Question

sebastian has researched the following scoring schemes: one that has 5 question choices, one that has 4 question choices, and two that have 3 question choices. which scoring scheme is the most favorable to the test taker?
scoring schemes of multiple - choice tests

scoring scheme	points for correct answer	points for incorrect answer
b (3 choices)	2	$-\frac{1}{2}$
c (4 choices)	1	$-\frac{1}{3}$
d (3 choices)	3	$-\frac{1}{3}$

Explanation:

Step1: Calculate expected value for A

For a 5 - choice question, probability of correct $p_A=\frac{1}{5}$ and incorrect $q_A = \frac{4}{5}$. Expected value $E_A=p_A\times1+q_A\times(-\frac{1}{4})=\frac{1}{5}\times1+\frac{4}{5}\times(-\frac{1}{4})=\frac{1}{5}-\frac{1}{5}=0$.

Step2: Calculate expected value for B

For a 3 - choice question, probability of correct $p_B=\frac{1}{3}$ and incorrect $q_B=\frac{2}{3}$. Expected value $E_B=p_B\times2 + q_B\times(-\frac{1}{2})=\frac{1}{3}\times2+\frac{2}{3}\times(-\frac{1}{2})=\frac{2}{3}-\frac{1}{3}=\frac{1}{3}$.

Step3: Calculate expected value for C

For a 4 - choice question, probability of correct $p_C=\frac{1}{4}$ and incorrect $q_C=\frac{3}{4}$. Expected value $E_C=p_C\times1+q_C\times(-\frac{1}{3})=\frac{1}{4}\times1+\frac{3}{4}\times(-\frac{1}{3})=\frac{1}{4}-\frac{1}{4}=0$.

Step4: Calculate expected value for D

For a 3 - choice question, probability of correct $p_D=\frac{1}{3}$ and incorrect $q_D=\frac{2}{3}$. Expected value $E_D=p_D\times3+q_D\times(-\frac{1}{3})=\frac{1}{3}\times3+\frac{2}{3}\times(-\frac{1}{3}) = 1-\frac{2}{9}=\frac{7}{9}$.

Answer:

D (3 choices)