This Exam P sample reference tests Discrete Random Variables. This problem subtracts constrained Store A frequencies from the combined frequency table. Store B must have frequency four at score 2 and at most two everywhere else, so it has exactly one mode and choice B.
How to solve this Discrete Random Variables question
Setup
Setup
A modal frequency of eight is possible only at combined-table scores whose total frequency is at least eight. Exactly three scores meet that requirement.
A0=A2=A5=8
Model
Model
Subtract those forced Store A modal frequencies from the corresponding combined frequencies.
B0=9−8=1
B2=12−8=4
B5=9−8=1
Compute
Compute
At each remaining score, Store A has frequency at least four while the combined frequency is six. Therefore Store B can have frequency at most two there.
B1≤6−4=2,B3≤2,B4≤2
B2=4>max(B0,B1,B3,B4,B5)
Answer
Answer
Store B has one and only one modal score.
1(B)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AEvery nonempty finite frequency table has at least one category attaining its largest frequency. Here score 2 explicitly has Store B frequency four.
CA second mode would require another Store B frequency of four. At scores 1, 3, and 4 that would force Store A frequency two, violating its minimum of four; scores 0 and 5 are fixed at one for Store B.
DThe three middle combined frequencies are all six, but Store B receives at most two at each of them. None can tie the forced frequency four at score 2.
EThis treats several possible response values as modes merely because they may have positive frequency. A mode must attain the table's maximum frequency, which only score 2 can do.
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
Variant answer in brief
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.
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