Population Mean and Standard Deviation

Oct 13, 2025 by magicalceylontours.com 3 views

Introduction

In statistics, a survey is a method of collecting data from a sample of individuals or households to make inferences about a larger population. Susan, in this case, is conducting a survey about the electric bills of households in her city. To understand the reliability of her survey results, it's essential to know the population mean and standard deviation of the electric bills. In this article, we will discuss the concept of sampling distribution and standard error, and how they relate to Susan's survey.

Population Mean and Standard Deviation

The population mean is the average value of a variable in a population. In this case, the population mean of the electric bills is $98.75. The population standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In this case, the population standard deviation of the electric bills is $10.45.

Sampling Distribution

A sampling distribution is a probability distribution of a statistic that is obtained from a sample of data. The sampling distribution of a statistic is a theoretical distribution that describes the possible values of the statistic and their probabilities. In this case, the sampling distribution of the sample mean is of interest. The sample mean is the average value of a variable in a sample of data.

Standard Error

The standard error (SE) is a measure of the variability of a sample statistic. It is a measure of how much the sample statistic is likely to differ from the population parameter. In this case, the standard error of the sample mean is of interest. The standard error of the sample mean is calculated as the population standard deviation divided by the square root of the sample size.

Calculating the Standard Error

To calculate the standard error of the sample mean, we need to know the population standard deviation and the sample size. In this case, the population standard deviation is $10.45, and the sample size is not given. However, we can still calculate the standard error of the sample mean using the given information.

The Formula for Standard Error

The formula for the standard error of the sample mean is:

SE = σ / √n

where σ is the population standard deviation, and n is the sample size.

Plugging in the Values

To calculate the standard error of the sample mean, we need to plug in the values of the population standard deviation and the sample size into the formula. However, we are given the population standard deviation as $10.45, but the sample size is not given. Let's assume that the sample size is n.

The Standard Error of the Sample Mean

The standard error of the sample mean is:

SE = $10.45 / √n

Interpretation of the Standard Error

The standard error of the sample mean is a measure of the variability of the sample mean. It is a measure of how much the sample mean is likely to differ from the population mean. In this case, the standard error of the sample mean is $10.45 / √n.

Implications for Susan's Survey

Susan's survey is designed to estimate the population mean of the electric bills in her city. The standard error of the sample mean is a measure of the variability of the sample mean. It is a measure of how much the sample mean is likely to differ from the population mean. In this case, the standard error of the sample mean is $10.45 / √n.

Conclusion

In conclusion, the standard error of the sample mean is a measure of the variability of the sample mean. It is a measure of how much the sample mean is likely to differ from the population mean. In this case, the standard error of the sample mean is $10.45 / √n. Susan's survey is designed to estimate the population mean of the electric bills in her city. The standard error of the sample mean is a measure of the reliability of her survey results.

References

[1] Moore, D. S., & McCabe, G. P. (2011). Introduction to the practice of statistics. W.H. Freeman and Company.
[2] Larson, R. E., & Farber, B. A. (2010). Elementary statistics: Picturing the world. Cengage Learning.

Appendix

A.1. Derivation of the Formula for Standard Error

The formula for the standard error of the sample mean is:

SE = σ / √n

where σ is the population standard deviation, and n is the sample size.

To derive this formula, we need to start with the definition of the sample mean:

x̄ = (1/n) * ∑x_i

where x_i is the i-th value in the sample, and n is the sample size.

The sample mean is a random variable, and its expected value is equal to the population mean:

E(x̄) = μ

where μ is the population mean.

The variance of the sample mean is equal to the population variance divided by the sample size:

Var(x̄) = σ^2 / n

where σ^2 is the population variance.

The standard error of the sample mean is the square root of the variance:

SE = √(σ^2 / n)

Simplifying this expression, we get:

SE = σ / √n

Q: What is the sampling distribution?

A: The sampling distribution is a probability distribution of a statistic that is obtained from a sample of data. It is a theoretical distribution that describes the possible values of the statistic and their probabilities.

Q: What is the standard error?

A: The standard error (SE) is a measure of the variability of a sample statistic. It is a measure of how much the sample statistic is likely to differ from the population parameter.

Q: How is the standard error calculated?

A: The standard error is calculated as the population standard deviation divided by the square root of the sample size.

Q: What is the formula for the standard error?

A: The formula for the standard error is:

SE = σ / √n

where σ is the population standard deviation, and n is the sample size.

Q: What is the difference between the standard error and the standard deviation?

A: The standard deviation is a measure of the amount of variation or dispersion of a set of values. The standard error, on the other hand, is a measure of the variability of a sample statistic.

Q: Why is the standard error important?

A: The standard error is important because it provides a measure of the reliability of a sample statistic. It helps to determine how much the sample statistic is likely to differ from the population parameter.

Q: How can I use the standard error in my research?

A: You can use the standard error to determine the margin of error for your sample statistic. This will give you an idea of how much the sample statistic is likely to differ from the population parameter.

Q: What is the margin of error?

A: The margin of error is the amount of error that is likely to occur when estimating a population parameter using a sample statistic.

Q: How is the margin of error calculated?

A: The margin of error is calculated as the standard error multiplied by a critical value from a standard normal distribution.

Q: What is the critical value?

A: The critical value is a value from a standard normal distribution that is used to determine the margin of error.

Q: How can I determine the critical value?

A: You can determine the critical value using a standard normal distribution table or calculator.

Q: What is the relationship between the standard error and the sample size?

A: The standard error is inversely proportional to the square root of the sample size. This means that as the sample size increases, the standard error decreases.

Q: How can I increase the precision of my sample statistic?

A: You can increase the precision of your sample statistic by increasing the sample size.

Q: What is the importance of the standard error in hypothesis testing?

A: The standard error is important in hypothesis testing because it provides a measure of the reliability of the sample statistic. It helps to determine whether the sample statistic is likely to differ from the population parameter.

Q: How can I use the standard error in hypothesis testing?

A: You can use the standard error to determine the test statistic and the p-value. This will help you to determine whether the sample statistic is likely to differ from the population parameter.

Q: What is the p-value?

A: The p-value is the probability of obtaining a sample statistic as extreme or more extreme than the one observed, assuming that the null hypothesis is true.

Q: How can I determine the p-value?

A: You can determine the p-value using a standard normal distribution table or calculator.

Q: What is the relationship between the standard error and the p-value?

A: The standard error is related to the p-value because it provides a measure of the reliability of the sample statistic. The p-value is a measure of the probability of obtaining a sample statistic as extreme or more extreme than the one observed, assuming that the null hypothesis is true.

Q: How can I use the standard error and the p-value in my research?

A: You can use the standard error and the p-value to determine the reliability of your sample statistic and to make inferences about the population parameter.