# Sampling distributions and test statistics

## Abstract

The fact that the sampling distribution of the mean approximates a normal distribution, which can be described exactly by a mathematical function, enables us to test certain hypotheses using statistical inference. In statistical inference, we make an assumption about a population parameter and then determine whether or not our sample statistics are consistent with that assumption, allowing for the effects of sampling error. The assumption that we wish to test is usually referred to as the null hypothesis. For example, we might want to determine whether or not the average family income of a particular minority group is equal to the average family income of the population as a whole. For the purposes of this example, we will assume that we know both the mean and the standard deviation of the distribution of family income in the population. Specifically, we shall assume that the population mean is \$24,000 and that the population standard deviation is \$2,000. In order to test this hypothesis, we will also assume that we have randomly sampled 1,600 minority group families and have found that their average family income is \$23,900. The question that we want to answer is whether or not the difference between the sample mean and the population mean can be attributed to sampling error or represents a real difference between the minority group and the population.

## Keywords

Null Hypothesis Sample Statistic Family Income Minority Group Sampling Error
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