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Chapter 6: Problem 75

Impact of Sample Size on Accuracy Compute the standard error for sample means from a population with mean \(\mu=100\) and standard deviation \(\sigma=\) 25 for sample sizes of \(n=30, n=200,\) and \(n=1000 .\) What effect does increasing the sample size have on the standard error? Using this information about the effect on the standard error, discuss the effect of increasing the sample size on the accuracy of using a sample mean to estimate a population mean.

Short Answer

Expert verified
The standard errors for sample sizes of 30, 200, and 1000 are \(25/\sqrt{30}\), \(25/\sqrt{200}\), and \(25/\sqrt{1000}\) respectively. The standard error decreases as the sample size increases, indicating that the sample mean provides a more accurate estimate of the population mean with increasing sample size.

Step by step solution

01

Calculation for Sample Size n=30

To find the standard error for a sample size of \(n=30\), we plug the given values into the standard error formula. So with \(\mu=100\), \(\sigma=25\), and \(n=30\), the standard error will be \(25/\sqrt{30}\).
02

Calculation for Sample Size n=200

For a sample size of \(n=200\), we get the standard error by applying the formula again, this time with \(n=200\). So we have \(25/\sqrt{200}\).
03

Calculation for Sample Size n=1000

For a sample size of \(n=1000\), the standard error becomes \(25/\sqrt{1000}\).
04

Analyzing the Impact of Sample Size on Standard Error

Comparing the calculated standard errors, it can be observed that as the sample size increases from 30 to 200 to 1000, the standard error decreases. This is because the denominator (\(n\)) in the standard error formula is under a square root - hence an increase in the sample size, \(n\), results in a smaller standard error.
05

Discussing the Impact of Sample Size on Accuracy of Population Mean Estimation

Because the standard error measures how much the sample mean will typically deviate from the population mean, a smaller standard error implies that a sample mean is a more accurate representation of the actual population mean. So, as sample size increases, the accuracy of the estimated population mean also increases.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Size
In statistical analysis, the sample size refers to the number of observations or data points collected from the population for a study. Choosing an appropriately sized sample is crucial as it directly influences the reliability of the results. The sample size affects the standard error, which is a measure of the variability or dispersion of the sample mean from the actual population mean.

In our example, we calculated the standard error for sample sizes of 30, 200, and 1000. The calculations showed that as the sample size increased, the standard error decreased. This happens due to the "Law of Large Numbers," which suggests that larger samples tend to yield results that are closer to the population parameters.

Here鈥檚 a quick takeaway about sample sizes:
  • Smaller sample sizes tend to have larger standard errors, indicating more variability
  • Larger sample sizes reduce the standard error, leading to more consistent estimates
  • Choosing an appropriate sample size is vital for effective statistical analysis
Population Mean
The population mean is the average of all individual values in a population set. It is denoted by the Greek letter \(\mu\). Understanding the population mean is essential for comprehending the general tendencies of a set of data.

In research, we often use a sample mean, calculated from a collected data subset, to estimate this population mean. However, knowing how close a sample mean is to the population mean involves considering the standard error. If the sample size is large enough, the sample mean can provide a good estimate of the population mean.

Quick insights about population mean:
  • The population mean \(\mu\) is a fixed value and represents the true mean of the entire set
  • A sample mean can be used to approximate the population mean if the sample is representative
  • The variability in this estimation depends on the sample size and the standard deviation
Accuracy of Estimation
Accuracy of estimation in statistics refers to how close the sample data鈥檚 findings are to the actual population parameters. A key factor influencing accuracy is the standard error, which decreases as sample size increases.

In our example, as the sample size increased from 30 to 200 to 1000, the standard error decreased, leading to a more accurate estimation of the population mean. When the standard error is smaller, the sample mean has a higher chance of being close to the population mean.

Key points about the accuracy of estimation:
  • Larger sample sizes provide more accurate estimates of population parameters
  • Reducing the standard error enhances the reliability of the estimation
  • Accurate estimations are essential for making informed decisions based on data

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Most popular questions from this chapter

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