At a certain hospital, 75% of the interns receive fewer than 6 hours of sleep and report feeling tired during their shifts. At the same time, 70% of the interns who receive 6 or more hours of sleep report no feelings of tiredness. If 80% of the interns receive fewer than 6 hours of sleep, what percent of the interns report no feelings of tiredness during their shifts?
a) 6
b) 14
c) 19
d) 20
e) 81
This is an EITHER/OR group problem.
Every intern EITHER receives fewer than 6 hours of sleep OR does not.
Every intern EITHER is tired OR is not.
To organize the data, use a GROUP GRID:
_______________<6________≥6_______Total
tired:
not tired:
total:
In the grid above, every row has to add up to the total, as does every column.
Looking at the top row, (fewer than 6 hours sleep and tired) + (6 or more hours sleep and tired) = total tired.
Looking at the left-most column, (fewer than 6 hours sleep and tired) + (fewer than 6 hours sleep and not tired) = total with fewer than 6 hours sleep.
Now let's fill in the data step by step.
Whenever we have 2 entries in a row or column, we can determine the remaining entry, since everything has to add up horizontally and vertically.
Let the total = 100:
_______________<6_______≥6_______Total
tired:
not tired:
total:_______________________________100
75% of the interns receive fewer than 6 hours of sleep and report feeling tired:
__________________<6_______≥6_______Total
tired:_____________75
not tired:
total:____________________________________100
80% of the interns receive fewer than 6 hours of sleep:
__________________<6_______≥6_______Total
tired:______________75
not tired:___________5
total:______________80_______20________ 100
70% of the interns who receive 6 or more hours of sleep report no feelings of tiredness:
Since .7(20) = 14, we get:
__________________<6_______≥6_______Total
tired:______________75_______6__________81
not tired:___________5_______14_________19
total:______________80_______20________ 100
Notice that everything adds up horizontally and vertically.
Total not tired = 19.
The correct answer is
C.
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