Explain why increasing the sample size tends to result in
Explain why increasing the sample size tends to result in a smaller sampling error when a sample mean is used to estimate a population mean.
Explain why increasing the sample size tends to result in
The Correct Answer and Explanation is:
Correct Answer:
Increasing the sample size tends to result in a smaller sampling error because larger samples provide more information and reduce the effect of random variation, leading to a more accurate estimate of the population mean.
Explanation:
Sampling error is the difference between a sample statistic (like the sample mean) and the true population parameter (like the population mean). It arises because a sample is only a subset of the population and may not perfectly represent it due to random variation. However, one of the key principles in statistics is that increasing the sample size helps reduce this sampling error.
When the sample size increases, each additional data point contributes more information about the population. With more observations, random fluctuations and outliers have less impact on the overall estimate. This leads to a sample mean that is more likely to be close to the actual population mean.
Mathematically, the standard error of the mean (SEM)—which measures the average distance between the sample mean and the population mean—decreases as sample size increases. The formula for SEM is:
$$
\text{SEM} = \frac{\sigma}{\sqrt{n}}
$$
Where:
- $\sigma$ is the population standard deviation,
- $n$ is the sample size.
This formula shows that as $n$ increases, SEM decreases, meaning the sample mean becomes a more precise estimate of the population mean.
Additionally, larger samples tend to follow the Central Limit Theorem, which states that the sampling distribution of the sample mean becomes approximately normal as the sample size increases, regardless of the population’s distribution. This enhances the reliability and predictability of the estimate.
In conclusion, increasing the sample size reduces sampling error by lowering the standard error, mitigating the impact of outliers and randomness, and improving the stability and accuracy of the sample mean as an estimate of the population mean.
