I quote from Webster's Dictionary of English Usage (Springfield, Mass.: Merriam-Webster Inc., 1989), 802:

Reason why is denounced as a redundancy by American Heritage 1969, 1982, Shaw 1970, 1975, 1987, and Prentice Hall 1978; it is defended by Evans 1957 and Bernstein 1971. Bernstein, in fact, finds instances when the why is required, and says that even when it is not required, it is never unidiomatic or wrong. Most other commentators who mention it are tolerant. Ther tolerance is well taken, for many usage writers themselves use reason why. . . . Mary Vaiana Taylor's survey of university teaching assistants (College English, April 1974) discovered that 70 percent of them marked reason why as an error—making TAs more hostile to the phrase than even the Heritage usage panel. So if you are taking freshman English, you had perhaps best avoid offending with this usage. Anyone else can use reason why freely.