Question

A random sample of n = 250 measurements is drawn from a binomial population with a probability of success of .85.

1. Find$E\left(\overbrace{P}\right)and{\sigma }_{\overbrace{p}}$
2. Describe the shape of the sampling distribution of$\stackrel{⏜}{p}$.
3. Find

Short answer

Random sampling is a sample strategy in which every sample has an equal probability of being selected. A random sample is intended to provide an impartial reflection of the overall population.

Step-by-step solution

 Step 1: (a) The data is given below

The calculation is given below:

Given,

$\begin{array}{rcl}\text{p}& =& 0.\text{85}\left(\text{Population Proportion}\right)\\ & & \text{= 1}-\text{P}\\ & =& \text{1}-0.\text{85}\\ & =& 0.\text{15}\\ & & \end{array}$

$\text{n}=\text{25}0\left(\text{Sample Size}\right)$

The sample proportionis $\stackrel{⏜}{p}$.

$$\begin{gathered} \text{Mean =}\mu \overbrace{p} \\ \text{P}=0.85 \end{gathered}$$

$\begin{array}{rcl}SD\left(\sigma \stackrel{⏜}{p}\right)& =& \sqrt{\text{p}\left(1-\text{p}\right)\text{/ n}}\\ & =& \sqrt{\frac{0.85\times 0.15}{250}}\\ & =& 0.02258317\\ & & \end{array}$

$$\begin{gathered} \mu \stackrel{⏜}{p}=0.85 \\ \sigma \stackrel{⏜}{p}=\text{0.02258317} \end{gathered}$$

(b) The data is given below

The calculation is given below:

Since both$\text{n}\times \text{p and n}\times \left(\text{1}-\text{p}\right)\text{are}>\text{=}10\text{, so}$

The $\stackrel{⏜}{p}$sampling distribution is about normal.

(c) The data is given below

The calculation is given below:

$\begin{array}{rcl}& & p\left(\left(\stackrel{⏜}{p}-\mu \stackrel{⏜}{p}\right)/\sigma \stackrel{⏜}{p}<\left(0.9-\mu \stackrel{⏜}{p}\right)/\mu \stackrel{⏜}{p}\right)\\ & =& \text{P}\left(\text{Z}\right)<\left(0.9-0.85\right)/0.02258317\\ & =& \text{P}\left(\text{Z}<2.21\right)\\ & =& 0.9864...\text{use z table}\\ & & \end{array}$

$p\left(\stackrel{⏜}{p}<0.9\right)=0.9864$