Reasons for tire for flying off my car?

Please delete if I'm violating any rules.I'm 25 with little car knowledge and have relied on my father and cousin for their car chops. I drive a 2001 Toyota Corolla. Great car, I'm pretty diligent with getting things replaced/oil changes. Back in mid-April, I got my front passenger tire changed for a brand new one at [reputable tire chain dealer here]. Dad noticed a little too much wear and tear on that one so we got a brand-new mid-grade tire.Fastforward to this past Saturday. I'm driving on the freeway and I noticed some LOUD rumbling, wheel vibrating a little too much. Instant thought is to get home ASAP (I'm 4 miles away anyway), change cars (Dad is out of town so I can use his car if need be), look at tire, get to my job, and have it looked at the next morning. Within 5 seconds of that thought, it felt like my tire blew out. Screeching, gripping wheel tightly yet calmly. I have to immediately get over to the freeway shoulder right when a slew of merging traffic came on. Luckily, everyone slowed down/accelerated to avoid my tire from flying at them. At the time, I just thought my tire blew out. A few cars stopped, checked to see if I was okay. A few of them asked if I had tire work done recently and that may have been the cause. Found my tire a hundred yards away atop the slope and rolled it to my car. Needless to say, it was the same new tire I had changed mid-April. Luckily, no one was hurt except for some skid marks my flying tire did to to a car in back of me when it flew and hit their back left side of their car. Exchanged insurance info and waited for a tow.I towed it to the Generic Tire Chain store with my receipt from the tire change that I still kept in my dashboard. My thinking was that they didn't tighten the lug nuts/use the right torque. Important to note that I also have a key for my lug nuts that I always hand over when anyone is doing any tire work for me. They told me they'd get back to me in a couple days, when the inspectors/supervisors come out to look at my car at their location.Called him today, and I asked when the inspectors would be there since I would be without a car. Dude gave me the runaround and said he wasn't sure. He said, "the thing is, that our work is only guaranteed a week after installation. If it came flying off, it would have happened within a week to a month." I'm stunned, I said that I thought their work was guaranteed for 90 days. He says that that's just for motor work. I stood my ground and said that it was too coincidental for that same tire to be flying off a few months later. So now, I'm worried as hell because my car has significant damage and I'm without a car.NOTE: From mid April-now, I did not take it in for any other car service. Ran normal until recently. I was due for an oil change within another 200 miles. Wish I could have taken it in earlier and maybe had more of a chance to catch the issue with my tire.Do I believe him? Is it their responsibility? What else could have caused my tire to fly off then? Help me, Reddit!EDIT: Pics via /r/cars http://ift.tt/2ub4dII