Updated on August 9, 2022 as the size of the sample increases what happens to a) it cannot be predicted in advance b) it approaches a normal distribution) as guaranteed by central limit theo- rem Central Limit Theorem: If large sized samples are drawn from any population, with mean-μ and standard deviation-σ, then the sampling distribution of sample means approximates a normal distribution, irrespective of the underlying distribution of population. The greater the sample size, the better the approximation (typically n 2 30) oThank youe Final SummaryThe Central Limit TheoremWe have examined in detail three components of the central limit theorem -- successive sampling, increasing sample size, and different populations. Let's review what we have learned from each and put them together into a final statement. Remember that the central limit theorem applies only to the mean and not to other statistics. General Procedure
Successive Sampling
Increasing Sample Size
Population Distributions
Central Limit TheoremThe central limit theorem states that when an infinite number of successive random samplesare taken from a population, the distribution of sample means calculated for each sample will become approximately normally distributed with mean � and standard deviation s/�N ( ~N(�,s/�N)) as the sample size (N) becomes larger, irrespective of the shape of the population distribution. Hypothesis TestsHow does the central limit theorem help us when we are testing hypotheses about sample means? Even if we do not know the distribution of scores in the original population, we know that the sampling distribution of the means will be approximately normally distributedwith mean� and standard deviation s/�N, if the sample is relatively large. Knowing the properties of the sampling distribution allows us to continue with the test, even if we don't know what the population distribution looks like.
How would increasing the size of the sample affect the shape of the curve?Sampling Distribution:
The increase in sample size increases the accuracy of the parametric values calculated from the samples and the distribution tends to approach a symmetric bell-shaped curve which is the shape of normal distribution.
What happens to spread When sample size increases?Let's put all of this together. As sample size increases, the sampling distribution more closely approximates the normal distribution, and the spread of that distribution tightens.
What happens to the shape of a sampling distribution of sample means as n increases it becomes narrower and more normal?Q. What happens to the shape of a sampling distribution of sample means as n increases? It becomes narrower and bimodal.
What is the shape of the distribution of sample means?The shape of the distribution of sample means tends to be normal. It is guaranteed to be normal if either a) the population from which the samples are obtained is normal, or b) the sample size is n = 30 or more.
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