Hi Ruth,
Don't be worried about bears. Toby was right that it is traffic and lightening that are significantly bigger threats.
I grew up in CO and have lived in British Columbia, Canada for the last 10+ years. I saw more bears my first field season here than all my years in CO. Read the books he recommended too. The author also produced 2 excellent videos: "Staying Safe in Bear Country" and "Working in Bear Country". There are likely to be copies of these at your local library or university. We watch the Working in Bear Country video every year or 2 for my job as a field geologist/engineer. The more you know the more comfort you will have! These books and videos describe the situations where you are likely to encounter bears, how to read their stress levels, and how to react should an encounter escalate into something more serious. But take comfort that most encounters either 1) you aren't aware there is a bear and 2) simply don't escalate with aggressive behavior from the bear.
Get at COSAR card, educate yourself and keep your food/smelly stuff (food, lip balm, trash, toothpaste, chamois cream too) away from your sleeping area. Get a bear bell for your bike. Outdoor stores sell bear bells that have a magnetic silencer for those times where you feel OK. Heck, get an air horn that you can recharge with your bike pump. Get bear spray and a holster. I've been known to sleep in my tent with the bear spray (as well as my rock hammer just 'cause it makes me feel better!!). Make noise on the trail in sections that look like favorable locations for bear habitat. Sing Mr. and Mrs. Bear a song. Let them know you are coming. Heck, if we're on the same section of the CTR at the same time this year, riding through bear terrain, then I'll teach you *my* bear song.
Think of the CTR as exposure therapy with very little chance of an actual encounter.
Em