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

0.648

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

Explanation:

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

• 
  `p=(3)/(15)=(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.