Question

Suppose a random sample of n = 500 measurements is selected from a binomial population with probability of success p. For each of the following values of p, give the mean and standard deviation of the sampling distribution of the sample proportion,$\hat{\mathbf{p}}$.

1. p= .1
2. p= .5
3. p= .7

Short answer

The mean and standard deviation of the sampling distribution of the sampling proportion are,

1. If p=0.1, mean = 0.1, standard deviation = 0.0134
2. If p=0.5, mean = 0.5, standard deviation = 0.0223
3. If p=0.7, mean = 0.7, standard deviation = 0.0204

Step-by-step solution

Given information

There is a random sample of n=500 from a binomial distribution with a probability of success p.

Calculate the mean and standard deviation when p=.1

The mean of the sampling distribution is equal to the true binomial proportion. So, and the standard deviation of the sampling distribution is ${\mathbf{\sigma }}_{\hat{\mathbf{p}}}\mathbf{=}\sqrt{\mathbf{p}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{p}\mathbf{)}\mathbf{/}\mathbf{n}}$.

Therefore,

a.

If p=0.1 then, Mean $E\left(\hat{p}\right)=p=0.1$.

Standard deviation${\sigma }_{\hat{p}}=\sqrt{0.1\left(1-0.1\right)/}500=0.0134$

Calculate the mean and standard deviation p=.5

b.

If p=0.5 then, Mean$E\left(\hat{p}\right)=p=0.5$

Standard deviation${\sigma }_{\hat{p}}=\sqrt{0.5\left(1-0.5\right)/500}=0.0223$

Calculate the mean and standard deviation p=.7

c.

If,p=0.7 then, Mean$E\left(\hat{p}\right)=p=0.7$

Standard deviation${\sigma }_{\hat{p}}=\sqrt{0.7\left(1-0.7\right)/500=}0.0204$