Children who develop unexpected difficulties with the spoken language are often diagnosed as specifically language impaired (SLI). A study published in the Journal of Speech, Language, and Hearing Research (Dec. 1997) investigated the incidence of SLI in kindergarten children. As an initial screen, each in a national sample of over 7,000 children was given a test for language performance. The percentages of children who passed and failed the screen were 73.8% and 26.2%, respectively. All children who failed the screen were tested clinically for SLI. About one-third of those who passed the screen were randomly selected and also tested for SLI. The percentage of children diagnosed with SLI in the "failed screen" group was 20.5%; the percentage diagnosed with SLI in the "pass screen" group was 2.8%.
a. For this problem, let "pass" represent a child who passed the language performance screen, "fail" represent a child who failed the screen, and "SLI" represent a child diagnosed with SLI. Now find each of the following probabilities: P(Pass), P(Fail), P(SLI | Pass), and P(SLI | Fail).
b. Use the probabilities from part a to find P(Pass ∩ SLI) and P(Fail ∩ SLI). What probability law did you use to calculate these probabilities?
c. Use the probabilities from part b to find P(SLI). What probability law did you use to calculate this probability?