Question

Suppose you take a 15 question True or False quiz and you guess on every problem. You only get 3 correct. (a) What is the probability of getting 3 or fewer correct guesses? Round your answer to 3 decimal places. (b) Would 3 be an unusually low number of correct guesses? Use the criteria that a number (x) is unusually low if P(x or fewer) ≤ 0.05. Yes, that is an unusually low number of correct guesses. No, that is not an unusually low number of correct guesses.

Answer 1

Answer:

0.648

No, that is not an unusually low number of correct guesses.

Step-by-step explanation:

Suppose you take a 15 question True or False quiz and you guess on every problem. You only get 3 correct. Then

• $p=\dfrac{3}{15}=\dfrac{1}{5}=0.2$ - correct;
• $q=1-p=1-0.2=0.8$ - o\incorrect.

The probability of getting 3 or fewer correct guesses is

$P(\text{3 or fewer})\\=P(0)+P(1)+P(2)+P(3)\\ \\=C^{15}_0p^0q^{15-0}+C^{15}_1p^1q^{15-1}+C^{15}_2p^2q^{15-2}+C^{15}_3p^0q^{15-3}\\ \\=1\cdot q^{15}+15pq^{14}+105p^2q^{13}+455p^3q^{12}\\ \\=(0.8)^{15}+15\cdot 0.2\cdot (0.8)^{14}+105\cdot (0.2)^2\cdot (0.8)^{13}+455\cdot (0.2)^3\cdot (0.8)^{13}\\ \\\approx 0.648$

Using the criteria that a number 3 is unusually low if P(3 or fewer) ≤ 0.05, we can state that 3 is not an unusually low number of correct guesses.