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Chapter 8: Hypothesis Testing “The average home value in the U.S. is less than $150,000.” The claim above is about ta population parameter. To determine if this statement is true, we would have to find the value of every home in the U.S. and calculate the mean. This task will be nearly impossible to accomplish, so we may never know for certain if this statement is true. We can estimate the above parameter by collecting sample data and calculating a sample mean. If the sample mean is less that $150,000, then we are tempted to say that the statement is true. However, we must remember that sample statistics are estimates of population parameters. Just because the sample mean is less that $150,000, we cannot conclude for certain that the population mean is less than $150,000. In this case, the data appears to support the claim, but we need to determine how strong the support is. This is the motivation behind the idea of hypothesis testing. Section 8.1: Steps in Hypothesis Testing – Traditional Method Definition: 1. Hypothesis testing is a formal approach for determining if data from a sample support a claim about a population. 2. A statistical hypothesis is a conjecture about a population parameter. This conjecture may or may not be true. Note, there are two types of statistical hypotheses for each situation: The null hypothesis and the alternative hypothesis. 3. The null hypothesis (symbolized by H0 ) is a statement about the value of the parameter we would expect to be true if there were no data indicating otherwise. 4. The alternative hypothesis (symbolized by H1 ) is a statement about the value of the parameter suggested by the data.

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Discuss the following three different statistical studies:

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Exercise 1. State the null and alternative hypotheses for each conjecture. a) A researcher things that if expectant mothers use vitamins pills, the birth weight of the babies will increase. The average birth weight of the population is 8.6 pounds. b) An engineer hypothesizes that the mean number of defects can be decreased in a manufacturing process of USB drives by using robots instead of humans for certain tasks. The mean number of defective drives per 1000 is 18. c) A psychologist feels that playing soft music during a test will change the results of the test. The psychologist is not sure where the grades will be higher or lower. In the past, the mean of the scores was 73.

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Note, a hypothesis test compares the merit of the two competing hypotheses by examining the data that are collected. The null hypothesis describes the currently accepted value for the population parameter, which is why it is the hypothesis that contains equality. Throughout a hypothesis test, the null hypothesis is assumed to be true. If the data disagree “sufficiently” with the null hypothesis, then the null is rejected in favor of the alternative hypothesis. After stating the hypothesis, the researcher designs a 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 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 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. 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. But fi the sample mean were say, 90, then in all likelihood the researcher would conclude that the medication increased the pulse rate of the users and would reject the null hypothesis. The question is, where does the researcher draw the line? This decision is not made on feelings or intuition; it is made statistically, that is, the difference must be significant and in all likelihood not due to chance. 5. A statistical test uses the data obtained from a sample to make a decision about whether the null hypothesis should be rejected.

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6. A test statistic is the value used to make a decision about the null hypothesis and is derived from the sample statistic. 7. A sample statistic is statistically significant if it is far enough away from the presumed value of the population parameter to conclude that it would be unlikely for that sample statistic to occur by chance if the null hypothesis is true.

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Note, the hypothesis-testing situation can be likened to a jury trial. In a jury trial, there are four possible outcomes. The defendant is either guilty or innocent, and he or she will be convicted or acquitted.

8. The level of significance, denoted by α , is the probability of making the error of rejecting a true null hypothesis in a hypothesis test. 9. The level of significance is the maximum probability of committing a type I error. This probability is symbolized by α . That is, P(type I error) =α . 10. The probability of a type II error is symbolized by β (beta). That is, P(type II error) = β . Note, α and β are related in that decreasing one increases the other. 11. The critical or rejection region is the range of test values that indicates that there is a significant difference an that the null hypotheses should be rejected. 12. The noncritical or nonrejection region is the range of test values that indicates that the difference was probably due to chance and that the null hypothesis should not be rejected. 13. The critical value separates the critical region from the non critical region. The symbol for critical value is C.V. 14. A one-tailed test indicates that the null hypothesis should be rejected when the test value is in the critical region on one side of the mean. A one-tailed test is either a right-tailed test or a left-tailed test, depending on the direction of the inequality of the alternative hypothesis. In a two-tailed test the null hypothesis should be rejected when the test value is in either of the two critical regions.

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Exercise 2. Using the z-table, find the critical value(s) for each situation and draw the appropriate figure, showing the critical region. a) A left-tailed test with α = 0.10 . b) A two-tailed test with α = 0.02 . c) A right-tailed test with α = 0.005 .

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Note, through the hypothesis test, the null hypothesis is assumed to be true. The null hypothesis always include equality. The data must provide overwhelming evidence to contradict the null hypothesis for the alternative hypothesis to be supported. Therefore, rejecting a null hypothesis is a stronger conclusion than failing to reject it. For the purpose of this chapter, a simplified version of the hypothesis-testing procedure will be used, since designing the study and collecting the data will be omitted. The steps are summarized in the Procedure Table. Hypothesis testing consists of five basic steps: 1. State the hypotheses and identify the claim. 2. Find the critical value(s). 3. Compute the test value. 4. Make the decision to reject or not reject the null hypothesis. 5. Summarize the result. Exercise 3. The incredible Edible Egg Company recently found that eating eggs does not increase a person’s blood serum cholesterol. Five hundred subjects participated in a study that lasted for 2 years. The participants were randomly assigned to either a no-egg group or a moderate-egg group. The blood serum cholesterol levels were checked at the beginning and at the end of the study. Overall, the groups’ levels were not significantly different. The company reminds us that eating eggs is healthy if done in moderation. Many of the pervious studies relating eggs and high blood serum cholesterol jumped to improper conclusions. Using this information, answer these questions. a) What prompted the study? b) What is the population under study?

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c) Was a sample collected? d) What was the hypothesis? e) Were data collected? f) Were any statistical test run? g) What was the conclusion?

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Z-table

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Z-table