Question

a) John is doing a math quiz, which has 10 questions where each question has four outcomes (A, B, C & D) but only one is correct. After the quiz, John told you he didn't study for the test at all and guessed all answers. i) If John guessed each and every item in the test, describe the probability distribution for the number of correct guesses. Present the probability distribution in the form of a table using 4 decimal ppints in your answers. (Use Excel to calculate the probabilities.) i) Using the formula: E(X) XP(X), find out the number of incorrect answers Mark would expect to get (You can use Excel to do the calculation. Just copy the Excel output onto your Word document.) il) What does expectation mean in statistics? Explain your answer in the context of the A to D Test. What is the probability that Mark gets 8 or more correct answers for this test? When the result was released, John got 3 out of 10. Would you be surprised? Do you think iv) v) Mark lied to you? Explain your answer.

Answer 1

(i)
p = 1/4, n = 10, q = 1 - p = 3/4
P(x) = C(n, x) p^x q^(n - x) = C(10, x) (1/4)^x (3/4)^(10 - x)
x	P(x)
0	0.0563
1	0.1877
2	0.2816
3	0.2503
4	0.1460
5	0.0584
6	0.0162
7	0.0031
8	0.0004
9	0.0000
10	0.0000
(ii)
p = 3/4, n = 10, q = 1 - p = 1/4
P(x) = C(n, x) p^x q^(n - x) = C(10, x) (3/4)^x (1/4)^(10 - x)
x	P(x)	x P(x)
0	0.0000	0.0000
1	0.0000	0.0000
2	0.0004	0.0008
3	0.0031	0.0093
4	0.0162	0.0649
5	0.0584	0.2920
6	0.1460	0.8760
7	0.2503	1.7520
8	0.2816	2.2525
9	0.1877	1.6894
10	0.0563	0.5631
Expected number of incorrect answers = ∑x P(x) = 7.5

(iii) Expected value or expectation is the long-run average of the statistic under investigation. In the present context, an expectation of 7.5 means if John takes several such 10-question quizzes, each with only one correct option, in the long run, he will get 7.5 answers incorrect on every quiz.

(iv) P(x ≥ 8) = P(8) + P(9) + P(10) = 0.0004 + 0 + 0 = 0.0004

(v) P(3) = 0.2503 is not unusual (it is not less than 0.05), so I wouldn't be surprised.