Guest: Problem #48:
In the August 2005 issue of Consumer Reports, readers suffering from depression reported that alternative treatments were less effective than prescription drugs. Suppose that 550 readers felt better taking prescription drugs, 220 felt better through meditation, and 45 felt better taking St. John's wort. Furthermore, 95 felt better using prescription drugs and meditation, 17 felt better using prescription drugs and St. John's wort, 35 felt better using meditation and St. John's wort, 15 improved using all three treatments, and 150 improved using none of these treatments. (Hypothetical results are partly based on percentages given in Consumer Reports.)
How many readers suffering from depression were included in the report? Of those included in the report.
b.How many felt better using prescription drugs or mediation?
c. How many felt better using St. John's wort only?
d. How many improved using prescription drugs and meditation, but not St. John's wort?
e. How many improved using prescription drugs or St. John's wort, but not meditation?
f. How many improved using exactly two of these treatments?
g.How many improved using at least one of these treatments?
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The question is a little bit ambiguous but I think that this is what it means
"220 felt better through meditation" includes all people who felt better who had used meditation, they may have used prescription drugs and/or St John's wart as well, that is not important.
It also tell us nothing about how many people used meditation and did not feel better. I have defined the other statements in a similar way.
So now what you start with is 3 overlapping circles. In the parts that don't overlap, write, presciptions, meditation and St John's wort respectively.
Now just start putting the numbers in starting in the very middle.
15 people felt better after using all 3.
now.
35 people felt better after using meditation and St john's wort. that goes in the correct overlap, but wait, there are already 3 in their. 35-3=32 so 32 goes in the other bit of the overlap.
And so on.
just keep subtracting and putting answers in the correct spots.
oh, 150 improved using none of these.
You will have to put the 3 circles inside a big rectangle and put 150 outside the circles but in the rectangle.
What you are drawing is called a Venn diagram (My brother's name is Ven so it was easy for me to remember.) It is a very useful tool for these types of question.
Now, if you have drawn it properly you should be able to work out the answers from the diagram.
Get back to me and let me know how you go.
+118653 
