Original practice · fully worked
Original variant: maximize modes after late survey entries
Five survey response categories currently have frequencies 7, 6, 6, 5, and 5. Three additional valid response forms remain to be entered, and each form may belong to any category. Calculate the largest possible number of modes in the completed frequency table.
- A
1 modal category - B
2 modal categories - C
3 modal categories - D
4 modal categories - E
5 modal categories
Answer and calculation
One form can raise each frequency-six category to seven, and the third can go to a frequency-five category. The final frequencies 7,7,7,6,5 have three modes; four modes would require at least four forms, so choice C.
Setup
Setup
Use the current leading frequency seven as the cheapest attainable common maximum for multiple categories.
(7,6,6,5,5) Model
Model
Assign one late form to each frequency-six category. Assign the remaining form to one of the frequency-five categories without creating a new leader.
(7,6,6,5,5)+(0,1,1,1,0)=(7,7,7,6,5) Compute
Compute
This construction produces three modes. To produce four modes at level seven would require filling deficits zero, one, one, and two, for a total of four forms.
0+1+1+2=4>3 maximum number of modes=3 Answer
Answer
At most three categories can share the completed table's largest frequency.
3(C) 