Key Concepts

• The procedure for forming a confidence interval for a mean when the sample size is small and the population standard deviation is not known is similar to forming a confidence interval for a large sample size with a known population standard deviation.
• Instead of basing the error bound on the normal distribution, the t-distribution is used with degrees of freedom (df) equal to the sample size minus 1.
• The error bound formula is [latex]\large t(\frac{s}{\sqrt{n}})[/latex]

Glossary

degrees of freedom (df): the number of objects in a sample that are free to vary

Student’s t-distribution: investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student; the major characteristics of the random variable (RV) are:

• It is continuous and assumes any real values.
• The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution.
• It approaches the standard normal distribution as n get larger.
• There is a “family” of t-distributions: each representative of the family is completely defined by the number of degrees of freedom, which is one less than the number of data.