A simple random sample size of 100 is selected from

Alyce Wilkinson

Alyce Wilkinson

Answered question

2021-09-26

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?

Answer & Explanation

izboknil3

izboknil3

Skilled2021-09-27Added 99 answers

Step 1
Consider p=0.40 and n=7
E(p)=p
=0.40
The expected value of p is 0.40
The expected value of p is the mean of the sampling distribution of the sample proportion.
Step 2
The standard error of p is calculated as follows:
σp=0.40(10.40)100
=0.40×0.60100
0.0024
=0.0490
The standard error of p is 0.0490.
The standard error of p is obtained by dividing of the product of pp and (1p) by the sample size n and then taking square root.
Step 3
Check whether the sampling distribution of p is approximately normal or not.
Obtain the value of np.
np=100(0.40)
=40[<5]
Obtain the value of n(1p)
n(1p)=100(10.40)
=60[<5]
Since the values of np and n(1p) is greater than 5, the sampling distribution of p is approximately normal.
The mean of the sampling distribution of the sample proportion is the population proportion p , which is 0.40.
The standard deviation of the sampling distribution of the sample proportion is 0.0490.
The sampling distribution of p has mean and standard deviation is 0.40 and 0.0490, respectively.
The general condition for the normality of the sampling distribution of the sample proportion is satisfied. The requirement for the sampling distribution of p to be approximately normal since npnp and n(1p) are greater than 5.
By the central limit theorem, the mean of the sampling distribution is same as proportion of the population distribution for the large sample size. The variance of the sampling distribution is obtained by taking the ratio of p(1p) and the sample size.
Step 4
According to the central limit theorem, the sampling distribution of sample proportion p shows the probability distribution for the sample proportion.
The sampling distribution of sample proportion p shows the probability distribution for the sample proportion.
The sampling distribution of the sample proportion is approximately normal when n30 using central limit theorem. It is important to obtain the probabilities about the sample proportions.

xleb123

xleb123

Skilled2023-06-17Added 181 answers

Step 1:
To solve the problem, we need to calculate the expected value of the sample mean (p), the standard error of the sample mean, and show the sampling distribution of p.
The expected value of the sample mean (p) can be calculated using the formula:
Expected Value of p=p
where p is the population proportion. In this case, p=0.40. Therefore, the expected value of p is:
Expected Value of p=0.40
Step 2:
The standard error of the sample mean (p) can be calculated using the formula:
Standard Error of p=p(1p)n
where n is the sample size. In this case, n=100 and p=0.40. Substituting these values into the formula, we get:
Standard Error of p=0.40(10.40)100
Step 3:
Simplifying the expression:
Standard Error of p=0.24100
Standard Error of p=0.0024
Standard Error of p0.049
The sampling distribution of p represents the distribution of sample means that would be obtained if repeated random samples of the same size were drawn from the population. It is often assumed to be approximately normal when the sample size is large enough due to the Central Limit Theorem. The mean of the sampling distribution is equal to the population mean, and the standard deviation (standard error) of the sampling distribution is equal to the population standard deviation divided by the square root of the sample size. It provides insights into the variability and precision of the sample mean estimate (p) and helps in making inferences about the population parameter (in this case, p).
Jazz Frenia

Jazz Frenia

Skilled2023-06-17Added 106 answers

Answer: 0.04899
Explanation:
The standard error of p can be calculated using the formula:
SEp=p(1p)n where n is the sample size. In this case, n=100, and p=0.40. Plugging these values into the formula:
SEp=0.40(10.40)100
Simplifying the expression:
SEp=0.24100
SEp=0.04899
The sampling distribution of p follows a normal distribution with a mean equal to the population proportion p and a standard deviation equal to the standard error SEp. It shows the range of possible sample means that could be obtained from repeated random sampling of the same size from the population.
Andre BalkonE

Andre BalkonE

Skilled2023-06-17Added 110 answers

The expected value of the sample mean, denoted as E(p), can be calculated using the formula:
E(p)=p where p is the population proportion.
Given that p=0.40, the expected value of p is:
E(p)=0.40
The standard error of the sample mean, denoted as SE(p), can be calculated using the formula:
SE(p)=p(1p)n where n is the sample size.
Given that p=0.40 and n=100, the standard error of p is:
SE(p)=0.40(10.40)100
SE(p)=0.24100
SE(p)=0.0024
SE(p)0.049
The sampling distribution of p represents the distribution of all possible sample means that could be obtained from the same population. It shows the variability of sample means around the population mean and provides insights into the precision of the sample mean as an estimator of the population mean.
To visualize the sampling distribution of p, we can plot a histogram or a probability density curve that represents the distribution of p values. The shape of the sampling distribution is often approximately normal when certain conditions are met (e.g., large sample size, random sampling).
In this case, since the sample size is 100 and the sampling is assumed to be random, the sampling distribution of p is expected to be approximately normal, centered around the population proportion p=0.40, with a standard error of approximately 0.049.

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