I have done an experiment to test the performance of a system. In this experiment I collected answers from people. These answers are categorical (they were able to select on a 5 points scale). Each subject participated to 2 conditions: in one they are giving evaluations for observations under an ideal situation and in the other they are giving evaluations for observations coming from the real results of my system. My hypothesis is that the distribution of the ideal condition fits well the distribution under real condition, namely I want to demonstrate that the results of my system are perceived equally good as in ideal condition by users.

My results for the 5 points of the scale are the following:

Observed frequencies (real condition): {16, 112, 42, 308, 100}

Expected frequencies (ideal condition): {9, 81, 53, 340, 95}

As I saw that most of the time the Chi squared goodness-of-fit test is used with an expected frequency that comes from laws and not from empirical observations, I was wondering if:

1. my approach of using chi squared test could be good for my aim of if there is another better tool;
2. if it is ok to use empirical frequencies given by the observation of the same population under another condition.

I have done an experiment to test the performance of a system. In this experiment I collected answers from people. These answers are categorical (they were able to select on a 5 points scale). Each subject participated to 2 conditions: in one they are giving evaluations for observations under an ideal situation and in the other they are giving evaluations for observations coming from the real results of my system. My hypothesis is that the distribution of the ideal condition fits the distribution well under the real condition, so I want to demonstrate that the results of my system are perceived equally good as in the ideal condition by users.

My results for the 5 points of the scale are the following:

Observed frequencies (real condition): {16, 112, 42, 308, 100}

Expected frequencies (ideal condition): {9, 81, 53, 340, 95}

As I saw that most of the time the chi squared goodness of fit test is used with an expected frequency that comes from laws and not from empirical observations, I was wondering if:

1. My approach of using the chi squared test could be good for my aim, or if there is another better tool;
2. If it is okay to use empirical frequencies given by the observation of the same population under another condition.