Population standard error vs subpopulation standard error

Question

Chapter 10 of Statistics by Witte and Witte (2016) describes the following experiment. (Paraphrased below)

The national mean SAT score is 500 and standard deviation is 110. 
We have one sample (of size 100) from the local university with a mean SAT score of 533. 
(The local university has much larger than 100 students, 100 is only the size one one sample)
Can we conclude if the local university population mean is higher than the national mean?

They go on to derive:

1. The null hypothesis is that the local population mean is equal to the national mean. $$H_0 : \mu_{loc} = 500$$
2. They then assume that the local population standard deviation is the same as the national population standard deviation (I am unclear why) $$\sigma_{loc} = 110$$
3. They now calculate the standard error $ = \frac{\sigma_{loc}}{\sqrt n} = 11 $ and we notice that the sample mean is 3 standard errors away, and reject the null hypothesis.
4. They conclude that the local university population must have a higher mean than the national mean.

The parts where I am confused:

1. How did we use the national standard deviation for the local population?
2. Even if we did, should we not conclude the following: Either the local population standard deviation is much higher than the national population standard deviation or the local mean is higher than the national mean or both rather than just a statement about the mean?

I have a feeling something can be concluded about the subpopulation standard error given the population standard error, but I am unsure.