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**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 $0.0353$.