Question

4.4/7.

7.The accompanying table contains the results from experiments with a polygraph instrument. Find the probabilities of the events in parts​ (a) and​ (b) below. Are these events​ unlikely?

_ No_(Did_Not_Lie) Yes_(Lied)

Positive_test_result 9 22

Negative_test_result 40 15

a. Four of the test subjects are randomly selected with​ replacement, and they all had true negative test results.

b. Four of the test subjects are randomly selected without​ replacement, and they all had true negative test results.

a. The probability that all four test subjects had a true negative test result when they are randomly selected with replacement is

(Round to three decimal places as​ needed.)

Is such an event​ unlikely?

Yes, because the probability of the event is greater than 0.05

No, because the probability of the event is greater than 0.05

Yes, because the probability of the event is less than 0.05

No, because the probability of the event is less than 0.05

B. The probability that all four test subjects had a true negative test result when they are randomly selected without replacement is____

​(Round to three decimal places as​ needed.)

Is such an event​ unlikely?

Yes, because the probability of the event is greater than 0.05

No, because the probability of the event is greater than 0.05

Yes, because the probability of the event is less than 0.05

No, because the probability of the event is less than 0.05

Answer 1

In the given problem, true negatives are cases for which Negative_test_result given for _ No_(Did_Not_Lie). So, there are 40 true negatives in total data = (9 + 22 + 40 + 15) = 86

a.

Probability of true negatives with replacement = 40 / 86 = 20 / 43

Probability that all four test subjects had a true negative test result when they are randomly selected with replacement = (20/43)* (20/43)* (20/43)* (20/43) = 0.047

Is such an event​ unlikely?

Yes, because the probability of the event is less than 0.05

b.

Total negatives at the start are 40 out of 86. When one true negatives is selected, remaining true negatives are 39 out of 85. Similarly, when second true negatives is selected, remaining true negatives are 38 out of 84 and so on.

Probability that all four test subjects had a true negative test result when they are randomly selected without replacement = (40/86)* (39/85)* (38/84)* (37/83) = 0.043

Is such an event​ unlikely?

Yes, because the probability of the event is less than 0.05