SECTION 7.1 Introduction to Hypothesis Testing 353 You will reject the null hypothesis when the sample statistic from the sampling distribution is unusual. You have already identified unusual events to be those that occur with a probability of 0.05 or less. When statistical tests are used, an unusual event is sometimes required to have a probability of 0.10 or less, 0.05 or less, or 0.01 or less. Because there is variation from sample to sample, there is always a possibility that you will reject a null hypothesis when it is actually true. In other words, although the null hypothesis is true, your sample statistic is determined to be an unusual event in the sampling distribution. You can decrease the probability of this happening by lowering the level of significance. In a hypothesis test, the level of significance is your maximum allowable probability of making a type I error. It is denoted by a, the lowercase Greek letter alpha. The probability of a type II error is denoted by b, the lowercase Greek letter beta. DEFINITION By setting the level of significance at a small value, you are saying that you want the probability of rejecting a true null hypothesis to be small. Three commonly used levels of significance are a = 0.10, a = 0.05, and a = 0.01. Statistical Tests and P-Values After stating the null and alternative hypotheses and specifying the level of significance, the next step in a hypothesis test is to obtain a random sample from the population and calculate the sample statistic (such as x, np, or s2) corresponding to the parameter in the null hypothesis (such as m, p, or s 2). This sample statistic is called the test statistic. With the assumption that the null hypothesis is true, the test statistic is then converted to a standardized test statistic, such as z, t, or x 2. The standardized test statistic is used in making the decision about the null hypothesis. In this chapter, you will learn about several one-sample statistical tests. The table shows the relationships between population parameters and their corresponding test statistics and standardized test statistics. Population parameter Test statistic Standardized test statistic m x z (Section 7.2, s known), t (Section 7.3, s unknown) p np z (Section 7.4) s 2 s2 x 2 (Section 7.5) One way to decide whether to reject the null hypothesis is to determine whether the probability of obtaining the standardized test statistic (or one that is more extreme) is less than the level of significance. If the null hypothesis is true, then a P@value (or probability value) of a hypothesis test is the probability of obtaining a sample statistic with a value as extreme or more extreme than the one determined from the sample data. DEFINITION Study Tip When you decrease a (the maximum allowable probability of making a type I error), you are likely to be increasing b. The value 1 - b is called the power of the test. It represents the probability of rejecting the null hypothesis when it is false. The value of the power is difficult (and sometimes impossible) to find in most cases.
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