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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.
Is normal distribution hypothetical?
The normal distribution is a theory, which works in practice (with a large enough sample). E.g if you were to plot the height of everyone in the country, you should end up with a normal distribution. Hence it is not usually considered hypothetical, in the same way that, say, imaginary numbers are hypothetical.
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.
When population distribution is right skewed is the sampling also with right skewed distribution?
If the population distribution is roughly normal, the sampling distribution should also show a roughly normal distribution regardless of whether it is a large or small sample size. If a population distribution shows skew (in this case skewed right), the Central Limit Theorem states that if the sample size is large enough, the sampling distribution should show little skew and should be roughly normal. However, if the sampling distribution is too small, the sampling distribution will likely also show skew and will not be normal. Although it is difficult to say for sure "how big must a sample size be to eliminate any population skew", the 15/40 rule gives a good idea of whether a sample size is big enough. If the population is skewed and you have fewer that 15 samples, you will likely also have a skewed sampling distribution. If the population is skewed and you have more that 40 samples, your sampling distribution will likely be roughly normal.
What is the definition of central limit theorem?
The central limit theorem basically states that as the sample size gets large enough, the sampling distribution becomes more normal regardless of the population 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.
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.
How can you compare means of two samples when the samples are chi square distributed?
According to the Central Limit Theorem if the sample size is large enough then the means will tend towards a normal distribution regardless of the distribution of the actual sample.
Is normal distribution hypothetical?
The normal distribution is a theory, which works in practice (with a large enough sample). E.g if you were to plot the height of everyone in the country, you should end up with a normal distribution. Hence it is not usually considered hypothetical, in the same way that, say, imaginary numbers are hypothetical.
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 happens to the distribution of the t-score as the sample size increases?
It approaches a 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.
When population distribution is right skewed is the sampling also with right skewed distribution?
If the population distribution is roughly normal, the sampling distribution should also show a roughly normal distribution regardless of whether it is a large or small sample size. If a population distribution shows skew (in this case skewed right), the Central Limit Theorem states that if the sample size is large enough, the sampling distribution should show little skew and should be roughly normal. However, if the sampling distribution is too small, the sampling distribution will likely also show skew and will not be normal. Although it is difficult to say for sure "how big must a sample size be to eliminate any population skew", the 15/40 rule gives a good idea of whether a sample size is big enough. If the population is skewed and you have fewer that 15 samples, you will likely also have a skewed sampling distribution. If the population is skewed and you have more that 40 samples, your sampling distribution will likely be roughly normal.
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 the definition of central limit theorem?
The central limit theorem basically states that as the sample size gets large enough, the sampling distribution becomes more normal regardless of the population distribution.
