9.1 Fundamentals of Hypothesis Testing

9.1 Fundamentals of Hypothesis Testing
LEARNING GOAL Understand the goal of hypothesis testing and the basic structure of a hypothesis test, including how to set up the null and alternative hypotheses, how to determine the possible outcomes of a hypothesis test, and how to decide between these possible outcomes. Page 370 Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc.
A company called ProCare Industries, Ltd. once claimed that its product, called Gender Choice, could increase a woman’s chance of giving birth to a baby girl. How could we test whether the Gender Choice claim is true? One way would be to study a random sample of, say, 100 babies born to women who used the Gender Choice product. If the product does not work, we would expect about half of these babies to be girls. If it does work, we would expect significantly more than half of the babies to be girls. The key question, then, is what constitutes “significantly more.” Page 370 Copyright © 2009 Pearson Education, Inc. Slide

In statistics, we answer such questions through hypothesis testing. Consider a sample in which there are 64 girls among 100 babies born to women who used the product. • We begin by assuming that Gender Choice does not work; that is, it does not increase the percentage of girls. If this is true, then we should expect about 50% girls among the population of all births to women using the product. That is, the proportion of girls is equal to 0.50. • We now use our sample (with 64 girls among 100 births) to test the above assumption. We conduct this test by calculating the likelihood of drawing a random sample of 100 births in which 64% (or more) of the babies are girls (from a population in which the overall proportion of girls is only 50%). Page 370 Copyright © 2009 Pearson Education, Inc. Slide

• If we find that a random sample of births is fairly likely to have 64% (or more) girls, then we do not have evidence that Gender Choice works. However, if we find that a random sample of births is unlikely to have at least 64% girls, then we conclude that the result in the Gender Choice sample is probably due to something other than chance—meaning that the product may be effective. Note, however, that even in this case we will not have proved that the product is effective, as there could still be other explanations for the result (such as that we happened to choose an unusual sample or that there were confounding variables we did not account for). Page 370 Copyright © 2009 Pearson Education, Inc. Slide

TIME OUT TO THINK Suppose Gender Choice is effective and you select a new random sample of 1,000 babies born to women who used the product. Based on the first sample (with 64 girls among 100 births), how many baby girls would you expect in the new sample? Next, suppose Gender Choice is not effective; how many baby girls would you expect in a random sample of 1,000 babies born to women who used the product? Page 370 Copyright © 2009 Pearson Education, Inc. Slide

Formulating the Hypotheses Definitions A hypothesis is a claim about a population parameter, such as a population proportion (p) or population mean (m). A hypothesis test is a standard procedure for testing a claim about a population parameter. Pages Copyright © 2009 Pearson Education, Inc. Slide

There are always at least two hypotheses in any hypothesis test. In the Gender Choice case, the two hypotheses are essentially: that Gender Choice works in raising the population proportion of girls from the 50% that we would normally expect and that Gender Choice does not work in raising the population proportion of girls. Page 371 Copyright © 2009 Pearson Education, Inc. Slide

H0 (null hypothesis): p = 0.50
• The null hypothesis is the claim that Gender Choice does not work, in which case the population proportion of girls born to women who use the product should be 50%, or 0.50. Using p to represent the population proportion, we write this null hypothesis as H0 (null hypothesis): p = 0.50 • The alternative hypothesis is the claim that Gender Choice does work, in which case the population proportion of girls born to women who use the product should be greater than 0.50. We write this alternative hypothesis as Page 371 Ha (alternative hypothesis): p > 0.50 Copyright © 2009 Pearson Education, Inc. Slide

population parameter < claimed value
In this chapter, the null hypothesis will always include the condition of equality, just as the null hypothesis for the Gender Choice example is the equality p = 0.50. However, the alternative hypothesis for the Gender Choice case (p > 0.50) is only one of three general forms of alternative hypotheses that we will encounter in this chapter: population parameter < claimed value population parameter > claimed value population parameter ≠ claimed value Page 371 Copyright © 2009 Pearson Education, Inc. Slide

It is useful to give names to the three different types of alternative hypotheses. The first form (“less than”) leads to what is called a left-tailed hypothesis test, because it requires testing whether the population parameter lies to the left (lower values) of the claimed value. Similarly, the second form (“greater than”) leads to a right-tailed hypothesis test, because it requires testing whether the population parameter lies to the right (higher values) of the claimed value. The third form (“not equal to”) leads to a two-tailed hypothesis test, because it requires testing whether the population parameter lies significantly far to either side of the claimed value. Page 371 Copyright © 2009 Pearson Education, Inc. Slide

Null and Alternative Hypotheses The null hypothesis, or H0, is the starting assumption for a hypothesis test. For the types of hypothesis tests in this chapter, the null hypothesis always claims a specific value for a population parameter and therefore takes the form of an equality: H0 (null hypothesis): population parameter = claimed value The alternative hypothesis, or Ha, is a claim that the population parameter has a value that differs from the value claimed in the null hypothesis. It may take one of the following forms: (left-tailed) Ha: population parameter < claimed value (right-tailed) Ha: population parameter > claimed value (two-tailed) Ha: population parameter ≠ claimed value Page 372 Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 1 Identifying Hypotheses In each case, identify the population parameter about which a claim is made, state the null and alternative hypotheses for a hypothesis test, and indicate whether the hypothesis test will be left-tailed, right-tailed, or two-tailed. a. The manufacturer of a new model of hybrid car advertises that the mean fuel consumption is equal to 62 miles per gallon on the highway. A consumer group claims that the mean is less than 62 miles per gallon. Solution: a. The population parameter about which a claim is made is a population mean (m)—the mean gas mileage of all new hybrid cars of this model. The null hypothesis must state that this population mean is equal to some specific value. Page 372 Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 1 Identifying Hypotheses Solution: (cont.) We therefore recognize the null hypothesis as the advertised claim that the mean mileage for the cars is 62 miles per gallon. The alternative hypothesis is the consumer group’s claim that the true gas mileage is less than advertised. To summarize: H0 (null hypothesis): m = 62 miles per gallon Ha (alternative hypothesis): m < 62 miles per gallon Because the alternative hypothesis has a “less than” form, the hypothesis test will be left-tailed. Page 372 Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 1 Identifying Hypotheses In each case, identify the population parameter about which a claim is made, state the null and alternative hypotheses for a hypothesis test, and indicate whether the hypothesis test will be left-tailed, right-tailed, or two-tailed. The Ohio Department of Health claims that the average stay in Ohio hospitals after childbirth is greater than the national mean of 2.0 days. Solution: The population is all women in Ohio who have recently given birth, and the population parameter is their mean (m) stay after childbirth in Ohio hospitals. The null hypothesis must be an equality, so in this case it is the claim that the mean hospital stay equals the national average of 2.0 days. Page 372 Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 1 Identifying Hypotheses Solution: (cont.) The alternative hypothesis is the Health Department’s claim that the mean stay in Ohio is greater than this national average. To summarize: H0 (null hypothesis): m = 2.0 days Ha (alternative hypothesis): m > 2.0 days Because the alternative hypothesis has a “greater than” form, the hypothesis test will be right-tailed. Page 372 Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 1 Identifying Hypotheses In each case, identify the population parameter about which a claim is made, state the null and alternative hypotheses for a hypothesis test, and indicate whether the hypothesis test will be left-tailed, right-tailed, or two-tailed. A wildlife biologist working in the African savanna claims that the actual proportion of female zebras in the region is different from the accepted proportion of 50%. Solution: In this case, the claim is about a population proportion (p)— the proportion of female zebras among the zebra population of the region. The accepted population proportion is p = 0.5, which becomes the null hypothesis. Pages Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 1 Identifying Hypotheses Solution: (cont.) The wildlife biologist claims that the actual population proportion is different from the value in the null hypothesis. Because “different” can be either greater or less than, the alternative hypothesis has a “not equal to” form: H0 (null hypothesis): p = 0.5 Ha (alternative hypothesis): p ≠ 0.5 The “not equal to” form of this alternative hypothesis leads to a two-tailed test. Pages Copyright © 2009 Pearson Education, Inc. Slide

Possible Outcomes of a Hypothesis Test Two Possible Outcomes of a Hypothesis Test There are two possible outcomes to a hypothesis test: Reject the null hypothesis, H0, in which case we have evidence in support of the alternative hypothesis. Not reject the null hypothesis, H0, in which case we do not have enough evidence to support the alternative hypothesis. Page 373 Copyright © 2009 Pearson Education, Inc. Slide

Notice that “accepting the null hypothesis” is not a possible outcome, because the null hypothesis is always the starting assumption. The hypothesis test may not give us reason to reject this starting assumption, but it cannot by itself give us reason to conclude that the starting assumption is true. The fact that there are only two possible outcomes makes it extremely important that the null and alternative hypotheses be chosen in an unbiased way. In particular, both hypotheses must always be formulated before a sample is drawn from the population for testing. Otherwise, data from the sample might inappropriately bias the selection of the hypotheses to test. Page 373 Copyright © 2009 Pearson Education, Inc. Slide

TIME OUT TO THINK The idea that a hypothesis test cannot lead us to accept the null hypothesis is an example of the old dictum that “absence of evidence is not evidence of absence.” As an illustration of this idea, explain why it would be easy in principle to prove that some legendary animal (such as Bigfoot or the Loch Ness Monster) exists, but it is nearly impossible to prove that it does not exist. Page 373 Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 2 Hypothesis Test Outcomes For each of the three cases from Example 1 (slides 12-17), describe the possible outcomes of a hypothesis test and how we would interpret these outcomes. Solution: Recall that the null hypothesis is the advertised claim that the mean mileage for the new cars is m = 62 miles per gallon. The alternative hypothesis is the consumer group’s claim that the true gas mileage is less than advertised, or m < 62 miles per gallon. The possible outcomes are • Reject the null hypothesis of m = 62 miles per gallon, in which case we have evidence in support of the consumer group’s claim that the mileage is less than advertised. Pages Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 2 Hypothesis Test Outcomes Solution: (cont.) • Do not reject the null hypothesis, in which case we lack evidence to support the consumer group’s claim. Note, however, that this option does not imply that the advertised claim is true. Pages Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 2 Hypothesis Test Outcomes For each of the three cases from Example 1 (slides 12-17), describe the possible outcomes of a hypothesis test and how we would interpret these outcomes. Solution: (cont.) The null hypothesis is that the mean hospital stay is the national average of 2.0 days. The alternative hypothesis is the Health Department’s claim that the mean stay in Ohio is greater than the national average. The possible outcomes are • Reject the null hypothesis of m = 2.0 days, in which case we have evidence in support of the Health Department’s claim that the mean stay in Ohio is greater than the national average. Pages Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 2 Hypothesis Test Outcomes Solution: (cont.) • Do not reject the null hypothesis, in which case we lack evidence to support the Health Department’s claim. Note, however, that this option does not imply that the Ohio average stay is actually equal to the national average stay of 2.0 days. Pages Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 2 Hypothesis Test Outcomes For each of the three cases from Example 1 (slides 12-17), describe the possible outcomes of a hypothesis test and how we would interpret these outcomes. Solution: (cont.) The null hypothesis is that the proportion of female zebras is the accepted population proportion of 50% (p = 0.5). The alternative hypothesis is the biologist’s claim that the accepted value is wrong, meaning that the actual proportion of female zebras is not 50% (it may be either more or less than 50%). The possible outcomes are • Reject the null hypothesis of p = 0.5, in which case we have evidence in support of the biologist’s claim that the accepted value is wrong. Pages Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 2 Hypothesis Test Outcomes Solution: (cont.) • Do not reject the null hypothesis, in which case we lack evidence to support the biologist’s claim. Note, however, that this does not imply that the accepted value is correct. Pages Copyright © 2009 Pearson Education, Inc. Slide

Drawing a Conclusion from a Hypothesis Test We will look at two ways to make the decision about rejecting or not rejecting the null hypothesis. Statistical Significance If the probability of a particular result is 0.05 or less, we say that the result is statistically significant at the 0.05 level; if the probability is 0.01 or less, the result is statistically significant at the 0.01 level. The 0.01 level therefore means stronger significance than the 0.05 level. Page 374 Copyright © 2009 Pearson Education, Inc. Slide

Hypothesis Test Decisions Based on Levels of Statistical Significance We decide the outcome of a hypothesis test by comparing the actual sample result (mean or proportion) to the result expected if the null hypothesis is true. We must choose a significance level for the decision. If the chance of a sample result at least as extreme as the observed result is less than 1 in 100 (or 0.01), then the test is statistically significant at the 0.01 level and offers strong evidence for rejecting the null hypothesis. If the chance of a sample result at least as extreme as the observed result is less than 1 in 20 (or 0.05), then the test is statistically significant at the 0.05 level and offers moderate evidence for rejecting the null hypothesis. Page 375 Copyright © 2009 Pearson Education, Inc. Slide

Hypothesis Test Decisions Based on Levels of Statistical Significance (cont.) If the chance of a sample result at least as extreme as the observed result is greater than the chosen level of significance (0.05 or 0.01), then we do not reject the null hypothesis. Page 375 Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 3 Statistical Significance in Hypothesis Testing Consider the Gender Choice example in which the sample size is n = 100 and the sample proportion is p = Using techniques that we will discuss later in this chapter, it is possible to calculate the probability of randomly choosing such a sample (or a more extreme sample with p > 0.64) under the assumption that the null hypothesis (p = 0.50) is true; the result is that the probability is Based on this result, should you reject or not reject the null hypothesis? p ˆ p ˆ Solution: The probability of means that if the null hypothesis is true (meaning that the true population proportion is 50%), the probability of drawing a random sample with a sample proportion of at least p = 0.64 is just slightly under 3 in 1,000. Page 375 p ˆ Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 3 Statistical Significance in Hypothesis Testing Solution: (cont.) Because this probability is less than 1 in 100, this result is statistically significant at the 0.01 level. It therefore gives us good reason to reject the null hypothesis, which means it provides support for the alternative hypothesis that Gender Choice raises the proportion of girls to more than 50%. Page 375 Copyright © 2009 Pearson Education, Inc. Slide

Drawing a Conclusion from a Hypothesis Test Statistical Significance P-Values P-value is short for probability value; notice the capital P, used to avoid confusion with the lowercase p that stands for a population proportion. We will discuss the actual calculation of P-values in Sections 9.2 and 9.3; here, we focus only on their interpretation. Page 375 Copyright © 2009 Pearson Education, Inc. Slide

Hypothesis Test Decisions Based on P-Values The P-value (probability value) for a hypothesis test of a claim about a population parameter is the probability of selecting a sample at least as extreme as the observed sample, assuming that the null hypothesis is true: A small P-value (such as less than or equal to 0.05) indicates that the sample result is unlikely, and therefore provides reason to reject the null hypothesis. A large P-value (such as greater than 0.05) indicates that the sample result could easily occur by chance, so we cannot reject the null hypothesis Page 375 Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 4 Fair Coin? You suspect that a coin may have a bias toward landing tails more often than heads, and decide to test this suspicion by tossing the coin 100 times. The result is that you get 40 heads (and 60 tails). A calculation (not shown here) indicates that the probability of getting 40 or fewer heads in 100 tosses with a fair coin is Find the P-value and level of statistical significance for your result. Should you conclude that the coin is biased against heads? Solution: The null hypothesis is that the coin is fair, in which case the proportion of heads should be about 50% (H0: p = 0.50). The alternative hypothesis is your suspicion that the coin is biased against heads, in which case the proportion of heads would be less than 50% (Ha: p < 0.50). Page 376 Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 4 Fair Coin? Solution: (cont.) Your 100 coin tosses represent a random sample of size n = 100, and the result of 40 heads is the sample proportion (p = 0.40) for the hypothesis test. The P-value for the test is the probability of getting a sample proportion at least as extreme as the one you found (p ≤ 0.40), assuming that the coin is fair and the population proportion of heads is 0.5. The given probability for this occurrence is , which is the P-value for the test. Because this P-value is smaller than 0.05, the result is statistically significant at the 0.05 level. Because it is not smaller than 0.01, the result is not significant at the 0.01 level. Statistical significance at the 0.05 level gives you moderate reason to reject the null hypothesis and conclude that the coin is biased against heads. p ˆ p ˆ Page 376 Copyright © 2009 Pearson Education, Inc. Slide

Putting It All Together The Hypothesis Test Process Step 1. Formulate the null and alternative hypotheses, each of which must make a claim about a population parameter, such as a population mean (μ) or a population proportion (p); be sure this is done before drawing a sample or collecting data. Based on the form of the alternative hypothesis, decide whether you will need a left-, right-, or two-tailed hypothesis test. Step 2. Draw a sample from the population and measure the sample statistics, including the sample size (n) and the relevant sample statistic, such as the sample mean (x) or sample proportion (p). Page 376 x ˆ p Copyright © 2009 Pearson Education, Inc. Slide

The Hypothesis Test Process (cont.) Step 3. Determine the likelihood of observing a sample statistic (mean or proportion) at least as extreme as the one you found under the assumption that the null hypothesis is true. The precise probability of such an observation is the P-value (probability value) for your sample result. Step 4. Decide whether to reject or not reject the null hypothesis, based on your chosen level of significance (usually 0.05 or 0.01, but other significance levels are sometimes used). Page 376 Copyright © 2009 Pearson Education, Inc. Slide

Again, be sure to avoid confusion among the three different uses of the letter p: • A lowercase p represents a population proportion—that is, the true proportion among a complete population. • A lowercase p (“p-hat”) represents a sample proportion—that is, the proportion found in a sample drawn from a population. • An uppercase P stands for probability in a P-value. p ˆ Page 376 Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 5 Mean Rental Car Mileage In the United States, the average car is driven about 12,000 miles each year. The owner of a large rental car company suspects that for his fleet, the mean distance is greater than 12,000 miles each year. He selects a random sample of n = 225 cars from his fleet and finds that the mean annual mileage for this sample is miles. A calculation shows that if you assume the fleet mean is the national mean of 12,000 miles, then the probability of selecting a 225-car sample with a mean annual mileage of at least 12,375 miles is Based on these data, describe the process of conducting a hypothesis test and drawing a conclusion. x = 12,375 x Page 377. The four step process is in the box on page 376. Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 5 Mean Rental Car Mileage Solution: We follow the four steps listed in slides 36 and 37: Step 1. The population parameter of interest is a population mean (μ)—the mean annual mileage for the population of all cars in the rental car fleet. The null hypothesis is that the population mean is the national average of 12,000 annual miles per car. The alternative hypothesis is the owner’s claim that the population mean for his fleet is greater than this national average. That is H0: m = 12,000 miles Ha: m > 12,000 miles Because the alternative hypothesis has a “greater than” form, the hypothesis test will be right-tailed. Page 377 Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 5 Mean Rental Car Mileage Solution: (cont.) Step 2. This step asks that we select the sample and measure the sample statistics; here, we are given the sample size of n = 225 and the sample mean of x = 12,375 miles. Step 3. This step asks us to determine the likelihood that, under the assumption that the fleet average is really 12,000 miles (the null hypothesis), we would by chance select a sample with a mean of at least 12,375 miles. Here, we don’t need to do the calculation, because we are told that the probability is (The calculation method is given in Section 9.2.) This probability is the P-value; it tells us that if the null hypothesis were true, there would be only a 0.01 probability of selecting a sample as extreme as the one observed. x = 12,375 x Page 377 Copyright © 2009 Pearson Education, Inc. Slide

EXAMPLE 5 Mean Rental Car Mileage Solution: (cont.) Step 4. The P-value of 0.01 tells us that the result is significant at the 0.01 level. We therefore reject the null hypothesis and conclude that the test provides strong support for the alternative hypothesis, implying that the mean annual mileage for the rental car fleet is greater than the national average of 12,000 miles. Page 377 Copyright © 2009 Pearson Education, Inc. Slide

A Legal Analogy of Hypothesis Testing In American courts of law, the fundamental principle is that a defendant is presumed innocent until proven guilty. Because the starting assumption is innocence, this represents the null hypothesis: The job of the prosecutor is to present evidence so compelling that the jury is persuaded to reject the null hypothesis and find the defendant guilty. If the prosecutor does not make a sufficiently compelling case, then the jury will not reject H0 and the defendant will be found “not guilty.” H0: The defendant is innocent. Ha: The defendant is guilty. Page 377 Copyright © 2009 Pearson Education, Inc. Slide

TIME OUT TO THINK Consider two situations. In one, you are a juror in a case in which the defendant could be fined a maximum of $2,000. In the other, you are a juror in a case in which the defendant could receive the death penalty. Compare the significance levels you would use in the two situations. In each situation, what are the consequences of wrongly rejecting the null hypothesis?. Page 378 Copyright © 2009 Pearson Education, Inc. Slide

9.1 Fundamentals of Hypothesis Testing

Similar presentations

Presentation on theme: "9.1 Fundamentals of Hypothesis Testing"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

9.1 Fundamentals of Hypothesis Testing

Similar presentations

Presentation on theme: "9.1 Fundamentals of Hypothesis Testing"— Presentation transcript:

Similar presentations

About project

Feedback