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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.
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.
How do you calculate distribution of sample means?
The sample mean is distributed with the same mean as the popualtion mean. If the popolation variance is s2 then the sample mean has a variance is s2/n. As n increases, the distribution of the sample mean gets closer to a Gaussian - ie Normal - distribution. This is the basis of the Central Limit Theorem which is important for hypothesis testing.
How does the number of repetitions effect the shape of the normal distribution?
When we discuss a sample drawn from a population, the larger the sample, or the large the number of repetitions of the event, the more certain we are of the mean value. So, when the normal distribution is considered the sampling distribution of the mean, then more repetitions lead to smaller values of the variance of the distribution.
What are the uses of Normal Distribution?
The normal distribution is very important in statistical analysis. A considerable amount of data follows a normal distribution: the weight and length of items mass-produced usually follow a normal distribution ; and if average demand for a product is high, then demand usually follows a normal distribution. It is possible to show that when the sample is large, the sample mean follows a normal distribution. This result is important in the construction of confidence intervals and in significance testing. In quality control procedures for a mean chart, the construction of the warning and action lines is based on the normal distribution.
What is the mean of the sampling distribution of the sample mean?
Frequently it's impossible or impractical to test the entire universe of data to determine probabilities. So we test a small sub-set of the universal database and we call that the sample. Then using that sub-set of data we calculate its distribution, which is called the sample distribution. Normally we find the sample distribution has a bell shape, which we actually call the "normal distribution." When the data reflect the normal distribution of a sample, we call it the Student's t distribution to distinguish it from the normal distribution of a universe of data. The Student's t distribution is useful because with it and the small number of data we test, we can infer the probability distribution of the entire universal data set with some degree of confidence.
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 sampling distribution of the mean?
Thanks to the Central Limit Theorem, the sampling distribution of the mean is Gaussian (normal) whose mean is the population mean and whose standard deviation is the sample standard error.
What does when the sample size and degrees of freedom is sufficiently large the difference between a t distribution and the normal distribution becomes negligible mean?
The t-distribution and the normal distribution are not exactly the same. The t-distribution is approximately normal, but since the sample size is so small, it is not exact. But n increases (sample size), degrees of freedom also increase (remember, df = n - 1) and the distribution of t becomes closer and closer to a normal distribution. Check out this picture for a visual explanation: http://www.uwsp.edu/PSYCH/stat/10/Image87.gif
We have a population with mean of 100 and standard deviation of 28 take repeated samples of size 49 and calculate the mean of each sample to form a sampling distribution Is it a Normal Distribution?
a) T or F The sampling distribution will be normal. Explain your answer. b) Find the mean and standard deviation of the sampling distribution. c) We pick one of our samples from the sampling distribution what is the probability that this sample has a mean that is greater than 109 ? Is this a usual or unusual event? these are the rest of the question.
PLOT A VARIABLE WITH NORMAL DISTRIBUTION WITH MEAN 200 AND DEVIATION 20. SUPERIMPOSED WITH THE PREVIOUS FIGURE, PLOT THE DISTRIBUTION OF THE ARITHMETIC MEAN OF SAMPLES OF SIZE N=4, 25 AND 100, OF THAT POPULATION?
A normal distribution with a mean of 200 and a deviation of 20 can be plotted as a bell-shaped curve, as shown in the figure below. Superimposed on the figure, the distribution of the arithmetic mean of samples of size n=4, 25 and 100 can be plotted as shown in the figure below. The arithmetic mean distribution for n=4 is a much narrower distribution than a normal distribution, since it is based on a small sample size. As the sample size increases, the distribution becomes wider and more similar to the normal distribution.
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.
