qxd 9/9/10 11:48 AM Page 399 C H A P T E Hypothesis Testing R 8 Objectives Outline After completing this chapter, you should be able to Introduction 1 Understand the definitions used in hypothesis testing. 8–1 Steps in Hypothesis Testing—Traditional Method 2 State the null and alternative hypotheses. 3 Find critical values for the z test. 8–2 z Test for a Mean 4 State the five steps used in hypothesis testing.
8–3 t Test for a Mean 5 Test means when s is known, using the z test. 8–4 z Test for a Proportion 6 Test means when s is unknown, using the t test. 7 Test proportions, using the z test. 8–5 X2 Test for a Variance or Standard Deviation 8 Test variances or standard deviations, using 8–6 Additional Topics Regarding Hypothesis the chi-square test.
Testing 9 Test hypotheses, using confidence intervals. Summary 10 Explain the relationship between type I and type II errors and the power of a test.qxd 9/9/10 11:48 AM Page 400 400 Chapter 8 Hypothesis Testing Statistics How Much Better Is Better? Today Suppose a school superintendent reads an article which states that the overall mean score for the SAT is 910. Furthermore, suppose that, for a sample of students, the average of the SAT scores in the superintendent’s school district is 960. Can the superintendent conclude that the students in his school district scored higher on average? At first glance, you might be inclined to say yes, since 960 is higher than 910.
But recall that the means of samples vary about the population mean when samples are selected from a specific population. So the question arises, Is there a real difference in the means, or is the difference simply due to chance (i., sampling error)? In this chapter, you will learn how to answer that ques- tion by using statistics that explain hypothesis testing. See Statistics Today—Revisited for the answer. In this chapter, you will learn how to answer many questions of this type by using statistics that are explained in the theory of hypothesis testing.
Introduction Researchers are interested in answering many types of questions. For example, a scien- tist might want to know whether the earth is warming up. A physician might want to know whether a new medication will lower a person’s blood pressure. An educator might wish to see whether a new teaching technique is better than a traditional one.
A retail merchant might want to know whether the public prefers a certain color in a new line of fashion. Automobile manufacturers are interested in determining whether seat belts will reduce the severity of injuries caused by accidents. These types of questions can be addressed through statistical hypothesis testing, which is a decision-making process for evaluating claims about a population. In hypothesis testing, the researcher must define the population under study, state the particular hypotheses that will be investigated, give the significance level, select a sample from the population, collect the data, perform the calculations required for the statistical test, and reach a conclusion.
Hypotheses concerning parameters such as means and proportions can be investigated. There are two specific statistical tests used for hypotheses concerning means: the z test 8–2 blu38582_ch08_399-470.qxd 9/9/10 11:48 AM Page 401 Section 8–1 Steps in Hypothesis Testing—Traditional Method 401 and the t test. This chapter will explain in detail the hypothesis-testing procedure along with the z test and the t test. In addition, a hypothesis-testing procedure for testing a single vari- ance or standard deviation using the chi-square distribution is explained in Section 8–5.
The three methods used to test hypotheses are 1. The traditional method 2. The P-value method 3. The confidence interval method The traditional method will be explained first.
It has been used since the hypothesis- testing method was formulated. A newer method, called the P-value method, has become popular with the advent of modern computers and high-powered statistical calculators. It will be explained at the end of Section 8–2. The third method, the confidence interval method, is explained in Section 8–6 and illustrates the relationship between hypothesis testing and confidence intervals.
8–1 Steps in Hypothesis Testing—Traditional Method Every hypothesis-testing situation begins with the statement of a hypothesis. A statistical hypothesis is a conjecture about a population parameter. This conjecture may or may not be true. Objective 1 There are two types of statistical hypotheses for each situation: the null hypothesis Understand the and the alternative hypothesis.
definitions used in hypothesis testing. The null hypothesis, symbolized by H0, is a statistical hypothesis that states that there is no difference between a parameter and a specific value, or that there is no difference between two parameters. The alternative hypothesis, symbolized by H1, is a statistical hypothesis that states the existence of a difference between a parameter and a specific value, or states that there is a difference between two parameters. (Note: Although the definitions of null and alternative hypotheses given here use the word parameter, these definitions can be extended to include other terms such as distri- butions and randomness.
This is explained in later chapters.) As an illustration of how hypotheses should be stated, three different statistical stud- ies will be used as examples. Situation A A medical researcher is interested in finding out whether a new medica- tion will have any undesirable side effects. The researcher is particularly concerned with the pulse rate of the patients who take the medication. Will the pulse rate increase, Objective 2 decrease, or remain unchanged after a patient takes the medication? Since the researcher knows that the mean pulse rate for the population under study State the null and is 82 beats per minute, the hypotheses for this situation are alternative hypotheses.
H0: m 82 and H1: m 82 The null hypothesis specifies that the mean will remain unchanged, and the alternative hypothesis states that it will be different. This test is called a two-tailed test (a term that will be formally defined later in this section), since the possible side effects of the med- icine could be to raise or lower the pulse rate.qxd 9/9/10 11:48 AM Page 402 402 Chapter 8 Hypothesis Testing Situation B A chemist invents an additive to increase the life of an automobile bat- tery. If the mean lifetime of the automobile battery without the additive is 36 months, then her hypotheses are H0: m 36 and H1: m 36 In this situation, the chemist is interested only in increasing the lifetime of the batteries, so her alternative hypothesis is that the mean is greater than 36 months. The null hypoth- esis is that the mean is equal to 36 months.
This test is called right-tailed, since the inter- est is in an increase only. Situation C A contractor wishes to lower heating bills by using a special type of Unusual Stat insulation in houses. If the average of the monthly heating bills is $78, her hypotheses Sixty-three percent of about heating costs with the use of insulation are people would rather H0: m $78 and H1: m $78 hear bad news before hearing the good This test is a left-tailed test, since the contractor is interested only in lowering heating costs. To state hypotheses correctly, researchers must translate the conjecture or claim from words into mathematical symbols.
The basic symbols used are as follows: Equal to Greater than Not equal to Less than The null and alternative hypotheses are stated together, and the null hypothesis con- tains the equals sign, as shown (where k represents a specified number). Two-tailed test Right-tailed test Left-tailed test H0: m k H0: m k H0: m k H1: m k H1: m k H1: m k The formal definitions of the different types of tests are given later in this section. In this book, the null hypothesis is always stated using the equals sign. This is done because in most professional journals, and when we test the null hypothesis, the assump- tion is that the mean, proportion, or standard deviation is equal to a given specific value.
Also, when a researcher conducts a study, he or she is generally looking for evidence to support a claim. Therefore, the claim should be stated as the alternative hypothesis, i., using or or . Because of this, the alternative hypothesis is sometimes called the research hypothesis. Table 8–1 Hypothesis-Testing Common Phrases Is greater than Is less than Is above Is below Is higher than Is lower than Is longer than Is shorter than Is bigger than Is smaller than Is increased Is decreased or reduced from Is equal to Is not equal to Is the same as Is different from Has not changed from Has changed from Is the same as Is not the same as 8–4 blu38582_ch08_399-470.qxd 9/9/10 11:48 AM Page 403 Section 8–1 Steps in Hypothesis Testing—Traditional Method 403 A claim, though, can be stated as either the null hypothesis or the alternative hypothesis; however, the statistical evidence can only support the claim if it is the alternative hypothe- sis.
Statistical evidence can be used to reject the claim if the claim is the null hypothesis. These facts are important when you are stating the conclusion of a statistical study. Table 8–1 shows some common phrases that are used in hypotheses and conjectures, and the corresponding symbols. This table should be helpful in translating verbal con- jectures into mathematical symbols.
Example 8–1 State the null and alternative hypotheses for each conjecture. A researcher thinks that if expectant mothers use vitamin pills, the birth weight of the babies will increase. The average birth weight of the population is 8. An engineer hypothesizes that the mean number of defects can be decreased in a manufacturing process of compact disks by using robots instead of humans for certain tasks.
The mean number of defective disks per 1000 is 18. A psychologist feels that playing soft music during a test will change the results of the test. The psychologist is not sure whether the grades will be higher or lower. In the past, the mean of the scores was 73.
H0: m 73 and H1: m 73 After stating the hypothesis, the researcher designs the study. The researcher selects the correct statistical test, chooses an appropriate level of significance, and formulates a plan for conducting the study. In situation A, for instance, the researcher will select a sample of patients who will be given the drug. After allowing a suitable time for the drug to be absorbed, the researcher will measure each person’s pulse rate.
Recall that when samples of a specific size are selected from a population, the means of these samples will vary about the population mean, and the distribution of the sample means will be approximately normal when the sample size is 30 or more.) So even if the null hypothesis is true, the mean of the pulse rates of the sample of patients will not, in most cases, be exactly equal to the population mean of 82 beats per minute. There are two possibilities. Either the null hypothesis is true, and the difference between the sample mean and the population mean is due to chance; or the null hypothesis is false, and the sample came from a population whose mean is not 82 beats per minute but is some other value that is not known. These situations are shown in Figure 8–1.
The farther away the sample mean is from the population mean, the more evidence there would be for rejecting the null hypothesis. The probability that the sample came from a population whose mean is 82 decreases as the distance or absolute value of the difference between the means increases. If the mean pulse rate of the sample were, say, 83, the researcher would probably conclude that this difference was due to chance and would not reject the null hypothesis.