1. We all recognize the difference between naughty and nice, right? What about children less than a year old - do they recognize the difference and show a preference for nice over naughty? In a study reported in the November 2007 issue of Nature, researchers investigated whether infants take into account an individual's actions towards others in evaluating that individual as appealing or aversive, perhaps laying the foundation for social interaction (Hamlin, Wynn, and Bloom, 2007). In one component of the study, infants were shown a "climber" character (a piece of wood with "google" eyes glued onto it) that could not make it up a hill in two tries. Then they were alternately shown two scenarios for the climber's next try, one where the climber was pushed to the top of the hill by another character ("helper") and one where the climber was pushed back down the hill by another character ("hinderer"). The infant was alternately shown these two scenarios several times. Then the child was presented with both pieces of wood (the helper and the hinderer) and asked to pick one to play with. Let the random variable X represent the number of 16 infants who choose the helper toy

a.) If the infants really had no preference between the toys and so were just choosing a toy at random, what probability distribution would X have? Identify its name and also its parameter values.

b.) The researchers found that 14 of the 16 infants chose the helper over the hinderer. Using your answer to part a), determine the probability that 14 or more of the 16 infants would have chosen the helper toy, assuming that they were just choosing a toy at random.

c.) Is this probability small enough to cast strong doubt on the "no preference" assumption, which would then suggest that infants really do show a preference for the helper toy? Explain.