We need to find out how many adults must the brand manager survey in order to be 90% confident that his estimate is within five percentage points of the true population percentage.
From the given data we know that our confidence level is 90%. From Standard Normal Table we know that the critical level at 90% confidence level is 1.645. In other words, 
.
We also know that E=5% or E=0.05
Also, since, 
is not given, we will assume that =0.5. This is because, the formula that we use will have 
in the expression and that will be maximum only when =0.5. (For any other value of , we will get a value less than 0.25. For example if, is 0.4, then 
and thus, 
.).
We will now use the formula
We will now substitute all the data that we have and we will get
which can approximated to n=271.
So, the brand manager needs a sample size of 271
Answer:
In statistics and econometrics, the first-difference (FD) estimator is an estimator used to address the problem of omitted variables with panel data. It is consistent under the assumptions of the fixed effects model. In certain situations it can be more efficient than the standard fixed effects (or "within") estimator.
First differences are the differences between consecutive y-‐values in tables of values with evenly spaced x-‐values. If the first differences of a relation are constant, the relation is _______________________________ If the first differences of a relation are not constant, the relation is ___________________________
