If I was a visitor to shul from an alien planet, you'd never be able to convince me that Orthodox Jews think prayer works.
1. 90 percent of the congregation comes late If you had an important meeting with someone far more important than you would you dream of shuffling in long after the start time? Of course not. The fact that the majority of nearly every Orthodox Jewish congregation arrives late to shul tells us something important about how the services are really viewed - at least by the latecomers.
2. There's widespread talking in shul In court, everyone is silent when the judge is being addressed, and I imagine that if a delegation of Jews were dispatched to petition the governor, or president, or even a local village mayor, for something important, no one from the group would carry on background conversations while the group leader made his case. In shul, however, there is plenty of talking right when the shat'z is supposedly addressing God. This tells me the talkers, at least, don't really believe that prayer works. If they did, they would shut up.
3. Along with the talking, some people always do other things during davening This shabos I observed men who supposedly believe that prayer works performing the following tasks and activities smack in the middle of the service: (1) studying (2) putting away books, and other types of straightening up (3) setting up for kiddush (4) passing around articles. (5) running out to fetch their hats or folding their talitot. This open disregard for the prayers and blatant violations of decorum are likewise strong suggestions that the service are not taken seriously, which in turn suggests that the offenders don't actually believe that the prayers work. If they did, they wouldn't be so easily distracted. If you had the ear of someone powerful would you duck out to fetch herring and cake? Would you pull out a biology text book? The fact that there are OJs who do this sort of thing is another reason why a visitor to shul from a foreign planet would quite logically conclude that OJs don't think prayer works.
4. The gesticulations and chanting. When you're requesting a raise do you sway back and forth? If you're asking a girl to marry you do you chant the question or say it sing-song? If you want your neighbor to lend you a power tool do you clench your hand into a fist and wave it around emphatically as you ask for the favor? Do you squeeze your eyes and do weird William Shatner things with your voice as you say "Tom, can I borrow your saw?" The fact that we do all these things - and more - when we're supposedly asking God for favors is another strong suggestion that OJs don't really think prayer works. We chant and sway and wave around our hands because this makes the experience more emotionally rewarding, that is, more pleasant, for us. If we honestly think God is listening, and that the prayers are genuine requests that might win God's favor why don't we articulate them in the way we articulate every other request?
All of these observations have been made by Rabbis and lay leaders who are dismayed with the way OJs pray but for the most part they have avoided the logical conclusion. Our shabby treatment of prayer isn't merely a sign that we don't take prayer seriously; taken as a whole these are strong suggestions that we don't believe prayer works at all.