Question

a group of 220 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.

\

$$\begin{tabular}{|c|c|c|} \\hline & no improvement & improvement \\\\ \\hline old medication & 77 & 55 \\\\ \\hline new medication & 22 & 66 \\\\ \\hline \\end{tabular}$$

a 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(\text{improvement}) = \\)

(b) find the probability that the patient showed improvement, given that he took the new medication.
\\(p(\text{improvement} \mid \text{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 greater than the probability found in part (a).
• no, because the probability found in part (b) is much greater than the probability found in part (a).
• yes, because the probability found in part (a) is much greater than the probability found in part (b).
• no, because the probability found in part (a) is much greater than the probability found in part (b).
• yes, because the probability found in part (b) is about the same as the probability found in part (a).
• no, because the probability found in part (b) is about the same as the probability found in part (a).

Explanation:

Calculate overall probability of improvement

$$
LATEXBLOCK0
$$

Calculate conditional probability of improvement given new medication

$$
LATEXBLOCK1
$$

Compare probabilities for evidence of effectiveness

$$
LATEXBLOCK2
$$

Answer:

(a) \(P(\text{improvement}) =\) <blank>0.55</blank>

(b) \(P(\text{improvement} \mid \text{new medication}) =\) <blank>0.75</blank>

(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 greater than the probability found in part (a). (Correct answer)
• No, because the probability found in part (b) is much greater than the probability found in part (a).
• Yes, because the probability found in part (a) is much greater than the probability found in part (b).
• No, because the probability found in part (a) is much greater than the probability found in part (b).
• Yes, because the probability found in part (b) is about the same as the probability found in part (a).
• No, because the probability found in part (b) is about the same as the probability found in part (a).