A Population Has A Mean Μ = 133 \mu=133 Μ = 133 And A Standard Deviation Σ = 20 \sigma=20 Σ = 20 . Find The Mean And Standard Deviation Of The Sampling Distribution Of Sample Means With Sample Size N = 59 N=59 N = 59 .The Mean Is Μ X ˉ = □ \mu_{\bar{x}}= \square Μ X ˉ = □ ,

Mar 8, 2025 by ADMIN 277 views

$A population has a mean $\mu=133$ and a standard deviation $\sigma=20$. Find the mean and standard deviation of the sampling distribution of sample means with sample size $n=59$.$

The Sampling Distribution of Sample Means

The sampling distribution of sample means is a probability distribution of the sample means that can be obtained from a large number of random samples of size $n$ drawn from a population with a given mean $\mu$ and standard deviation $\sigma$ . The sampling distribution of sample means is an important concept in statistics, as it provides a way to understand the variability of sample means and to make inferences about the population mean.

The Mean of the Sampling Distribution of Sample Means

The mean of the sampling distribution of sample means is denoted by $\mu_{\bar{x}}$ and is equal to the population mean $\mu$ . This is because the sampling distribution of sample means is a probability distribution of sample means, and the mean of a probability distribution is the expected value of the random variable. In this case, the expected value of the sample mean is the population mean.

The Standard Deviation of the Sampling Distribution of Sample Means

The standard deviation of the sampling distribution of sample means is denoted by $\sigma_{\bar{x}}$ and is equal to the population standard deviation $\sigma$ divided by the square root of the sample size $n$ . This is because the sampling distribution of sample means is a probability distribution of sample means, and the standard deviation of a probability distribution is a measure of the variability of the random variable. In this case, the variability of the sample mean is reduced by a factor of $\sqrt{n}$ .

Calculating the Standard Deviation of the Sampling Distribution of Sample Means

To calculate the standard deviation of the sampling distribution of sample means, we can use the formula:

\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}

where $\sigma$ is the population standard deviation and $n$ is the sample size.

Substituting the Given Values

We are given that the population standard deviation $\sigma$ is equal to 20 and the sample size $n$ is equal to 59. Substituting these values into the formula, we get:

\sigma_{\bar{x}} = \frac{20}{\sqrt{59}}

Simplifying the Expression

To simplify the expression, we can calculate the square root of 59, which is approximately equal to 7.68. Substituting this value into the expression, we get:

\sigma_{\bar{x}} = \frac{20}{7.68}

Calculating the Value

To calculate the value of the standard deviation of the sampling distribution of sample means, we can divide 20 by 7.68, which is approximately equal to 2.60.

Conclusion

In conclusion, the mean of the sampling distribution of sample means is equal to the population mean, and the standard deviation of the sampling distribution of sample means is equal to the population standard deviation divided by the square root of the sample size. In this case, the standard deviation of the sampling distribution of sample means is approximately equal to 2.60.

The Final Answer

The final answer is: $\boxed{2.60}$

Frequently Asked Questions

The sampling distribution of sample means is a fundamental concept in statistics, and it can be a bit confusing at first. Here are some frequently asked questions and answers to help clarify the concept.

Q: What is the sampling distribution of sample means?

A: The sampling distribution of sample means is a probability distribution of the sample means that can be obtained from a large number of random samples of size n drawn from a population with a given mean μ and standard deviation σ.

Q: What is the mean of the sampling distribution of sample means?

A: The mean of the sampling distribution of sample means is equal to the population mean μ.

Q: What is the standard deviation of the sampling distribution of sample means?

A: The standard deviation of the sampling distribution of sample means is equal to the population standard deviation σ divided by the square root of the sample size n.

Q: How do I calculate the standard deviation of the sampling distribution of sample means?

A: To calculate the standard deviation of the sampling distribution of sample means, you can use the formula: σ_{\bar{x}} = \frac{σ}{\sqrt{n}}.

Q: What is the significance of the sampling distribution of sample means?

A: The sampling distribution of sample means is important because it provides a way to understand the variability of sample means and to make inferences about the population mean.

Q: How do I determine the sample size n for a given population standard deviation σ?

A: To determine the sample size n for a given population standard deviation σ, you can use the formula: n = \frac{σ^2}{E2}, where E is the desired margin of error.

Q: What is the relationship between the sampling distribution of sample means and the population distribution?

A: The sampling distribution of sample means is a probability distribution of sample means, and it is related to the population distribution by the formula: P(\bar{x} ≤ x) = P(X ≤ x), where X is a random variable from the population distribution.

Q: Can I use the sampling distribution of sample means to make inferences about the population mean?

A: Yes, you can use the sampling distribution of sample means to make inferences about the population mean. For example, you can use the sampling distribution to calculate the probability that the sample mean is within a certain range of the population mean.

Q: What are some common applications of the sampling distribution of sample means?

A: Some common applications of the sampling distribution of sample means include:

Estimating the population mean
Testing hypotheses about the population mean
Calculating confidence intervals for the population mean
Making inferences about the population mean based on sample data

Conclusion

In conclusion, the sampling distribution of sample means is a fundamental concept in statistics that provides a way to understand the variability of sample means and to make inferences about the population mean. By understanding the sampling distribution of sample means, you can make more informed decisions and draw more accurate conclusions from your data.

Additional Resources

If you want to learn more about the sampling distribution of sample means, here are some additional resources that you may find helpful:

"Statistics for Dummies" by Deborah J. Rumsey
"Statistics: The Art and Science of Learning from Data" by Alan Agresti and Christine Franklin
"The Sampling Distribution of the Sample Mean" by the American Statistical Association

I hope this Q&A article has been helpful in clarifying the concept of the sampling distribution of sample means. If you have any further questions or need additional clarification, please don't hesitate to ask.