A philosophy professor decides to give a 20 question multiple-choice quiz to det

Question

A philosophy professor decides to give a 20 question multiple-choice quiz to determine who has read an assignment. Each question has 5 choices. Let Y be the random variable that counts the number of questions that a student guesses correctly. You can assume that questions and answers are independent.

a. Find the probability distribution function of Y by making a table of the possible values of Y and their corresponding probabilities. Hint: what type of distribution does Y have?

b. Add a column for the Cumulative Distribution Function (CDF) for each value of X.

c. Find the expected value and standard deviation of Y.

d. If she wants to choose a passing grade such that the probability of passing a student who guesses at every question is less than 0.05. What score should she set as the lowest passing grade?

Explanation / Answer

Ans:

a)n=20

probability of guessing correctly for each question,p=1/5=0.2

Y will be Binomial distribution with n=20,p=0.2

Y can take values from 0,1,2,3........20

b)

c)Expected value=np=20*0.2=4

standard deviation=sqrt(np(1-p))=sqrt(4*(1-0.2))=sqrt(3.2)=1.79

d)see the cumulatve probability column

P(Y<=0)=0.012

P(Y<=1)=0.069

she should set 1 question(as passing grade condition) as lowest passing grade.

Y BINOMDIST(Y,20,0.2,FALSE) Cumulative 0 0.012 0.012 1 0.058 0.069 2 0.137 0.206 3 0.205 0.411 4 0.218 0.630 5 0.175 0.804 6 0.109 0.913 7 0.055 0.968 8 0.022 0.990 9 0.007 0.997 10 0.002 0.999 11 0.000 1.000 12 0.000 1.000 13 0.000 1.000 14 0.000 1.000 15 0.000 1.000 16 0.000 1.000 17 0.000 1.000 18 0.000 1.000 19 0.000 1.000 20 0.000 1.000

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