Sampling distribution of a proportion

Practise sampling distribution of a proportion with an original VectorPaths question, worked explanation and related VCE Methods practice.

4 practice marks

Original practice question

A population proportion is $p=0.4$. A random sample of 100 independent observations is taken. X is the number of successes and $\hat p=X/100$. (a) State the distribution of X and the mean of $\hat p$. [2 marks] (b) Find the variance of $\hat p$ and describe the effect of quadrupling n. [2 marks]

Worked explanation

• Part (a)

  The fixed independent trials have common probability 0.4, so X is binomial. $E(\hat p)=E(X)/100=np/100=p$.

• Part (b)

  $\operatorname{Var}(\hat p)=p(1-p)/n=(0.4)(0.6)/100=0.0024$. Standard deviation is proportional to $1/\sqrt n$, so four times the sample size halves it.

(a) $X\sim\operatorname{Bin}(100,0.4)$; $E(\hat p)=0.4$.; (b) $0.0024$; standard deviation halves.

Original VectorPaths practice; not an official VCAA question or marking rubric.