2379-chapter11 - MAT 2379 Introduction to Biostatistics...

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MAT 2379, Introduction to Biostatistics, Lecture Notes for Chapter 11 1 MAT 2379, Introduction to Biostatistics Chapter 11. Hypothesis Testing Hypothesis testing is a statistical method that is used when one wants to gain support (or evidence) towards a desired statement, called the research hypothesis (denoted with H 1 ). The other hypothesis, which the researcher would like to reject is called the null hypothesis and is denoted with H 0 . When you are using this method, formulate your statistical hypotheses in hopes of being able to reject H 0 and hence gain evidence towards H 1 . Hypotheses testing can result in two types of errors: 1. Type I error We have rejected hypothesis H 0 when H 0 was true. 2. Type II error We have failed to reject hypothesis H 0 when H 1 was true. In class we will draw the table which contains all the 4 possible situations. Statisticians working in the pharmaceutical industry encounter this all the time when they are trying to promote a new drug and to convince the population that this drug is efficient. Example 1. (adapted from p. 190 of the the book ”Biostatistics. How it works” by Selvin) The systolic blood pressure level in a certain hypertensive population is approximately equal to the value 130 mm Hg. A new drug is developed to reduce the systolic blood pressure levels in this population under the value 130. Set up the two hypotheses. Explain when the two errors occur. Let μ denote the mean level of systolic blood pressure for the patients who were administered the drug. The null hypothesis (that we would like to reject) is that the drug didn’t change anything: H 0 : μ = 130 The research hypothesis (that we would like to gain support for) is that μ is below 130: H 1 : μ < 130 A type I error occurs when we decide that the new drug has reduced the systolic blood pressure level in that population when, in fact, it has not. A type II error occurs when we are unable to gain enough evidence that the new drug has reduced the systolic blood pressure level in that population when, in fact, it has. To perform a test of hypotheses for μ , we need to consider 2 cases: the variance σ 2 is known (Section 11.1). We skip this case (it is never encountered in practice). the variance σ 2 is unknown (Section 11.2). We also discuss hypothesis testing for the proportion p (Section 11.3).
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MAT 2379, Introduction to Biostatistics, Lecture Notes for Chapter 11 2 11.2 Hypothesis Testing for the Mean: σ 2 unknown In this section we will study the problem of testing a hypothesis on the mean μ of a population.
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