# Hypothesis Testing with One Sample

In statistics, hypothesis testing is a procedure for determining whether a statistical hypothesis is supported by the data. The hypothesis is typically a mathematical model that predicts the outcome of an experiment, and the sample is a subset of the data used to test the hypothesis. If the sample data does not provide sufficient information to determine the truth of the hypothesis, the null hypothesis is rejected in favor of an alternative hypothesis. In the formal hypothesis testing procedure, the null hypothesis is not rejected unless the test is significant. A test is considered significant if the p-value is less than the alpha level (?). The p-value is the probability of observing data at least as extreme as the data in the sample, assuming that the null hypothesis is true. The alpha level is the threshold for determining whether or not the test is significant. The significance level is the probability of mistakenly rejecting the null hypothesis when it is true. The use of hypothesis testing has widespread applications in statistics. For example, hypothesis testing can be used for hypothesis testing in two-sample tests of hypothesis, such as t-tests which are often used in sample surveys. In hypothesis testing, a null hypothesis is a proposition that the hypothesized effect is not present. It is usually set up in the form of an alternative hypothesis to the effect, and is used to decide whether or not the effect is present. The null hypothesis is usually considered to be the starting point for a scientific investigation, since it is the hypothesis that the effect is not present. The most common null hypothesis test of a hypothesis is: The null hypothesis is usually set up in the form of an alternative hypothesis. The alternative hypothesis is usually set up in the form of a testable implication of the effect, and is used to decide whether or not the effect is present. The null hypothesis is a statement of the effect that is being hypothesized. The alternative hypothesis is a statement of the effect that is not being hypothesized. The null hypothesis is a statement of "no effect", while the alternative hypothesis is a statement of "an effect". The null hypothesis is often set up in the form of a statement of the effect, but it may also be set up as a statement of the effect's absence. For example, the null hypothesis in a one-sample test of the sign of the population standard deviation of a normally distributed population might be set up as the statement, "The population standard deviation is zero". Similarly, the null hypothesis in a one-sample test of the sign of the population of a normally distributed population might be set up as the statement, "The population is zero". The alternative hypothesis is usually set up in the form of a testable implication of the effect, but it may also be set up as a statement of the effect's absence. For example, the alternative hypothesis in a one-sample test of the sign of the population standard deviation of a normally distributed population might be set up as the statement, "The population standard deviation is not zero". Similarly, the alternative hypothesis in a one-sample test of the sign of the population of a normally distributed population might be set up as the statement, "The population is not zero". The p-value of a hypothesis test is the probability of observing data as extreme as the data in the sample, assuming that the null hypothesis is true. This probability is called the "p-value" of the test. Some authors use the term significance level instead of p-value, with the intent to differentiate it from the term "significance level" used in hypothesis tests for testing whether the null hypothesis is true. However, this usage is ambiguous, as "significance level" may refer to either the probability of observing data as extreme as the data in the sample, or the probability of observing data more extreme than the data in the sample, or both. In fact, "significance level" is not a term used in hypothesis testing (it is called "level of significance" in the abovementioned hypothesis test for testing the sign of the population standard deviation of a normally distributed population), but it is often thought of as the probability of observing data as extreme as the data in the sample, or the probability of observing data more extreme than the data in the sample, or both. The test statistic is the sum of the test statistic plus the (inverse) square root of the sample variance of the test statistic. The test statistic is a function of the test statistic plus the sample variance of the test statistic, so it cannot be used without taking the square root of the sample variance of the test statistic first. This is one reason why the test statistic and the sample variance of the test statistic are usually calculated and presented together. The test statistic is calculated as follows. The formula for the sample variance of the test statistic is and the formula for the test statistic is The