A random sample obtained from a population has a mean of µ=100 and a standard deviation of ó = 20. The error between the sample mean and the population mean for a sample of n = 16 is 5 points and the error between a sample men and population mean for a sample of n = 100 is 2 points. Explain the difference in the standard error for the two samples.
The measure of variability for distribution of means is SEm.
SEm = SD/√n
As n increases, the SEm decreases.
To explain the difference in the standard error for the two samples, we first need to understand what standard error represents. Standard error is a measure of the variability or dispersion of sample means around the population mean. It tells us how accurately the sample mean represents the population mean.
The formula to calculate the standard error is:
Standard Error = Standard Deviation / Square Root of Sample Size
Let's calculate the standard errors for the two given samples:
For the sample with n = 16:
Standard Error = 20 / √16 = 20 / 4 = 5
For the sample with n = 100:
Standard Error = 20 / √100 = 20 / 10 = 2
Now we can observe that the standard error for the sample with n = 16 is 5, while the standard error for the sample with n = 100 is 2.
The difference in standard error arises due to the larger sample size in the second case (n = 100). As the sample size increases, the standard error decreases.
This happens because with a larger sample size, there is more information available, resulting in a more precise estimate of the population mean. The larger sample reduces the variability of sample means and, thus, decreases the standard error.
In other words, when you have more data points in your sample, the estimate of the population mean becomes more accurate, leading to a smaller standard error. Conversely, with a smaller sample size, there is less information available, resulting in a larger standard error.