1. In thinking about doing statistical analysis, the sample mean should be interpreted as:
A.)a constant value that is equal to the population mean.
B.) a constant value that is approximately equal to the population mean.
C.) a random variable that is approximately equal to the population mean when sampling is done without replacement.
D.)a random variable that is approximately equal to the population mean if n > 30 and when sampling is done without replacement.
E.)a random variable that when averaged across many samples is approximately equal to the population mean.
The correct answer is E.) a random variable that when averaged across many samples is approximately equal to the population mean.
To understand why this is the correct interpretation, let's break it down:
When conducting statistical analysis, we often work with a sample from a larger population. The sample mean is the average value of the observations in the sample.
Now, the sample mean is considered a random variable because it can vary from sample to sample. This is because each sample is typically taken from the population using a random sampling method.
However, the key concept to understand is that when we calculate the mean across many different samples (i.e., take the average of the sample means), the average tends to be very close to the population mean. This is known as the Law of Large Numbers.
In other words, even though the sample mean may differ from the population mean in any particular sample, the variation tends to even out when we look at the averages across multiple samples. This is why the sample mean can be interpreted as a random variable that, when averaged across many samples, is approximately equal to the population mean.
Therefore, option E is the correct interpretation.