No, I'm pretty sure this is a scam. Legitimate mystery shopping companies do not contact strangers and offer to pay them to do mystery shops. To work with a legitimate mystery shopping company, you must sign up on their website and provide your SS# or EIN# and your personal information, because any work you perform is taxable. Legitimate companies, with few exceptions, do not pay in advance. You do the work and you wait to get paid back. And pay for shops generally ranges from $5-$30.
Let me guess - they send you money. You cash the money gram or deposit it in your bank account. You send some of the money to another person through Western Union. Then you do a mystery shop and keep the rest of the money. And when you find out the money gram is no good, you will owe the bank the whole amount.
If you have already deposited a check make sure to go back to your bank and explain you have been the victim of a scam. It is the only way you have a chance of the bank working with you on the bounced check fees resulting from the bogus check and not coming after you for fraud. If you have withdrawn money put it back right now.
Equal rights for others does not mean fewer rights for you. It's not pie.
Never try to teach a pig to sing. It's a waste of your time and it really annoys the pig.