An educational consulting firm is trying to decide whether high school students who have never before used a handheld calculator can solve a certain type of problem more easily with a calculator that uses reverse Polish logic or one that does not use this logic. A sample of 25 students is selected and allowed to practice on both calculators. Then each student is asked to work one problem on the reverse Polish calculator and a similar problem on the other. Let p =P(S), where ?S ?indicates that a student worked the problem more quickly using reverse Polish logic than without, and let X = number of S’s. a.?? f p = .5 , what is P( 7 ? X ? 18)? b.?? f p = .8, what is P( 7 ? X ? 18)? c. ?If the claim that p = .5 is to be rejected when either x ? 7 or x ? 8 , what is the probability of rejecting the claim when it is actually correct? d. ?If the decision to reject the claim is made as in part (c), what is the probability that the claim is not rejected when p = .6? When p = .8? e. ?What decision rule would you choose for rejecting the claim p = .5 if you wanted the probability in part (c) to be at most .01?

Solution : Step 1: An experiment is conducted by an educational firm to find whether high school students who have never before used a handheld calculator can solve a certain type of problem more easily with a calculator that uses reverse Polish logic or one that does not use this logic. They gave 2 type calculators to a sample 25 students. If p =P(S), where S indicates that a student worked the problem more quickly using reverse Polish logic than without, and let X = number of S’s .Then we have to find probabilities for different values of X . Step 2 a) We have to find the probability P( 7 X 18) ,when p= 0.5. It is given that they selected a sample of 25 students. So here n=25, this 25 students are allowed to practice on two type of calculators. That means there is only two choices for this students. And it is also given that p =P(S), where S indicates that a student worked the problem more quickly using reverse Polish logic than without, and X is the number of students who worked more quickly in reverse polish logic calculator. So the range of X can be X= 0,1,2,....,25 with different probabilities. From all this given information it is clear that X~ binomial distribution. With n=25 and X= 0,1,...,25 So the probability mass function of binomial distribution. n x n-x f(x) = C Px(1-P) , x=0,1,2,..,n Here f(X) = 25Cx (1-P) 25-x,x=0,1,2,..,25 So P( 7 X 18), when p=0.5 will be 18 25 x 25-x P( 7 X 18) = Cx0.5) (1-0.5) x=7 = P(X=7)+P(X=8)+P(X=9)+... +P(X=18) = 0.143+ 0.0322+0.0608+...+0.01433 = 0.9853 We can find this large number of probability values also from excel by using the function, BINOMDIST(X,TRIALS,PROBABILITY,CUMULATIVE=FALSE),THEN SUM(CELL NUMBERS). b) we have to find the same probability P( 7 X 18) ,when p=0.8 18 25 x 25-x P( 7 X 18) = Cx0.8) (1-0.8) x=7 = P(7)+P(8)+P(9)+...+P(18) = 0.00+0.00+....+0.0623+0.11084 = .21996