Question

A teacher gives a student a make-up test consisting of 20 true-false questions. The intent of the test is to determine whether the student answers the questions correctly through knowledge of the material or merely by making lucky guesses. Assume the correct answers are a random sequence of “true” and “false” and that the student’s guesses are also random.

a. State a null hypothesis based on the probability of guessing the correct answer to a question.

b. State a one-tailed alternative hypothesis based on the probability of arriving at the correct answer through knowledge.

c. Find the region of rejection when a is set as close to 0.05 as possible. (Remember that the null hypothesis will be rejected only if an extreme value occurs on one side of the distribution.)

d. If the student correctly answers 16 of the 20 questions:

i. What is the P value?

ii. What should the teacher conclude?

Answer 1

(a) The hypothesis being tested is:

H₀: p = 1/2 = 0.5

(b) H_a: p > 0.5

(c) The rejection region is reject H_o z > 1.96 for = 0.05.

(d) (i) p̂ = 16/20 = 0.8

The test statistic, z = (p̂ - p)/√p(1-p)/n

z = (0.8 - 0.5)/√0.5(1-0.5)/20

z = 2.68

The upper tail p-value for z = 2.68 is 0.0036.

(ii) Since the p-value (0.0036) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the student answers the questions correctly by making lucky guesses.