Let me first set the scene. This is how we learned the variables and their names and symbols.
We have the sugar content of 77 different cereal brands. It firsts asks "estimate the true average sugar content of these 77 cereal brands using a SRS of size 10". These next two questions are what confuses me.
- What is the true variance of the estimator above?
- Using this sample data, estimate the true variance of this estimator
I am really confused on the difference between these two questions. I thought the first one is asking var(y_hat) and the second one is asking var_circumflex(y_hat). If someone could please explain the difference preferable in pretty simple terms since more will just confuse me more.
The "true" value of anything is the value you calculate if you have perfect information about the entire population. When you have to calculate a value from a specific sample and use that to infer the true value, then that's the "estimate".
The true variance of the estimator is $\mathrm{var}(\bar{y})$ - notice that its formula uses $\sigma^2$ which is a property of the whole population. The estimator of the variance is $\widehat{\mathrm{var}}(\bar{y})$ - notice that it uses $s^2$, which is a value that depends on the specific sample that has been taken.