As the sample size increases, the standard error of the mean:
A. increases.
B. decreases.
C. may increase or decrease.
The correct answer and explanation is :
The correct answer is:
B. decreases.
Explanation:
The standard error of the mean (SE) refers to the measure of how much the sample mean (average) is expected to vary from the true population mean. It provides a way to quantify the uncertainty or variability associated with the estimate of the population mean based on the sample data.
The formula for the standard error of the mean is: SE=σnSE = \frac{\sigma}{\sqrt{n}}
Where:
- σ is the population standard deviation,
- n is the sample size.
From this formula, it’s clear that the standard error (SE) is inversely related to the square root of the sample size (n). As the sample size increases, the denominator of this formula increases, which causes the overall value of SE to decrease.
This makes intuitive sense because a larger sample size provides more information about the population, leading to a more precise estimate of the population mean. As we gather more data, the variability of the sample mean decreases, meaning the sample mean is likely to be closer to the true population mean.
Here’s a breakdown of the relationship:
- Smaller sample sizes: When the sample size is small, there’s greater variability in the sample mean, which means the standard error is larger. This results in more uncertainty about where the true population mean lies.
- Larger sample sizes: With larger sample sizes, the estimate of the mean becomes more reliable because the effect of random sampling fluctuations diminishes. Therefore, the standard error decreases, and the estimate of the mean becomes more accurate.
In summary, as the sample size increases, the standard error of the mean decreases, leading to more precise estimates of the population mean.