# How to check hypothesis in statistical data?

I have a statistical problem.

In a city there are some hostels which differ by the number of rooms. The input data are the following.

In a table there is information about hostels and corresponding number of rooms.

\begin{array}{|c|c|c|c|} \hline Room \ count & 0-30& 30-60& 60-90 & 90-120 & 120-150 & 150-180 \\ \hline Number \ of \ objects & 66& 28 & 15 & 9 & 1 & 1\\ \hline \end{array}

In this problem I have to calculate some statistical parameters such as mean, standard deviation and so on. This is not interesting.

But the thing I can't do is the following. How to check hypothesis that average object has more than 30 rooms with level of confidence 0.95?

I think I have to use somehow integral function of the Laplace, but I'm not sure that it's correct... And by the way I don't think that data has normal distribution law.

How should I check hypothesis?

• I think the problem is that your hypothesis is not an hypothesis, but a probability event. To compute its probability, you need an underlying probability distribution. There might be ways of getting that from your data but it seems to me that this is more complicated than what you want. Aug 15, 2014 at 12:38
• @fabee I believe it should be quite easy, cause it's just "student" problem Aug 15, 2014 at 12:45
• Are you sure, the problem is posed like this? If yes, than it is badly posed. First of all, hypotheses are about statistics which are computed from the data. For example, an hypothesis could be that the mean number of hotel rooms is $50$. You case sounds more like "What is the probability that a randomly drawn object has more than 30 rooms". If that is the case, it is badly posed as well since you only have a sample from the probability distribution, but not the distribution itself. These might be very different things. Aug 15, 2014 at 12:53
• @fabee A new information has come. Not "random" object, but average object. Aug 15, 2014 at 14:08
• I think Eupraxis1981 sums it up pretty nicely. Aug 15, 2014 at 17:34

You can formulate this as a binomial test of a proportion and use either the Wilson score interval or the Clopper-Pearson interval.

Specifically: let our sample space be all hostels under study and assume random sampling. Let X be a random variable that takes the value 1 if a selected hostel has at most 30 rooms and 0 otherwise. For a sample of N rooms the sum if N X's will have a binomial(n,p) distribution. What we want to test is:

H0: p = 0.5 vs Ha: p<0.5

If p <0.5 then the average hostel has more than 30 rooms.

Now you may think I have ignored the actual magnitude of the rooms per hostel, but that information is relevant to a different hypothesis:

H: what is the average number of rooms per hostel.

Contrast this with your hypothesis: does a hostel, on average, have more than 30 rooms. This is a categorical hypothesis.

To answe the second hypothesis you need to either assume something about the distribution of rooms per hostel or test it unde a very pessimistic scenario, where each hostel has either 0 or 31 rooms and then simulate the null distribution if the sample average and compare it to your actual sample value. If you reject at the 95% level then you are good to go.