A simple random sample size of 100 is selected from a population with p= 0.40. What is the expected value of p? What is the standard error of p? Show the sampling distribution of p? What does the sampling distribution of p show?

This article discusses 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

1. The expected value of $\bar p $ is $0.40$.
2. The standard error of $\bar p$ is $0.0490$.
3. 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 $0.0353$.

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