## How to apply Chi Squared test according to this Simulation?

Hi there,

I have been wondering about some points of Chi squared test.

I would like to introduce the following scenario to better clarify my question:

Let's say that one day while doing research on the city of Villy God has appeared to me and told me that city there are 10,000 adults, of which 5036 are women and 4964 are men. He also told me how tall each one measures (in meters).

As God spoke to me about the city, he told me that asked that Mother Nature attributed the following probability of tall for women in that city are born, as follows: for every 7 women who are born there 01 will be 1.60m as an adult; 03 will be 1.65m; 02 will be 1.70m and 01 will be 1.75m.

Table 01

Note about the simulation: To develop the scenario of Table 01 I did a simulation of 10,000 random number according to the probability of the height of each women.

A little suspicious about what God told me, I decided to go into the field to collect data.

Since I can not go into every home and measure the height of all women, I drew 300 adults randomly to make the measurement and of them 151 were women.

As I am interested to know if indeed the height of women follows the proportion that God has given to Mother Nature, so I did the Chi-squared test to see if it is indeed true.

Table 02

Note about the simulation: The sample collected randomly through an algorithm that generated 300 different random numbers that may be between the adult of number 01 to the number of adult 10,000.

Now following questions:
1. What does that value of .498461 Chi-squared test mean?
2. Can I interpret the value as "49.84% of chance that the actual observed value in W is actually the expected value E [W], and that any difference is only the chance"?
3. Since the height distribution follows that probability shown in Table 01 and also my collection was completely random, should not be the value of Chi-Square close to 01? For example, a slight change that I have made on the collected real values (Table 03) has completely changed the test value Chi-Squared to .7235.

Table 03

4. What is the F Test and T Test in this scenario?

Thank you, guys!