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Suppose you have a theory about some random variable and you want to check if your theory is correct. You take some observations and summarise them using a test statistic. This may be their mean, maximum, a measure of spread - whatever. You want to check if the result that you got is consistent with your theory.

You face a problem, though. Because the variable has a random element to it, it is always possible that the result that you got was pure chance. So you work out what the probability distribution of that test statistics would be IF your theory (hypothesis) were true. You use this distribution and the value of your test statistic to decide how likely your result was if your hypothesis were true.

If that probability is very small, you conclude that your theory is not reasonable and you reject your hypothesis. Otherwise, you continue with the assumption that your theory is not unreasonable.

Q: What is test statistic Why do you have to know the distribution of a test statistic?

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Because under the null hypothesis of no difference, the appropriate test statistic can be shown to have a t-distribution with the relevant degrees of freedom. So you use the t-test to see how well the observed test statistic fits in with a t-distribution.

The sampling distribution for a statistic is the distribution of the statistic across all possible samples of that specific size which can be drawn from the population.

With either test, you have a number of categories and for each you have an expected number of observations. The expected number is based either on the variable being independent of some other variable, or determined by some know (or hypothesised) distribution. You will also have a number of observations of the variable for each category. The test statistic is based on the observed and expected frequencies and has a chi-squared distribution. The tests require the observations to come from independent, identically distributed variables.

It is the sampling distribution of that variable.

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.

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A test statistic is used to test whether a hypothesis that you have about the underlying distribution of your data is correct or not. The test statistic could be the mean, the variance, the maximum or anything else derived from the observed data. When you know the distribution of the test statistic (under the hypothesis that you want to test) you can find out how probable it was that your test statistic had the value it did have. If this probability is very small, then you reject the hypothesis. The test statistic should be chosen so that under one hypothesis it has one outcome and under the is a summary measure based on the data. It could be the mean, the maximum, the variance or any other statistic. You use a test statistic when you are testing between two hypothesis and the test statistic is one You might think of the test statistic as a single number that summarizes the sample data. Some common test statistics are z-score and t-scores.

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.

It is a value of a test statistic based on the Student's t distribution.

Because under the null hypothesis of no difference, the appropriate test statistic can be shown to have a t-distribution with the relevant degrees of freedom. So you use the t-test to see how well the observed test statistic fits in with a t-distribution.

A chi-squared test is any statistical hypothesis test in which the sampling distribution of the test statistic is a chi-squared distribution when the null hypothesis is true.

with mean and standard deviation . Once standardized, , the test statistic follows Standard Normal Probability Distribution.

Any decision based on the test statistic is marginal in such a case. It is important to remember that the test statistic is derived on the basis of the null hypothesis and does not make use of the distribution under the alternative hypothesis.

The s2 statistic is used to test to test whether the assumption of normality is reasonable for a given population distribution. The sample consists of 5000 observations and is divided into 6 categories (intervals). The degrees of freedom for the chi-square statistic is:

You can calculate a result that is somehow related to the mean, based on the data available. Provided that you can work out its distribution under the null hypothesis against appropriate alternatives, you have a test statistic.

The statement is true that a sampling distribution is a probability distribution for a statistic.

A test statistic is a value calculated from a set of observations. A critical value depends on a null hypothesis about the distribution of the variable and the degree of certainty required from the test. Given a null hypothesis it may be possible to calculate the distribution of the test statistic. Then, given an alternative hypothesis, it is may be possible to calculate the probability of the test statistic taking the observed (or more extreme) value under the null hypothesis and the alternative. Finally, you need the degree of certainty required from the test and this will determine the value such that if the test statistic is more extreme than the critical value, it is unlikely that the observations are consistent with the hypothesis so it must be rejected in favour of the alternative hypothesis. It may not always be possible to calculate the distribution function for the variable.

The sampling distribution for a statistic is the distribution of the statistic across all possible samples of that specific size which can be drawn from the population.