Abstract: This chapter discuses two-sample t -tests. In a two-sample t -test, the samples are taken from two distinct populations. The chapter presents examples to explain the two-sample t -distribution. To find the mean vector nearest the observation vector, the concept of the orthogonal projection of a vector onto a given subspace and a method for finding this vector are needed. The orthogonal projection of a vector onto a subspace is given geometrically by dropping a perpendicular from the vector onto the subspace. The vector minus the orthogonal projection of the vector is perpendicular to the subspace. This property is used in defining the orthogonal projection operator and the orthogonal projection of a vector onto a subspace. Subspaces are orthogonal if any two vectors chosen from two distinct subspaces are orthogonal. This is another key idea for multinormal observations with a covariance matrix. The chapter also discusses the noncentral chi-squared distribution.