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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