We’ll look at the recipe for computing confidence intervals for means first, then return to this discussion about probabilities for hypotheses.
Our goal is to define a symmetric interval about the population mean that will contain all potentially measured values of with a probability of .
Typically will be
The assumptions that we need in order to use the -distribution to compute confidence intervals for means are :
- The population standard deviation, , is known (a somewhat artificial assumption since it is usually not known in an experimental situation) or
- The sample size is greater than (or equal to) 30, and we use , the sample standard deviation in our confidence interval formula.
Definition : Let where be the -value, from the Standard Normal Distribution Table that corresponds to an area, between 0 and of as shown in Figure 8.1.
To get our confidence interval we simply inverse -transform the picture of Figure 8.1, taking the mean of 0 to the sample mean and the standard deviation of 1 to the standard error as shown in Figure 8.2.
So here is our recipe from Figure 8.2. The -confidence interval for the mean, under one of the two assumptions given above, is :
or using notation that we will use as a standard way of denoting symmetric confidence intervals
The notation is more convenient for us than because we will use the t Distribution Table in the Appendix to find very quickly. We could equally well write
but we will use Equation (8.1) because it explicitly gives the bounds for the confidence interval.
Notice how the confidence interval is backwards from the picture that the central limit theorem gives, the picture shown in Figure 8.3. We actually had no business using the inverse -transformation to arrive at Figure 8.2. It reverses the roles of and . We’ll return to this point after we work through the mechanics of an example.
Example 8.2 : What is the 95 confidence interval for student age if the population is 2 years, sample , ?
Solution : So . First write down the formula prescription so you can see with numbers you need:
First determine . With the tables in the Appendices, there are two ways to do this. The first way is to use the Standard Normal Distribution Table noting that we need the associated with a table area of . Using the table backwards we find . The second way, the recommended way especially during exams, is to use the t Distribution Table. Simply find the column for the 95 confidence level and read the from the last line of the table. We quickly find .
Either way we now find
with 95 confidence.
- Because of this issue about probabilities of hypotheses, many prefer to say "confidence" and not probability. But we will learn enough about Bayesian probability to say "probability". ↵