A coaching company believes that coaching improves the SAT scores of high school students. To test their theory, they get two samples of students, one that was coached and one that was not. They record their improvement in test scores over two separate tests. Use the traditional method and test to see if coaching improves their test scores more than not coaching using α = 0.05.

coached:          = 29

                        S₁= 59

                        n₁= 427

not coached:     = 21

                            S₂= 52

                            n₂ = 2733