Hi SAT, nice to meet you again!
In my opinion,
@KenS did make a very good job here. Bible never said that prayers by unbelievers to heal sickness would even be considered.
The Bible points out that there is a difference between prayers issued by believers and the rest.
It's really just as Ken said, I think.
Let me add a few points: In my opinion, many many Christians would opt for not participating in such a study. For the following reasons:
* Bible says there is evidence for the Creator. Romans 1:20.
So why ask for a second bit of evidence? A survey that would find out that prayers to God work... would constitute a second bit of evidence.
* the survey puts God under pressure.
If it doesn't work out, some atheists might feel tempted to mock and potentially also say "no evidence".
Some Christians, as I see it, do have issues with setting God under pressure instead of giving him honor.
Other Christians say setting God under pressure is ok, though. They compare it to demanding something from a loving father.
* why unite with atheists in prayer? Some might ask. Many Christians believe that their own conversion is what atheists need most.... instead of prayer success concerning minor issues. Minor in comparison to salvation.
* a successful study would constitute a sign from God.
There is scripture against demanding such thing as a sign: Matthew 16:4.
Yet Gideon also demanded a sign and it was ok.
So, sometimes signs are ok, sometimes demanding a sign is not.
Let's assume 50% of all Christians assume the latter when asked to participate in such a study... declining the demand.
* Many Christians, such as myself, don't love people they never met or heard anything from. So, why pray for a person you don't love? It would be dishonest, at least in my case
* if I remember right, there are cases in the Bible in which illness served as punishment.
If God punishes, some Christians might not be wanting to interfere.
Others might ask themselves: why mess with situations you have nothing to do with.. and opt out
* in my opinion, useful illness does exist, I know of elderly people who do not want to hear well, even if they could. Some do want to, others don't.
Long story short, reasons for Christians not to partake in such a study are numerous.
If you don't have believers participating, there is no Biblical reason to believe that prayers (concerning a matter other than declaring Jesus as Lord) actually work.
However, it was good to see your thoughts.