The answer depends on what character is used for the variable that is used for the population values.
N is neither the sample or population mean. The letter N represents the population size while the small case letter n represents sample size. The symbol of sample mean is x̄ ,while the symbol for population mean is µ.
The population data may be skewed and thus the mean is not a valid statistic. If mean > median, the data will be skewed to the right. If median > mean, the data is skewed to the left.
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.
What symbol
a "T" or a "Z" score. A "T" Score if comparing a sample. A "Z" Score when comparing a population. Remember, a population includes all observation, and a sample includes only a random selection of the population.
When the null hypothesis is true, the expected value for the t statistic is 0. This is because the t statistic is calculated as the difference between the sample mean and the hypothesized population mean, divided by the standard error, and when the null hypothesis is true, these values should be equal, resulting in a t statistic of 0.
Hypothesis
The null hypothesis is that there is no change in the population mean while the alternative hypothesis is that there is a change in the mean. The null hypothesis is stated as Ho:Mu=? in statistics while the alternative hypothesis is stated as Ho:Mu(<,>,≠)? depending on whether you are looking for mu to be greater, less than, or not equal to population mean.
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.
Your question is a bit difficult to understand. I will rephrase: In hypothesis testing, when the sample mean is close to the assumed mean of the population (null hypotheses), what does that tell you? Answer: For a given sample size n and an alpha value, the closer the calculated mean is to the assumed mean of the population, the higher chance that null hypothesis will not be rejected in favor of the alternative hypothesis.
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.
Yes.
It depends on whether the hypothesis concerns the mean or the standard error (or variance) or something else.
N is neither the sample or population mean. The letter N represents the population size while the small case letter n represents sample size. The symbol of sample mean is x̄ ,while the symbol for population mean is µ.
The relations depend on what measures. The sample mean is an unbiased estimate for the population mean, with maximum likelihood. The sample maximum is a lower bound for the population maximum.
The sample standard error.
F is the test statistic and H0 is the means are equal. A small test statistic such as 1 would mean you would fail to reject the null hypothesis that the means are equal.