I was reading my textbook and had a question,
$P\left(\hat{p}=\frac{x}{n}\right)=P(X=x)={ }^{n} \mathrm{C}_{x} p^{x} q^{n-x}$, so the various graphs of a sample proportion distribution are just the graphs of the corresponding binomial distribution stretched horizontally by a factor of $\frac{1}{n}$.
I understand why the graph of sample proportion are stretched horizontally by a factor of $\frac{1}{n}$ because you are dividing by $n$.
But to stretch the graph horizontally by a factor of $\frac{1}{n}$ wouldn't you have to replace $x$ by $\frac{x}{\frac{1}{n}}=nx$? This is what I learned as the rule for dilating a function, so I am a bit confused. Thanks..
If $y=\frac xn$ then $P\left(\hat{p}=y\right)={ }^{n} \mathrm{C}_{ny} \,\,p^{ny} q^{n-ny}$ replacing $x$ with $ny$.
Is this what you meant?