This **article discuss****es** the sample **mean**, **standard error**, and the **sampling distribution** of p, the random sample with a size of 100. This article uses the concept of sample mean and standard error. **The sample mean** is defined as the average of a set of data.** Standard error** is how different the population mean is from the **sample mean**.

**Expert Answer**

**Step 1**

**Consider** $p=0.40$ and $n=7$.

\[E(\bar p)=p\]

\[=0.40\]

The **expected value** of $\bar p$ is$0.40$.

**The expected value** $\bar p$ is the sample proportion’s mean of the **sampling distribution.**

**Step 2**

The** standard error** of $\bar p$ is calculated as follows:

\[\sigma _{\bar p}=\sqrt{\dfrac{0.40 (1-0.40)}{100}}\]

\[=\sqrt {\dfrac{0.40\times 0.60}{100}}\]

\[=\sqrt{0.0024}\]

\[=0.0490\]

The** standard error** of $\bar p$ is $0.0490$.

The **standard error **of $\bar p $ is obtained by **dividing the product** $p$ and $(1-p)$ by the** sample size** $n$ and then taking the **square root.**

**Step 3**

Check whether the **sample distribution** of $\bar p$ is **normal.**

**Obtain** the value of $np$.

\[np=100 (0.40) \]

\[=40 <5\]

**Obtain** the value of $n(1-p)$

\[n(1-p)=100(1-0.40)\]

\[= 60<5\]

Since the values of $np$ and $n(1-p)$ are **greater than** $5$, the sampling distribution of $\bar p$ is **approximately normal**. The** sample proportion’s mean** of the sampling distribution is the population proportion $p$, which is $0.40$. The **sample proportion’s standard deviation** of the sampling distribution is $0.0490$.

The **sample distribution** of $\bar p$ has **a mean** and **standard deviation** is $0.40$ and $0.0490$, respectively.

The **general condition of normality** of the sampling distribution of the **sample proportion** is fulfilled. The sampling distribution of $\bar p$ is approximately **normal** since $np$ and $n(1−p)$ are **greater** than $5$.

By **central limit theorem,** mean of the **sample distribution** is equal to the proportion of the population distribution for a **large sample.** The variance of sampling distribution is obtained from the ratio of $p(1−p)$ and the **sample size.**

**Step 4**

According to **central limit theorem,** **sampling distribution** of sample proportion $\bar p$ shows the probability distribution for the sample proportion.

The **sampling distribution** of sample proportion $\bar p$ shows the probability distribution for the **sample proportion.**

Sampling distribution of sample proportion is approximately normal when $n≥30$ using the **central limit theorem**. It is important to obtain the probabilities of the **sample proportions.**

**Numerical Results**

- The
**expected value**of $\bar p $ is $0.40$. - The
**standard error**of $\bar p$ is $0.0490$. - The
**sample distribution**of $\bar p$ is**approximately normal.**

**Example**

**A simple random sample size of 200 is selected from a population with p=0.50.What is expected value of $\bar p$? What is standard error of $\bar p$?**

**Solution**

**Step 1**

**Consider** $ p = 0.50 $ and $ n = 7$.

\[E(\bar p) = p \]

\[=0.50\]

The **expected value** of $\bar p$ is $0.50$.

**Step 2**

The** standard error** of $\bar p$ is calculated as follows

\[\sigma _{\bar p} = \sqrt {\dfrac{0.50 (1-0.50)}{200}}\]

\[ = \sqrt {\dfrac{0.50 \times 0.50 }{200}}\]

\[= \sqrt {0.00125}\]

\[= 0.0353\]

The** standard error** of $\bar p$ is $0.0353$.