Art of Electronics Ex.2.20 Series negative feedback amplifier
I will try to describe the method used by the book. 

First, we need to find the open-loop gain. 

And I will use this circuit diagram

 

The Q2 voltage gain is 

$$A_{V2} \approx \frac{R_5+R_4}{re2} \approx \frac{11\textrm{k}\Omega}{25\Omega} \approx 440 \: V/V $$

The first stage gain is: 

$$ A_{V1} \approx \frac{R_3||(\beta +1)re2}{R_4||R_5} \approx \frac{500\Omega}{1\textrm{k} \Omega} \approx 0.5 \: V/V$$ 

Therefore the open-loop gain is: 

\$ A_{OL} = A_{V1} * A_{V2} = 220 \:V/V \$ 

From the diagram we see that: 

$$Z_{out(OL)} = R_4+R_5 = 11\textrm{k}\Omega$$

And 

$$ Z_{out(CL)} = \frac{Z_{out(OL)}}{1+ A_{OL}\times\frac{R_4}{R_4+R_5} } = \frac{11k \Omega}{1 + 220 \times \frac{1}{11}} \approx 520\Omega $$  

And now the closed-loop gain:

$$A_{CL} = \frac{A_{OL}}{1+ A_{OL}\times\frac{R_4}{R_4+R_5} } \approx 10.47 \:V/V$$

The LTspice is showing \$Z_{out(CL)} = 529\Omega\$ and \$A_{CL} = 10.47\: V/V\$ quite good.