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When is the sample mean over repeated samples from the same population or process not normally distributed?
Provided the samples are independent, the Central Limit Theorem will ensure that the sample means will be distributed approximately normally with mean equal to the population mean.
Will the sampling distribution of the mean always be approximatelly normally distributed?
Yes, and more so for larger samples. (It follows from the Central Limit Theorem.)
Will sample means be nearly normally distributed if the distribution of the measurement among the individuals are not from a normal distribution?
Yes, as you keep drawing more and more samples and the number of samples become sufficiently large. This is known as the Central Limit Theorem.
When do you use a normal distribution?
When studying the sum (or average) of a large number of independent variables. A large number is necessary for the Central Limit Theorem to kick in - unless the variables themselves were normally distributed. Independence is critical. If they are not, normality may not be assumed.
Does the central limit theorem allows the use of the normal distribution to analyze the sample mean if the sample sizes are large enough?
Yes. Roughly, very large samples are very likely to have subsets data points having very similar means and distributions. Large numbers of such subsets will tend to be normal distributed (Why?) and will tend to make the total sample be normally distributed.
When is the sample mean over repeated samples from the same population or process not normally distributed?
Provided the samples are independent, the Central Limit Theorem will ensure that the sample means will be distributed approximately normally with mean equal to the population mean.
What does the Central Limit Theorem state?
The Central Limit Theorem (abbreviated as CLT) states that random variables that are independent of each other will have a normally distributed mean.
Will the sampling distribution of the mean always be approximatelly normally distributed?
Yes, and more so for larger samples. (It follows from the Central Limit Theorem.)
When do you use the central limit theorem?
You use the central limit theorem when you are performing statistical calculations and are assuming the data is normally distributed. In many cases, this assumption can be made provided the sample size is large enough.
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