My school has a 30-question mulitple choice test to place students in English courses. Each question has 4 choices (a,b,c,d). If you get 7 answers right, you pass. What is the statistical probability of getting 7 or more correct answers by filling out the answer sheet randomly? Do the chances of passing in this way make the test valid or invalid?
What is the statistical probability of passing this test randomly?
You can use Pascal's Triangle to determine the probability of randomly getting 0, 1, 2, ... 6 answers right, beginning p(0) = 0.25^30, p(1) = 30 * 0.25^29 * 0.75, p(2) = 435 * 0.25^28 * 0.75^2, . . . then add those up to get the random probability of failing with 6 or less correct answers, then subtract it from 1 to get the random probability of passing.
As an approximation, the distribution of random scores is binomial with a mean of 7.5 and a standard deviation of 2.37 (= sqrt (30 * 0.25 * 0.75)). About 68% of the corresponding normal distribution would be above the value 6.5, so that's my estimate of what the procedure in the first paragraph would give.
The test is clearly invalid.
Reply:P of getting right the question: 1/4 = 25%
P of getting at least 7 questions right: 7/30 = 23%
P of passing the test randomly= (1/4) x (7/30) = 7/120= 6%
Validity of the test? 6 out of 100 will pass the test randomly in theory with some guts and good luck. I will say the test is good enough ust for placing students in an English course.
Reply:P = probability of right answer = 1/4
Q = probability of wrong answer = ¾
n = 30
probability of passing
= sigma( 30CrP^r*Q^n-r), r = 7,8, .... 30
Reply:If you answered the questions completely at random you would get an average of 7.5 questions correct.
The laws of probability show that a monkey has more chance of passing than failing so it's not a particularly valid way of evaluating a child.
Reply:filling it out randomly? 25%, thats the chance you have on every question
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