In practical applications, we replace the population standard deviation in ( 7.2) by S, the standard deviation of the sample. However, this substitution changes the coverage probability . Fortunately, there is a simple adjustment that allows us to maintain the desired coverage level : replace the normal distribution critical value z by the slightly larger t-distribution critical value t. The resulting confidence interval is the primary result of this section.
where t is a critical value determined from the tn-1 distribution in such a way that there is area between t and -t.
The value n-1 is called
degrees of freedom , or df for short.
It is a parameter of the t-curve in the sense that changing the value of
n-1 changes the shape of the t-curve, though usually not by much.
Here are appropriate t critical values for selected
The t critical values are always larger than the z, and get progressively closer as n-1 gets larger (they are equal at ). For a 95% confidence interval, the t values are 2.06, 2.03, 2.01, 1.98, and 1.96 for respective sample sizes n= 26,36, 51, 101, and 501.
Recall that the term in equation ( 7.5) is the (estimated) standard error of the mean. With .68 chance, misses by less than this amount. To generalize, misses by less than with certainty. Thus, the term is called the margin of error with confidence level .
If , then t is close to 2.0. For this reason, the 95% margin of error is often written as . When working with a random sample, the exact critical value t is read from a table or calculator, and depends on the sample size. However, for sample size calculations (see next section), the approximate critical value 2.0 is typically used.
We end this section with a reminder that the confidence interval ( 7.5) works well ONLY when at least one of the following conditions hold: (i) the population histogram looks like the normal curve, or (ii) the sample size is relatively large ( is a frequently used thumb rule . The following table summarizes the performance of the t-intervals under four situations.
|Normal curve||Not normal curve|
|Small sample size [n<30]||Good||Poor|
|Larger sample sizes ||Good||Fair|