group of 200 patients tested a new medication.Some tried the new medication, and the rest took the old medication.The results are reported in the following two-way frequency table.Improvement No improvementNew medication 27 63Old medication 37 73A patient is chosen at random from this group.Complete the following. Write your answers as decimals.(a)Find the probability that the patient showed improvement.=P(improvement)(b)Find the probability that the patient showed improvement, given that he took the new medication.=P(improvement | new medication)(c)Is there evidence that a patient who takes the new medication is more likely to show improvement than a randomly chosen patient from the group?Yes, because the probability found in part (b) is much
Question
group of 200 patients tested a new medication.Some tried the new medication, and the rest took the old medication.The results are reported in the following two-way frequency table.Improvement No improvementNew medication 27 63Old medication 37 73A patient is chosen at random from this group.Complete the following. Write your answers as decimals.(a)Find the probability that the patient showed improvement.=P(improvement)(b)Find the probability that the patient showed improvement, given that he took the new medication.=P(improvement | new medication)(c)Is there evidence that a patient who takes the new medication is more likely to show improvement than a randomly chosen patient from the group?Yes, because the probability found in part (b) is much
Solution
(a) To find the probability that the patient showed improvement, we need to add the number of patients who showed improvement with both the new and old medication and divide it by the total number of patients.
So, the total number of patients who showed improvement is 27 (new medication) + 37 (old medication) = 64.
The total number of patients is 200.
Therefore, P(improvement) = 64/200 = 0.32.
(b) To find the probability that the patient showed improvement, given that he took the new medication, we need to divide the number of patients who showed improvement with the new medication by the total number of patients who took the new medication.
So, the number of patients who showed improvement with the new medication is 27.
The total number of patients who took the new medication is 27 (improvement) + 63 (no improvement) = 90.
Therefore, P(improvement | new medication) = 27/90 = 0.3.
(c) To determine if there is evidence that a patient who takes the new medication is more likely to show improvement than a randomly chosen patient from the group, we need to compare the probabilities found in part (a) and (b).
The probability of improvement given the new medication (0.3) is less than the probability of improvement in general (0.32).
Therefore, there is no evidence to suggest that a patient who takes the new medication is more likely to show improvement than a randomly chosen patient from the group.
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