How do I compare means when I have a sample and the whole population?

Question

I have the following didactical exercise to carry out:

Scenario

I have the avg room price for all the hotels of my chain (32 observations of the 32 hotels). Then I have the avg room price for a sample of competitors (60 observations taken from a larger population).

Problem

I would like to understand whether the avg room price of the hotels of my chain is equal to that of the competitors

Proposed solution

First, I computed the average room price across all the hotels of my chain $p_a$. Since the entire population is known, I would say that there is no uncertainty here, this is the exact average.

Then I computed the average room price of the sample of competitors and the related std deviation ($\bar{p}_c$ and $\tilde\sigma_c$).

I performed a hypothesis testing with the null $H_0: p_c = p_a$ against the alternative $H_1: p_c \neq p_a$. The test statistic is then ($n=60$): $$ t = \frac{\bar{p}_c-p_a}{\frac{\tilde\sigma_c}{\sqrt{n}}} $$ If the associated p-value is sufficently low, I reject the null hypothesis.

Basically here I'm considering that the average room price of the hotels of my chain is known and well-established, hence I'm doing the hypothesis testing for a single population mean (the competitors' population). Do you think this is the right approach or shall I test a hypothesis about two population mean (this is the alternative that comes to my mind)

Thanks for any help, T.

Answer 1

This seems completely reasonable to me. It is what I would have done.

Answer 2

It depends on how you're really planning to use the statistic.

One thing that immediately comes to mind is that hotel prices depend heavily on location. So the price of a hotel room in NYC is apt to be much higher than the price of a hotel in Bismark, ND.

So a better statistic might be to do a paired T test, where the price for each of your hotels is compared to a comparable hotel in the same market place.