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.
SEm = SD/√n
To explain the difference in the standard error for the two samples, we first need to understand what standard error represents. The standard error measures the variability or uncertainty in the sampling distribution of the sample mean. It tells us how much we can expect the sample mean to deviate from the population mean.
The formula for calculating the standard error is:
Standard Error = σ / √n
where σ (sigma) is the population standard deviation, and n is the sample size.
In the given scenario, the population mean is µ = 100, and the population standard deviation is σ = 20. We are given two different sample sizes: n = 16 and n = 100. We are also given the error between the sample mean and the population mean for each sample size.
For the sample with n = 16, the error is 5 points. This means that the sample mean can vary from the population mean by up to 5 points. Now, let's calculate the standard error for this sample:
Standard Error (n = 16) = 20 / √16 = 20 / 4 = 5
So, with a sample size of 16, the standard error is 5. This means that on average, we can expect the sample mean to deviate from the population mean by around 5 units.
For the sample with n = 100, the error is 2 points. This means that the sample mean can vary from the population mean by up to 2 points. Let's calculate the standard error for this sample:
Standard Error (n = 100) = 20 / √100 = 20 / 10 = 2
So, with a sample size of 100, the standard error is 2. This means that on average, we can expect the sample mean to deviate from the population mean by around 2 units.
Comparing the two standard errors, we can see that the standard error for the sample with n = 16 is larger (5) than the standard error for the sample with n = 100 (2). This difference occurs because the sample size (n) is in the denominator of the standard error formula. As the sample size increases, the denominator gets larger, resulting in a smaller standard error. In other words, larger sample sizes provide more precise estimates of the population mean, leading to smaller standard errors.