When Larry and The Beaver go to the movies to see "The Mad Hypnotist', the scene is introduced by some stock footage showing children in front of the Village Theatre in Westwood, Los Angeles, California, USA. Very similar footage is used in season 2 in Beaver's Sweater (1959).
Ward makes a comment that hints at his being in the advertising industry. He tells June that a survey from work is meant to get the "pulse" of the female shopper.
In School Sweater (1960) that originally aired one week prior to this episode and anticipates the plot of this episode, Beaver at one point looks into his bedroom mirror and practices being a hypnotist.
Beaver's malapropism for the washer he ties on a string and uses in his attempts to hypnotize is 'omelet'; he, of course, means 'amulet'.