That is a badly formulated question. You introduce completly irrelevant Details, and leave out all the important context.
To avoid mixups with who “you” refers to, I will call the theoretical Person in your Scenario “Bob” (Bob the Blob to avoid the gender question).
A. How many children Bob has is irrelevant, if he lives in a modern Country. The question is not supposed to be “Should Bob work as a cop, or his children will starve?” because in modern countries Bob will get unemployment and Food stamps for his children.
So first missing context is “Why does Bob want to become a cop and not work at Walmart?”
If Bob lives in a poor Country, where the only well-paid Job is Cop, then the question is no longer about integrity, but the “Steal medicine” Dilemma.
B. In what Country is Bob living? Because if he’s in the US, he can’t Claim integrity. The current state of US Police is that it’s a rotten Organisation to the core, not just a few bad apples. Black Lives Matter has shown how often not just cops shoot civilians because they are black - but that the whole Group Closes ranks and nothing happens. The cop isn’t automatically suspended until after Trial; evidence is planted by the cop or other cops, and nothing happens when it’s discovered; cops lie in court, and nothing happens; cops are sentenced, but not fired.
If there is any Kind of selection on who not to make a cop, it’s against People with too much intelligence, not on who has the personality of a bully. There is no established Standard of Training, so even without evidence by the FBI that supremacist Groups have deliberatly infiltrated law enforcment in the US, it’s obvious by behaviour alone that bullies feel attracted to a Job where a badge protects them from consequences.
C. What Kind of cop School does Bob go to that “Integrity” is the most important part? Integrity in itself is not a value; People can have integrity to be racist jerks.
The first duty of a cop in a democratic state is to protect citizens by keeping the peace. Not protecting the established order: not protect their own lives by Shooting first; not protecting the (White) majority by Shooting (black) People first; not by dividing People into “Us Cops” and “those civilians” instead of “we citizens”.
D. What Kind of test is it - Multiple choice or Essay questions?
Essay questions are graded individually, so Points may be awarded for what Bob doesn’t see.
E. Who wrote that test? A federal board with not only cops, but also experts in law, and advocacy Groups, to test not just knowledge, but also personality of a cop (Essay questions)?
Or is it the local Sheriff, and the questions are “Write your Name, how you are related to the Sheriff, What do you think of the Job the Sheriff is doing, what’s your favourite colour?”
Is it the equivalence to the Alabama voter literacy test for one Group of Testees, and “Write your Name” for another Group?
F. How many Points? Passing by one Point out of 10 is different from one Point out of 400.
G. Who graded the test - his own instructors? Or neutral federal testers?
H. Apparently this does take place only in the US, because Bob finishes after 24 weeks, instead of 2 years Minimum.
My question was about how Bob could have studied hard for his dream Job all this time and not notice he was failing the half-year Progress tests, but in only 6 months, there might not have been Interim tests.
My conclusion is:
If Bob only got 24 weeks, supposedly studied hard because we’re supposed to believe he takes his future Job as cop serious, and yet just scrapes by with one Point, he should quit any way, regardless if this one Point was in error or not.
Scraping by with only one Point after studying hard means that Bob is not cut out for the difficult, responsible, demanding Job of a Police officer. If he values integrity, then he must look for a Job he is actually suited for - which might still be in law enforcement: there are lots of Support positions. But not as cop.
Additionally, since this is the US: Obviously this is a trap. There are too many stories out there of how People in power in the US love to screw with those under them (And get away because there is no legal protection).
Passing by only one Point, and obviously given in mistake? It’s a trap disguised as “test of character”. If Bob doesn’t speak up, they will tell him he failed the character test and he won’t get the Job anyway.