DC BJT Circuit Allowable collecter swing discepency
If Vc can swing by +/- 1v, then as Vb is constant (0.7v assumption!), by definition Vcb can also swing by +/- 1v. Note this constant 0.7v assumption is false, but it's sufficiently true to be useful. If it's stated as true, then it's true for this exercise.

The next assumption to make is what collector voltage you want at DC. With a 3v power supply, the maximum collector voltage is 3v at zero emitter current (less a gnat's for Ib). Do you take as the minimum voltage 0.7v, so the current through \$R_B\$ never goes -ve, or do you take 0.2/0.3(ish), which is a reasonable \$VCE_{sat}\$? Although this would imply a negative current through Rb, so what? The only way to get collector voltages down below Vc(dc) is to be injecting an external base current from the signal input anyway, so I would argue that a negative current through \$R_B\$ is of no consequence, and Vc can go below 0.7v.

Let's assume \$Vc_{min}\$ = 0.4v. Almost any BJT ought to be able to get down here, and it allows the 3v to 0.4v = 2.6v total Vc range to easily be divided by 2 (this is my engineers' 'if a signal is not well defined, then assume it so it's easy to calculate' thang). A 2.6v swing is +/- 1.3v, so the mid point is 3v(rail)-1.3v = 1.7v, and the range 0.7v to 2.7v. Note the lower excursion of 0.7v has nothing directly to do with VBE=0.7v, it's the mid-point of Vcc and a reasonable VCEsat less a -1v signal.

For 0.5mA Ie at DC, we need 0.5mA though Rc with a 1.3v voltage drop, Rc = 2.6k. There's 1v across Rb, and we need 5uA through this, so that's 200k. You can grind through the 5% resistor and changing beta variation yourself.

Any difference between the above answer, your answer, and your book answer is down to assumptions about what the Vdc should be. When you make assumptions, they should be a) reasonable, and b) stated, like I have. Adding the justification for the assumptions doesn't hurt as well, especially if this is an exercise. 

A lot of what an engineer does on the back-of-an-envelope is about making reasonable assumptions, and this is why numbers are shaded if necessary to make them easy to do without a calculator. It's better to have an approximate calculation that you understand, can find your way around, can modify easily, can understand why it's different to the 'one in the book', than to match the book solution to the 5th decimal place. Though some professors disagree, know yours.

A further comparison of our answers. Mine contains a lot of explanatory words, and very few numbers. Yours, lots of numbers and equations, without explanation. I had difficulty in following it, and so will your prof. You will in six months time.

When you come to do the beta variation, it says 'what is the allowable variation at the collector?' This is another assumption, what is the allowable variation? If there's no further collector load stated (what is this stage driving?), then presumably we can go all the way to +ve rail and down to VCEsat? Or do we stick at +/-1v swing. It doesn't really matter which you choose, as long as you state what you have assumed.