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Population mean vs sample mean watch

1. Hi guys

I'm having a bit of a thinking blockade regarding the following question:

"When the sample size is large, the sample mean is as likely to lie above the population mean as below it."

When I first saw this question I thought that this cannot be true because the population size will always be bigger than a sample. Then I thought that it wouldn't matter as the sample mean is based on the data obtained which until the CLT become apparent it could be less or greater than the population mean . However the question states that the sample is large...

Can anyone help?

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2. How are you supposed to answer it?

True/false?
3. (Original post by Fyer1234)
Hi guys

I'm having a bit of a thinking blockade regarding the following question:

"When the sample size is large, the sample mean is as likely to lie above the population mean as below it."

When I first saw this question I thought that this cannot be true because the population size will always be bigger than a sample. Then I thought that it wouldn't matter as the sample mean is based on the data obtained which until the CLT become apparent it could be less or greater than the population mean . However the question states that the sample is large...

Can anyone help?

Posted from TSR Mobile
As you mentioned, a population, by definition, will always be larger than a sample taken from it. To explain this answer, we should first consider what a sample is. A sample is a set of data representative of the sample as a whole.

With this in mind, and considering the Central Limit theorem (CLT), which states:

...the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed, regardless of the underlying distribution.
It is statistically justified to assume that a random sample from this defined "large" sample would be approximately normally distributed and so the sample mean is as likely to lie above the population mean as below it.

Hope this helps.
4. Thanks guys

Yes I can see that now, thanks.

At first it I just saw the sample mean as always rising as the sample size does . Then I considered that it could also be decreasing the closer the sample size becomes to the sample mean. As the mean is the average of all data obtained then it can also be assumed that these data values could , by chance all be quite high....that's about right, right...?:-)

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