Melissaa:
Doing a radon test is a perfectly allowed test procedure and, unless you have a big problem, should be of little or no impact on your pending sale. Especially considering that very little of Michigan is considered to be Zone 1 (greater than 4 pCi/L - picocurries per liter).
In fact, if the radon test does turn up positive, wouldn't it be nice for YOU to know so that you don't endanger you and yours???
From the concerns you have over a simple cannister test, I can't help but believe that you are already aware of the 'radon' results.
Here is a link to a VERY important EPA site discussing the need for Radon testing by the seller or buyer of any existing home.
http://www.epa.gov/iaq/radon/pubs/hmbyguid.html
Now that we have completed the reasons for why a test is recommended (which you have already heard), I will offer the following to your continued request:
"does buyer need special permission to conduct radon test where a unti is left in home 48hrs?"
It depends on the EXACT wording of your contract. Does it say that the buyer has the right to perform an 'inspection'?? If so, and that clause doesn't specifically exempt a radon test, then the buyer has the right to do his 'inspection'. And if you refuse or hinder that 'inspection', the buyer COULD have a justification for rescinding the contract and getting a refund of the earnest money. So, this pretty much comes down to... are you willing to lose the home sale in order to enforce your concern about possible radon???