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No, many sample statistics do not have a normal distribution. In most cases order statistics, such as the minimum or the maximum, are not normally distributed, even when the underlying data themselves have a common normal distribution. The geometric mean (for positive-valued data) almost never has a normal distribution. Practically important statistics, including the chi-square statistic, the F-statistic, and the R-squared statistic of regression, do not have normal distributions. Typically, the normal distribution arises as a good approximation when the sample statistic acts like the independent sum of variables none of whose variances dominates the total variance: this is a loose statement of the Central Limit Theorem. A sample sum and mean, when the elements of the sample are independently obtained, will therefore often be approximately normally distributed provided the sample is large enough.
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The distribution of sample means is not always a normal distribution Under what circumstances will the distribution of sample means not be normal?
The distribution of sample means will not be normal if the number of samples does not reach 30.
Is the distribution of sample means always a normal distribution If not why?
It need not be if: the number of samples is small; the elements within each sample, and the samples themselves are not selected independently.
Is it possible for sample not normal to be from normal population?
Yes. You could have a biased sample. Its distribution would not necessarily match the distribution of the parent population.
What is the meaning of Sampling distribution of the test statistic?
Given any sample size there are many samples of that size that can be drawn from the population. In the population is N and the sample size in n, then there are NCn, but remember that the population can be infinite. A test statistic is a value that is calculated from only the observations in a sample (no unknown parameters are estimated). The value of the test statistic will change from sample to sample. The sampling distribution of a test statistic is the probability distribution function for all the values that the test statistic can take across all possible samples.
What happens to the distribution of the t-score as the sample size increases?
It approaches a normal distribution.
The distribution of sample means is not always a normal distribution Under what circumstances will the distribution of sample means not be normal?
The distribution of sample means will not be normal if the number of samples does not reach 30.
Is the distribution of sample means always a normal distribution If not why?
It need not be if: the number of samples is small; the elements within each sample, and the samples themselves are not selected independently.
What is the expected shape of the distribution of the sample mean?
The distribution of the sample mean is bell-shaped or is a normal distribution.
Is it possible for sample not normal to be from normal population?
Yes. You could have a biased sample. Its distribution would not necessarily match the distribution of the parent population.
What is the purpose of a sampling distribution?
A statistic based on a sample is an estimate of some population characteristic. However, samples will differ and so the statistic - which is based on the sample - will take different values. The sampling distribution gives an indication of ho accurate the sample statistic is to its population counterpart.
What is the meaning of Sampling distribution of the test statistic?
Given any sample size there are many samples of that size that can be drawn from the population. In the population is N and the sample size in n, then there are NCn, but remember that the population can be infinite. A test statistic is a value that is calculated from only the observations in a sample (no unknown parameters are estimated). The value of the test statistic will change from sample to sample. The sampling distribution of a test statistic is the probability distribution function for all the values that the test statistic can take across all possible samples.
Can one treat sample means as a normal distribution?
Not necessarily. It needs to be a random sample from independent identically distributed variables. Although that requirement can be relaxed, the result will be that the sample means will diverge from the Normal distribution.
What distribution does the F distribution approach as the sample size increases?
The F distribution is used to test whether two population variances are the same. The sampled populations must follow the normal distribution. Therefore, as the sample size increases, the F distribution approaches the normal distribution.
What happens to the distribution of the t-score as the sample size increases?
It approaches a normal distribution.
What is the nearly normal condition?
The nearly normal condition refers to the assumption in statistical analysis that the sampling distribution of a statistic is nearly normal if the sample size is large enough, typically greater than 30. This condition allows for the use of methods that assume normality, even if the population distribution is not exactly normal.
What is a difference between F statistics F distribution?
An F-statistic is a measure that is calculated from a sample. It is a ratio of two lots of sums of squares of Normal variates. The sampling distribution of this ratio follows the F distribution. The F-statistic is used to test whether the variances of two samples, or a sample and population, are the same. It is also used in the analysis of variance (ANOVA) to determine what proportion of the variance can be "explained" by regression.
What is sample distribution?
Theoretically, it is the distribution of a statistic based on all possible samples of a given size. In practice, it may be the distribution under repeated samples.
