An article describes a survey of 2,096 American adults. Survey participants were asked if they were football
fans and also if they agreed or disagreed that the rules that the National Football League adopted in 2010
designed to limit head injuries have been effective. Data from the survey are summarized in the table
below.
Agree Disagree Total
Football Fan 683 533 1,216
Not a Football Fan 219 661 880
Total 902 1,194 2,096
Suppose that a survey participant is to be selected at random. Consider the following events.
(a) Calculate the following probabilities. (Round your answers to four decimal places.)
1. P(A)
2. P(D)
3. P(A|F)
4. P(A|F^C)
(b) Are the events F and A independent events? Justify your answer using relevant probabilities.
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To calculate the probabilities, we can use the information from the table:
(a)
1. P(A) = (Number of people who agree)/(Total number of people) = 902/2096 = 0.4301
2. P(D) = (Number of people who disagree)/(Total number of people) = 1194/2096 = 0.5699
3. P(A|F) = (Number of football fans who agree)/(Total number of football fans) = 683/1216 = 0.5617
4. P(A|F^C) = (Number of non-football fans who agree)/(Total number of non-football fans) = 219/880 = 0.2489
(b) To determine if F and A are independent events, we need to compare P(A|F) with P(A) and P(A|F^C) with P(A). If P(A|F) = P(A) and P(A|F^C) = P(A), then F and A are independent.
Comparing P(A|F) = 0.5617 with P(A) = 0.4301, we see that they are not equal. Therefore, F and A are not independent events. Football fans' agreement/disagreement with the rule is dependent on whether they are football fans or not.
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