Students in grades 3 through 8 in New York State are required to take a state ma
ID: 3070410 • Letter: S
Question
Students in grades 3 through 8 in New York State are required to take a state mathematics exam. To meet the state's proficiency standards, a student must demonstrate an under- standing of the mathematics expected at his or her grade level. The following data show the number of students tested in the New York City school system for grades 3 through 8 and the number who met and did not meet the proficiency standards on the exam (New York City Department of Education website, January 16, 2010). 38. Met Proficiency Standards? Grade Yes 47,401 35,020 36,062 36,361 40,945 40,720 No 23,975 34,740 33,540 32,929 29,768 31.931 Develop a joint probability table for these data. dent meeting or not meeting the proficiency standards on the exam? met the proficiency standards? If the student is a fourth grader, what is the probabil- a. b. What are the marginal probabilities? What do they tell about the probabilities of a stu- c. If a randomly selected student is a third grader, what is the probability that the student d. If a randomly selected student is known to have met the proficiency standards on the exam, ity that the student met the proficiency standards? what is the probability that the student is a third grader? What is the probability if the student is a fourth grader?Explanation / Answer
(a-b)
Following table shows the row total and column total:
To find the joint and marginal distribution we need to divide each number by 423392. Following table shows the joint and marginal probability distributions:
It does not seem that student meet proficiency standards because the marginal probability of "No" (0.4414) is large.
(c)
The probability that the student met the proficiency standards given that student is third grader is
P(yes | 3rd grader) = P(yes and 3rd grader) / P(3rd grader) = 0.112 / 0.1686 = 0.6643
The probability that the student met the proficiency standards given that student is fourth grader is
P(yes | 4th grader) = P(yes and 4th grader) / P(4th grader) = 0.0827 / 0.1648 = 0.5018
(d)
P(3rd grader | yes) = P(yes and 3rd grader) / P(yes) = 0.112 / 0.5586 = 0.2005
P(4th grader | yes) = P(yes and 4th grader) / P(yes) = 0.0827 / 0.5586 = 0.1480
Met Proficiency Standards? Grade Yes No Total 3 47401 23975 71376 4 35020 34740 69760 5 36062 33540 69602 6 36361 32929 69290 7 40945 29768 70713 8 40720 31931 72651 Total 236509 186883 423392