GER1000: Quiz 8

GER100 QUIZ 8
1) Population 1000 answers: 80% and 30%
2) Chinese Male answers: 0.023
3) New Drug Test Answers: (II) ONLY
4) Dependent Events: (I) ONLY
1) In a population of 1000, 750 people test positive and 250 people test negative for a specific disease. If a randomly chosen person has a negative test result, the probability that he or she actually has the disease is 72%. The specificity of the test is 70%. The sensitivity of the test is closest to______.
Let A be the event of Disease
Let ~A be the event of No Disease
Let B be the event of Positive
Let ~B be the event of Negative
B = 0.75 and ~B = 0.25
A and ~B = 0.72 * 0.25 = 0.18
~A and ~B = P(~B) - P(A and ~B) = 0.25 - 0.18 = 0.07
~B | ~A = ( ~B and ~A) / ~A = 0.7
Thus, ~A = (~B and ~A) | (~B | ~A) = 0.07 / 0.7
~A = 0.1
A = 1 - 0.1 = 0.9
A and B = A - (A and ~B) = 0.9 - 0.18 = 0.72
P(A and B) | P(A) = P(B | A) = 0.72 / 0.9
= 0.8
ANSWER: 0.8

4) In a population of 1000, 750 people test positive and 250 people test negative for a specific disease. If a randomly chosen person has a negative test result, the probability that he or she actually has the disease is 72%. The specificity of the test is 70%. The probability of a randomly chosen person having a positive test result given that he or she does not have the disease is closest to ______.
From Question 1,
P(~A and B) = 1 - P(A and B) - P(A and ~B) - P(~A and ~B)
= 1 - 0.72 - 0.18 - 0.07 = 0.03
P(B | ~A) = P(~A and B) / P(~A) = 0.03 / 0.10 = 0.30
ALTERNATE ANSWER
P(B | ~A) = 1 - P(~B | ~A) = 1 - 0.7 = 0.30
Answer: 30%

2) Refer to the following table adapted from Yearbook of Statistics Singapore, 2017. What is the probability that a randomly selected person is male and between 0-4 years old, given that the person is Chinese?
Let A be the event that the person is Chinese
Let B be the event that the person is Male
Let C be the event that the person is 0 to 4 years old
P(A) = 2923172 / 3933559 = 0.7431
P(A and B and C) = 66 943 / 3 933 559 = 0.0170
P(C and B | A) = P(A and B and C) / P(A) = 0.0170 / 0.7431
= 0.0229
ANSWER = 0.023

3) Refer to the new drug test in unit 4 (slides 9-12). Suppose six patients took the new drugs and one of them dies of the disease, which of the following is/are true?
(I) The null hypothesis says that the chance of survival is 5/6.
(II) The p value is more than 0.10.
Let S be the event where someone survives is 60% aka 0.60
Let D be the event where someone dies is 0.40
(I) is False, event statistics do not affect null hypothesis
(ii) is true
P(SSSSSD) = 0.60 ^ 5 * 0.40 = 0.0311
P(SSSSSS) = 0.60 ^ 6 = 0.4666
p-value = 6 * P(SSSSSD) + P(SSSSSS) = 0.1866 + 0.4666 = 0.6532 > 0.1
Answer: (ii) is true

5) Suppose A and B are two dependent events. Let P(A) denote the probability that A occurs, P(A and B) the probability that both A and B occur, and P(A|B) the conditional probability that A occurs given B has occurred. Which of the following is/are true?
(I) P(A and B) is always less than or equal to P(A).
(II) P(A | B) is always less than or equal to P(A).
(i) True. P(A and B) = P(A) - P(A and ~B). Thus P(A and B) < P(A) and P(A and B).
(ii) False. P(A | B) = P(A and B) / P(B)
Let P(A) = 0.5, P(B) = 0.5, P(A and B) = 0.3
P(A | B) = 0.3 / 0.5 = 0.6 and P(A } B) > P(A)
Answer: (I) is true