Question

Please can anyone help
A true-false test consists of 20 questions, each of which has one correct answer: true, or false. One point is awarded for every correct answer, but one point is taken off for each wrong answer. Suppose a student answers every question by guessing at random, independently of other questions. Let S be the student’s score on the test.Find E(S)? and Find SE(S)?
My Solution:
E(S)=Expected Score=20*(1/2)=?
SE(S)=SQRT((20)*(1/2)*(1/2))=?

My solution for SE(S)=S=M1+M2+⋯+M20.
It is standard that the standard error of Xi is SQRT((1/2)(1−1/2))=1/2. So the standard error of 2Xi−1 is (2)(1/2)=1.
The standard error of the sum M1+M2+⋯+M20 is therefore SQRT (20)*(1)=SQRT(20)= 4.472135955 which is correct answer!

Answer 1

expected score for one question
\[=\sum_x xp(x)\]\[=1P(correct) + (-1)P(incorrect)\]\[=1(1/2) + (-1)(1/2)\]\[=0\]
expected score for 20 questions = 20 * expected score foe one question = ...

Answer 2

So E(s)=20*(0)=0

Answer 3

since the Avereage zero for one the expected score zero for the test

Answer 4

Yup

Answer 5

Thanks !!

Answer 6

The SE(S)?

Answer 7

SE(S) = \(\sqrt{p(1-p)/n}\), isn't it?

Answer 8

P = 0.5
n = 20

Answer 9

Thanks!

Answer 10

welcome

Answer 11

Rolypoly Please recheck above formula? the correct one is SE(S):S=M1+M2+⋯+M20.
It is standard that the standard error of Xi is SQRT((1/2)(1−1/2))=1/2. So the standard error of 2Xi−1 is (2)(1/2)=1.
The standard error of the sum M1+M2+⋯+M20 is therefore SQRT (20)*(1)=SQRT(20)= 4.472135955 which is correct answer!

Answer 12

My bad :(
Standard error = standard derivation / \(\sqrt{n}\), n= sample size

Answer 13

Rolypoly: Remark: when you divided the (not quite right) standard error per question by SQRT(20), you were finding the standard error of the average mark per question, not the standard error of the total mark.
We could also find the SE of each Mi directly. The variance of Mi is E(M2i)−(μi)2, where μi is the mean of Mi. But the mean of Mi is 0. And M2i=1 always, so E(M2i)=1. It follows that the variance of Mi is 1, and therefore the SE of Mi is 1.
The Mi are assumed independent. The total mark is the sum of the Mi, so has variance (20)(1), and therefore standard deviation SQRT(20).

Answer 14

Alright.. I'm sorry :(

Answer 15

Accepted and Thanks for the First part E(S) Your Assistance! Regards!

Answer 16

Haha! I've only learnt expected value and variance! Thanks for teaching me standard error! :D