A psychologist studying the senses of taste and smell has carried out many studies in which students are given each of 20 different foods (apricot, chocolate, cherry, coffee, garlic, and so on). She administers each food by dropping a liquid on the tongue. Based on her past research, she knows that for students overall at the university, the mean number of the 20 foods that students can identify correctly is 14, with a standard deviation of 4, and the distribution of scores follows a normal curve. The psychologist wants to know whether people's accuracy on this task has more to do with smell than with taste. In other words, she wants to test whether people do worse on the task when they are only able to taste the liquid compared to when they can both taste and smell it (this is a directional hypothesis). Thus, she sets up special procedures that keep a person from being able to use the sense of smell during the task. The psychologist then tries the procedure on one randomly selected student. This student is able to identify only 5 correctly.
(a) Using the .05 significance level, what should the psychologist conclude? Solve this problem explicitly using all five steps of hypothesis testing and illustrate your answer with a sketch showing the comparison distribution, the cutoff (or cutoffs), and the score of the sample on this distribution.
(b) Then explain your answer to someone who has never had a course in statistics (but who is familiar with mean, standard deviation, and Z scores).