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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 happens to the distribution of the t-score as the sample size increases?
It approaches a normal distribution.
Does a sample statistic always have a normal distribution?
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.
Why do population follow normal distribution?
The form of this question incorportates a false premise. The premise is that the data are normally distributed. Actually, is the sample mean which, under certain circumstances, is normally distributed.
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.
Will the sampling distribution of x ̅ always be approximately normally distributed?
The sampling distribution of the sample mean (( \bar{x} )) will be approximately normally distributed if the sample size is sufficiently large, typically due to the Central Limit Theorem. This theorem states that regardless of the population's distribution, the sampling distribution of the sample mean will tend to be normal as the sample size increases, generally n ≥ 30 is considered adequate. However, if the population distribution is already normal, the sampling distribution of ( \bar{x} ) will be normally distributed for any sample size.
Which estimator will consistently have an approximately normal distribution?
The sample mean is an estimator that will consistently have an approximately normal distribution, particularly due to the Central Limit Theorem. As the sample size increases, the distribution of the sample mean approaches a normal distribution regardless of the original population's distribution, provided the samples are independent and identically distributed. This characteristic makes the sample mean a robust estimator for large sample sizes.
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.
When the population standard deviation is known the sample distribution is a?
When the population standard deviation is known, the sample distribution is a normal distribution if the sample size is sufficiently large, typically due to the Central Limit Theorem. If the sample size is small and the population from which the sample is drawn is normally distributed, the sample distribution will also be normal. In such cases, statistical inference can be performed using z-scores.
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.
Does a sample statistic always have a normal distribution?
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.
What is a t distribution?
The t distribution is a probability distribution that is symmetric and bell-shaped, similar to the normal distribution, but has heavier tails. It is used in statistics, particularly for small sample sizes, to estimate population parameters when the population standard deviation is unknown. The t distribution accounts for the additional uncertainty introduced by estimating the standard deviation from the sample. As the sample size increases, the t distribution approaches the normal distribution.
Why do population follow normal distribution?
The form of this question incorportates a false premise. The premise is that the data are normally distributed. Actually, is the sample mean which, under certain circumstances, is normally distributed.
