Researchers from the educational testing service (ets) found that providing immediate feedback to students answering openended questions can dramatically improve students’ future performance on exams (educational and psychological measurement, feb. 2010). the ets researchers used questions from the graduate record examination (gre) in the experiment. after obtaining feedback, students could revise their answers. consider one of these questions. initially, 50% of the students answered the question correctly. after providing immediate feedback to students who answered incorrectly, 70% answered correctly. consider a bank of 100 open-ended questions similar to those on the gre. a. in a random sample of 20 students, what is the probability that more than half initially answer the question correctly? b. refer to part a . after providing immediate feedback, what is the probability that more than half of the students answer the question correctly?

a. P(X>10)= 0.5881

b. P(X>5) = 0.9527

Step-by-step explanation:

Hello!

According to the study of the ETS, providing immediate feedback to students answering open-ended questions can dramatically improve student's future performance in exams.

They used questions from the graduate record examination in the experiment. Considering one of the questions, they observed that 50% of the students answered the question correctly. After receiving immediate feedback, of those that answered incorrectly, 70% answered correctly, over a bank of 100 open-ended questions.

You have two situations to take into account, the first one is "the number of students that answered the question correctly in the first try" and the second one "the number of students, that answered the question wrong in the first try but answered it correctly after receiving feedback"

a. Considering a sample of 20 students, what is the probability that more than half initially answer the question.

For this item the study variable is

X: Number of students that answer the question correctly in the first try, in a sample of 20"

Taking into account the previous information, that of 100 open-ended questions, 50% were initially answered correct, then you cond say that the probability of answering it correctly in the first try is 50%, symbolically: p= 0.5

Binomial criteria:

1. The number of observation of the trial is fixed (In this case n = 20)

2. Each observation in the trial is independent, this means that none of the trials will affect the probability of the next trial (The fact that one student gives the correct answer does not affect the probability of another one doing so)

3. The probability of success in the same from one trial to another (Our "success" is that the student answered correctly in the first try)

Since it meets the binomial criteria, you can say this variable is discrete with binomial distribution X~Bi (np)

P(X>10)= 1 - P(X≤9)= 1 - 0.4119= 0.5881

The probability that more than a half answer the question correctly is 0.5881

b. Refer to part a. After providing immediate feedback, what is the probability that more than half of the students answer the question correctly?

For this instance, you have to work with the "half" that didn't answer the question correctly. Then your study variable is:

X: The number of students, that answered the question wrong in the first try but answered it correctly after receiving feedback, in a sample of 10"

This time the variable is also discrete and has a binomial distribution, but the probability of success will be p= 0.70.

P(X>5) = 1 - P(X≤4)= 1 - 0.0473= 0.9527

The probability that more than a half will answer the question correctly after obtaining feedback is 0.9527.

I hope it helps!