Ebook Elementary statistics: A step by step approach (Eighth edition) - Part 2 presents the following content: Chapter 8 - hypothesis testing; chapter 9 - testing the difference between two means, two proportions, and two variances; chapter 10 - correlation and regression; chapter 11 - other chi-square tests; chapter 12 - analysis of variance; chapter 13 - nonparametric statistics; chapter 14 - sampling and simulation.
blu38582_ch08_399-470.qxd 9/9/10 11:48 AM Page 399 C H A P T E R Hypothesis Testing Objectives After completing this chapter, you should be able to Outline Introduction Understand the definitions used in hypothesis testing State the null and alternative hypotheses Test means when s is unknown, using the t test Test proportions, using the z test 8–5 X2 Test for a Variance or Standard Deviation Test variances or standard deviations, using the chi-square test 8–6 Additional Topics Regarding Hypothesis Testing Test hypotheses, using confidence intervals 8–1 Steps in Hypothesis Testing—Traditional Method Find critical values for the z test 8–2 z Test for a Mean State the five steps used in hypothesis testing 8–3 t Test for a Mean Test means when s is known, using the z test 10 Explain the relationship between type I and 8–4 z Test for a Proportion Summary type II errors and the power of a test 8–1 blu38582_ch08_399-470.qxd 400 9/9/10 11:48 AM Page 400 Chapter Hypothesis Testing Statistics Today How Much Better Is Better? 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.e., sampling error)? In this chapter, you will learn how to answer that question 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 scientist 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 variance or standard deviation using the chi-square distribution is explained in Section 8–5 The three methods used to test hypotheses are The traditional method The P-value method The confidence interval method The traditional method will be explained first It has been used since the hypothesistesting 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 Understand the definitions used in hypothesis testing There are two types of statistical hypotheses for each situation: the null hypothesis and the alternative hypothesis 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 distributions and randomness This is explained in later chapters.) As an illustration of how hypotheses should be stated, three different statistical studies will be used as examples Objective State the null and alternative hypotheses Situation A A medical researcher is interested in finding out whether a new medication 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, decrease, or remain unchanged after a patient takes the medication? Since the researcher knows that the mean pulse rate for the population under study is 82 beats per minute, the hypotheses for this situation are H0: m ϭ 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 medicine could be to raise or lower the pulse rate 8–3 blu38582_ch08_399-470.qxd 402 9/9/10 11:48 AM Page 402 Chapter Hypothesis Testing Situation B A chemist invents an additive to increase the life of an automobile battery If the mean lifetime of the automobile battery without the additive is 36 months, then her hypotheses are H0: m ϭ 36 and H1: m Ͼ 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 hypothesis is that the mean is equal to 36 months This test is called right-tailed, since the interest is in an increase only Unusual Stat Sixty-three percent of people would rather hear bad news before hearing the good news Situation C A contractor wishes to lower heating bills by using a special type of insulation in houses If the average of the monthly heating bills is $78, her hypotheses about heating costs with the use of insulation are H0: m ϭ $78 H1: m Ͻ $78 and 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 Not equal to ϭ Ͼ Ͻ Greater than Less than The null and alternative hypotheses are stated together, and the null hypothesis contains the equals sign, as shown (where k represents a specified number) Two-tailed test Right-tailed test Left-tailed test H0: m ϭ k H1: m k H0: m ϭ k H1: m Ͼ k H0: m ϭ k H1: m Ͻ 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 assumption 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.e., using Ͻ or Ͼ or Because of this, the alternative hypothesis is sometimes called the research hypothesis Table 8–1 Hypothesis-Testing Common Phrases Ͼ Is greater than Is above Is higher than Is longer than Is bigger than Is increased ϭ Is equal to Is the same as Has not changed from Is the same as 8–4 Ͻ Is less than Is below Is lower than Is shorter than Is smaller than Is decreased or reduced from Is not equal to Is different from Has changed from Is not the same as 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 hypothesis 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 conjectures into mathematical symbols Example 8–1 State the null and alternative hypotheses for each conjecture a 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 8.6 pounds b 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 c 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 Solution a H0: m ϭ 8.6 and H1: m Ͼ 8.6 b H0: m ϭ 18 and H1: m Ͻ 18 c H0: m ϭ 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 (See Section 6–3.) 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 But if 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 Here is where the concepts of statistical test and level of significance are used 8–5 blu38582_ch08_399-470.qxd 404 9/9/10 11:48 AM Page 404 Chapter Hypothesis Testing Figure 8–1 (a) H is true Distribution of sample means Situations in Hypothesis Testing X X = 82 (b) H is false Distribution of sample means 82 X X = ? A statistical test uses the data obtained from a sample to make a decision about whether the null hypothesis should be rejected The numerical value obtained from a statistical test is called the test value In this type of statistical test, the mean is computed for the data obtained from the sample and is compared with the population mean Then a decision is made to reject or not reject the null hypothesis on the basis of the value obtained from the statistical test If the difference is significant, the null hypothesis is rejected If it is not, then the null hypothesis is not rejected In the hypothesis-testing situation, there are four possible outcomes In reality, the null hypothesis may or may not be true, and a decision is made to reject or not reject it on the basis of the data obtained from a sample The four possible outcomes are shown in Figure 8–2 Notice that there are two possibilities for a correct decision and two possibilities for an incorrect decision Figure 8–2 H true H false Error Type I Correct decision Correct decision Type II Possible Outcomes of a Hypothesis Test Reject H0 Do not reject H0 8–6 Error blu38582_ch08_399-470.qxd 9/9/10 11:48 AM Page 405 Section 8–1 Steps in Hypothesis Testing—Traditional Method 405 If a null hypothesis is true and it is rejected, then a type I error is made In situation A, for instance, the medication might not significantly change the pulse rate of all the users in the population; but it might change the rate, by chance, of the subjects in the sample In this case, the researcher will reject the null hypothesis when it is really true, thus committing a type I error On the other hand, the medication might not change the pulse rate of the subjects in the sample, but when it is given to the general population, it might cause a significant increase or decrease in the pulse rate of users The researcher, on the basis of the data obtained from the sample, will not reject the null hypothesis, thus committing a type II error In situation B, the additive might not significantly increase the lifetimes of automobile batteries in the population, but it might increase the lifetimes of the batteries in the sample In this case, the null hypothesis would be rejected when it was really true This would be a type I error On the other hand, the additive might not work on the batteries selected for the sample, but if it were to be used in the general population of batteries, it might significantly increase their lifetimes The researcher, on the basis of information obtained from the sample, would not reject the null hypothesis, thus committing a type II error A type I error occurs if you reject the null hypothesis when it is true A type II error occurs if you not reject the null hypothesis when it is false 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 See Figure 8–3 Now the hypotheses are H0: The defendant is innocent H1: The defendant is not innocent (i.e., guilty) Next, the evidence is presented in court by the prosecutor, and based on this evidence, the jury decides the verdict, innocent or guilty If the defendant is convicted but he or she did not commit the crime, then a type I error has been committed See block of Figure 8–3 On the other hand, if the defendant is convicted and he or she has committed the crime, then a correct decision has been made See block If the defendant is acquitted and he or she did not commit the crime, a correct decision has been made by the jury See block However, if the defendant is acquitted and he or she did commit the crime, then a type II error has been made See block Figure 8–3 Hypothesis Testing and a Jury Trial H 0: The defendant is innocent H 1: The defendant is not innocent H true (innocent) H false (not innocent) Type I error Correct decision The results of a trial can be shown as follows: Reject H0 (convict) Do not reject H (acquit) Type II error Correct decision 8–7 blu38582_ch08_399-470.qxd 406 9/9/10 11:48 AM Page 406 Chapter Hypothesis Testing The decision of the jury does not prove that the defendant did or did not commit the crime The decision is based on the evidence presented If the evidence is strong enough, the defendant will be convicted in most cases If the evidence is weak, the defendant will be acquitted in most cases Nothing is proved absolutely Likewise, the decision to reject or not reject the null hypothesis does not prove anything The only way to prove anything statistically is to use the entire population, which, in most cases, is not possible The decision, then, is made on the basis of probabilities That is, when there is a large difference between the mean obtained from the sample and the hypothesized mean, the null hypothesis is probably not true The question is, How large a difference is necessary to reject the null hypothesis? Here is where the level of significance is used Unusual Stats Of workers in the United States, 64% drive to work alone and 6% of workers walk to work The level of significance is the maximum probability of committing a type I error This probability is symbolized by a (Greek letter alpha) That is, P(type I error) ϭ a The probability of a type II error is symbolized by b, the Greek letter beta That is, P(type II error) ϭ b In most hypothesis-testing situations, b cannot be easily computed; however, a and b are related in that decreasing one increases the other Statisticians generally agree on using three arbitrary significance levels: the 0.10, 0.05, and 0.01 levels That is, if the null hypothesis is rejected, the probability of a type I error will be 10%, 5%, or 1%, depending on which level of significance is used Here is another way of putting it: When a ϭ 0.10, there is a 10% chance of rejecting a true null hypothesis; when a ϭ 0.05, there is a 5% chance of rejecting a true null hypothesis; and when a ϭ 0.01, there is a 1% chance of rejecting a true null hypothesis In a hypothesis-testing situation, the researcher decides what level of significance to use It does not have to be the 0.10, 0.05, or 0.01 level It can be any level, depending on the seriousness of the type I error After a significance level is chosen, a critical value is selected from a table for the appropriate test If a z test is used, for example, the z table (Table E in Appendix C) is consulted to find the critical value The critical value determines the critical and noncritical regions The critical value separates the critical region from the noncritical region The symbol for critical value is C.V The critical or rejection region is the range of values of the test value that indicates that there is a significant difference and that the null hypothesis should be rejected The noncritical or nonrejection region is the range of values of the test value that indicates that the difference was probably due to chance and that the null hypothesis should not be rejected The critical value can be on the right side of the mean or on the left side of the mean for a one-tailed test Its location depends on the inequality sign of the alternative hypothesis For example, in situation B, where the chemist is interested in increasing the average lifetime of automobile batteries, the alternative hypothesis is H1: m Ͼ 36 Since the inequality sign is Ͼ, the null hypothesis will be rejected only when the sample mean is significantly greater than 36 Hence, the critical value must be on the right side of the mean Therefore, this test is called a right-tailed test 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 righttailed test or left-tailed test, depending on the direction of the inequality of the alternative hypothesis 8–8 blu38582_ch08_399-470.qxd 9/9/10 11:48 AM Page 407 Section 8–1 Steps in Hypothesis Testing—Traditional Method 407 Figure 8–4 Finding the Critical Value for A ؍0.01 (Right-Tailed Test) z 0.9900 0.00 0.01 0.02 0.03 0.04 0.05 0.0 Critical region 0.01 0.1 0.2 0.3 Find this area in table as shown z 2.1 2.2 2.3 0.9901 Closest value to 0.9900 2.4 (a) The critical region Objective Find critical values for the z test (b) The critical value from Table E To obtain the critical value, the researcher must choose an alpha level In situation B, suppose the researcher chose a ϭ 0.01 Then the researcher must find a z value such that 1% of the area falls to the right of the z value and 99% falls to the left of the z value, as shown in Figure 8–4(a) Next, the researcher must find the area value in Table E closest to 0.9900 The critical z value is 2.33, since that value gives the area closest to 0.9900 (that is, 0.9901), as shown in Figure 8–4(b) The critical and noncritical regions and the critical value are shown in Figure 8–5 Figure 8–5 Critical and Noncritical Regions for A ؍0.01 (Right-Tailed Test) 0.9900 Noncritical region Critical region 0.01 +2.33 Now, move on to situation C, where the contractor is interested in lowering the heating bills The alternative hypothesis is H1: m Ͻ $78 Hence, the critical value falls to the left of the mean This test is thus a left-tailed test At a ϭ 0.01, the critical value is Ϫ2.33, since 0.0099 is the closest value to 0.01 This is shown in Figure 8–6 When a researcher conducts a two-tailed test, as in situation A, the null hypothesis can be rejected when there is a significant difference in either direction, above or below the mean 8–9 blu38582_ch08_399-470.qxd 408 9/9/10 11:48 AM Page 408 Chapter Hypothesis Testing Figure 8–6 Critical and Noncritical Regions for A ؍0.01 (Left-Tailed Test) Noncritical region Critical region 0.01 –2.33 In a two-tailed test, the null hypothesis should be rejected when the test value is in either of the two critical regions For a two-tailed test, then, the critical region must be split into two equal parts If a ϭ 0.01, then one-half of the area, or 0.005, must be to the right of the mean and onehalf must be to the left of the mean, as shown in Figure 8–7 In this case, the z value on the left side is found by looking up the z value corresponding to an area of 0.0050 The z value falls about halfway between Ϫ2.57 and Ϫ2.58 corresponding to the areas 0.0049 and 0.0051 The average of Ϫ2.57 and Ϫ2.58 is [(Ϫ2.57) ϩ (Ϫ2.58)] Ϭ ϭ Ϫ2.575 so if the z value is needed to three decimal places, Ϫ2.575 is used; however, if the z value is rounded to two decimal places, Ϫ2.58 is used On the right side, it is necessary to find the z value corresponding to 0.99 ϩ 0.005, or 0.9950 Again, the value falls between 0.9949 and 0.9951, so ϩ2.575 or 2.58 can be used See Figure 8–7 Figure 8–7 Finding the Critical Values for A ؍0.01 (Two-Tailed Test) 0.9900 0.9950 0.005 0.005 0.4950 –z +z The critical values are ϩ2.58 and Ϫ2.58, as shown in Figure 8–8 Figure 8–8 Critical and Noncritical Regions for A ؍0.01 (Two-Tailed Test) Noncritical region Critical region Critical region –2.58 8–10 ϩ2.58 blu38582_ans_IS1-IS76.qxd 9/28/10 8:24 PM Page 71 Instructor’s Section Answers rs ϭ 0.471; H0: r ϭ and H1: r 0; C.V ϭ Ϯ0.886; not reject There is no significant linear relationship rs ϭ 0.817; H0: r ϭ and H1: r 0; C.V ϭ Ϯ0.700; reject There is a significant relationship between the number of new releases and the gross receipts rs ϭ 0.893; H0: r ϭ and H1: r 0; C.V ϭ Ϯ0.786; reject There is a significant relationship between the number of hospitals and the number of nursing homes in a state rs ϭ 0.048; H0: r ϭ and H1: r 0; C.V ϭ Ϯ0.738; not reject There is not enough evidence to say that a significant correlation exists between calories and the cholesterol amounts in fast-food sandwiches 10 rs ϭ 0.8857; H0: r ϭ and H1: r 0; C.V ϭ Ϯ0.886 Very close! There is not a significant relationship between the number of books published in 1980 and in 2004 in the same subject area Since r is not significant, no relationship can be predicted 20 years from now Even if r is significant, you should not make a prediction for 20 years from now That would be extrapolating 11 rs ϭ 0.624; H0: r ϭ and H1: r 0; C.V ϭ Ϯ0.700; not reject There is no significant relationship between gasoline prices paid to the car rental agency and regular gasoline prices One would wonder how the car rental agencies determine their prices 12 rs ϭ 0.714; H0: r ϭ and H1: r 0; C.V ϭ Ϯ0.886; not reject There is not sufficient evidence to conclude a significant relationship between the number of motor vehicle thefts and burglaries 13 rs ϭ Ϫ0.10; H0: r ϭ and H1: r 0; C.V ϭ Ϯ0.900; not reject There is no significant relationship between the number of cyber school students and the cost per pupil In this case, the cost per pupil is different in each district 14 rs ϭ 0.542; H0: r ϭ and H1: r 0; C.V ϭ Ϯ0.643; not reject There is no significant relationship between the costs of the drugs 15 H0: the number of cavities in a person occurs at random and H1: the null hypothesis is not true There are 21 runs; the expected number of runs is between 10 and 22 Therefore, not reject the null hypothesis; the number of cavities in a person occurs at random 16 H0: the numbers occur at random and H1: the null hypothesis is not true There are 14 runs Since the expected number of runs is between and 20, not reject The numbers occur at random 17 H0: the purchases of soft drinks occur at random and H1: the null hypothesis is not true There are 16 runs, and the expected number of runs is between and 22, so not reject the null hypothesis Hence the purchases of soft drinks occur at random 18 H0: the integers generated by a calculator occur at random and H1: the null hypothesis is not true There are 13 runs, and the expected number of runs is between and 17, so the null hypothesis is not rejected The integers occur at random 19 H0: the seating occurs at random and H1: the null hypothesis is not true There are 14 runs Since the expected number of runs is between 10 and 23, not reject The seating occurs at random 20 H0: the gender of the shoppers in line at the grocery store is random (claim) and H1: the null hypothesis is not true There are 10 runs Since the expected number of runs is between and 16, the null hypothesis should not be rejected There is not enough evidence to reject the hypothesis that the gender of the shoppers in line is random 21 H0: the number of absences of employees occurs at random over a 30-day period and H1: the null hypothesis is not true There are only runs, and this value does not fall within the 9-to-21 range Hence, the null hypothesis is rejected; the absences not occur at random 22 H0: the days customers are able to ski occur at random (claim) and H1: the null hypothesis is not true There are runs Since this number is not between and 20, the decision is to reject the null hypothesis There is enough evidence to reject the claim that the days customers are able to ski occur at random 23 Answers will vary 24 Ϯ0.28 25 Ϯ0.479 26 Ϯ0.400 27 Ϯ0.215 28 Ϯ0.413 Review Exercises H0: median ϭ 36 years and H1: median 36 years; z ϭ Ϫ0.548; C.V ϭ Ϯ1.96; not reject There is insufficient evidence to conclude that the median differs from 36 H0: median ϭ 40,000 miles (claim) and H1: median 40,000 miles; z ϭ Ϫ0.913; C.V ϭ Ϯ1.96; not reject There is not enough evidence to reject the claim that the median is 40,000 miles H0: there is no difference in prices and H1: there is a difference in prices; test value ϭ 1; C.V ϭ 0; not reject There is insufficient evidence to conclude a difference in prices Comments: Examine what affects the result of this test H0: there is no difference in the record high temperatures of the two cities and H1: there is a difference in the record high temperatures of the two cities (claim); z ϭ Ϫ1.24; P-value ϭ 0.2150; not reject There is not enough evidence to support the claim that there is a difference in the record high temperatures of the two cities H0: there is no difference in the hours worked and H1: there is a difference in the hours worked; R ϭ 85; mR ϭ 110; sR ϭ 14.2009; z ϭ Ϫ1.76; C.V ϭ Ϯ1.645; reject There is sufficient evidence to conclude a difference in the hours worked C.V ϭ Ϯ1.96; not reject IS–71 blu38582_ans_IS1-IS76.qxd 9/28/10 8:24 PM Page 72 Instructor’s Section Answers H0: the additive did not improve the gas mileage and H1: the additive did improve the gas mileage (claim); C.V ϭ 14; ws ϭ 14; reject There is enough evidence to support the claim that the additive improved the gas mileage H0: there is no difference in the amount spent and H1: there is a difference in the amount spent; ws ϭ 1; C.V ϭ 2; reject There is sufficient evidence of a difference in amount spent at the 0.05 level of significance H0: there is no difference in the breaking strengths of the ropes and H1: there is a difference in the breaking strengths of the ropes (claim); C.V ϭ 5.991; H ϭ 28.02; reject There is enough evidence to support the claim that there is a difference in the breaking strengths of the ropes H0: there is no difference in beach temperatures and H1: there is a difference in temperatures; H ϭ 15.524; C.V ϭ 7.815; reject There is sufficient evidence to conclude a difference in beach temperatures (Without the Southern Pacific: H ϭ 3.661; C.V ϭ 5.991; not reject.) 10 rs ϭ 0.933; H0: r ϭ and H1: r 0; C.V ϭ Ϯ0.700; reject There is a significant relationship between the rankings 11 rs ϭ 0.891; H0: r ϭ and H1: r 0; C.V ϭ Ϯ0.648; reject There is a significant relationship in the average number of people who are watching the television shows for both years 12 H0: the books are arranged at random and H1: the null hypothesis is not true There are 12 runs Since the expected number of runs is between 10 and 22, not reject The books are arranged at random 13 H0: the grades of students who finish the exam occur at random and H1: the null hypothesis is not true Since there are runs and this value does not fall in the 9-to-21 interval, the null hypothesis is rejected The grades not occur at random Chapter Quiz False False True True a c d Nonparametric 11 Sign b 10 Nominal, ordinal 12 Sensitive 13 H0: median ϭ $177,500; H1: median $177,500 (claim); C.V ϭ 2; test value ϭ 3; not reject There is not enough evidence to say that the median is not $177,500 14 H0: median ϭ 1200 (claim) and H1: median 1200 There are 10 minus signs Do not reject since 10 is greater than the critical value There is not enough evidence to reject the claim that the median is 1200 IS–72 15 H0: there will be no change in the weight of the turkeys after the special diet and H1: the turkeys will weigh more after the special diet (claim) There is plus sign; hence, the null hypothesis is rejected There is enough evidence to support the claim that the turkeys gained weight on the special diet 16 H0: there is no difference in the amounts of money received by the teams and H1: there is a difference in the amounts of money each team received; C.V ϭ Ϯ1.96; z ϭ Ϫ0.79; not reject There is not enough evidence to say that the amounts differ 17 H0: the distributions are the same and H1: the distributions are different (claim); z ϭ Ϫ0.14434; C.V ϭ Ϯ1.65; not reject the null hypothesis There is not enough evidence to support the claim that the distributions are different 18 H0: there is no difference in the GPA of the students before and after the workshop and H1: there is a difference in the GPA of the students before and after the workshop (claim); test statistic ϭ 0; C.V ϭ 2; reject the null hypothesis There is enough evidence to support the claim that there is a difference in the GPAs of the students 19 H0: there is no difference in the amounts of sodium in the three sandwiches and H1: there is a difference in the amounts of sodium in the sandwiches; C.V ϭ 5.991; H ϭ 11.795; reject There is enough evidence to conclude that there is a difference in the amounts of sodium in the sandwiches 20 H0: there is no difference in the reaction times of the monkeys and H1: there is a difference in the reaction times of the monkeys (claim); H ϭ 6.9; 0.025 Ͻ P-value Ͻ 0.05 (0.032); reject the null hypothesis There is enough evidence to support the claim that there is a difference in the reaction times of the monkeys 21 rs ϭ 0.683; H0: r ϭ and H1: r 0; C.V ϭ Ϯ0.600; reject There is enough evidence to say that there is a significant relationship between the drug prices 22 rs ϭ 0.943; H0: r ϭ and H1: r 0; C.V ϭ Ϯ0.829; reject There is a significant relationship between the amount of money spent on Head Start and the number of students enrolled in the program 23 H0: the births of babies occur at random according to gender and H1: the null hypothesis is not true There are 10 runs, and since this is between and 19, the null hypothesis is not rejected There is not enough evidence to reject the null hypothesis that the gender occurs at random 24 H0: there is no difference in the rpm of the motors before and after the reconditioning and H1: there is a difference in the rpm of the motors before and after the reconditioning (claim); test statistic ϭ 0; C.V ϭ 6; not reject the null hypothesis There is not enough evidence to support the claim that there is a difference in the rpm of the motors before and after reconditioning blu38582_ans_IS1-IS76.qxd 9/28/10 8:24 PM Page 73 Instructor’s Section Answers 25 H0: the numbers occur at random and H1: the null hypothesis is not true There are 16 runs, and since this is between and 21, the null hypothesis is not rejected There is not enough evidence to reject the null hypothesis that the numbers occur at random Chapter 14 Exercises 14–1 Random, systematic, stratified, cluster Samples can save the researcher time and money They are used when the population is large or infinite They are used when the original units are to be destroyed, such as in testing the breaking strength of ropes A sample must be randomly selected Random numbers are used to ensure every element of the population has the same chance of being selected Talking to people on the street, calling people on the phone, and asking your friends are three incorrect ways of obtaining a sample Over the long run each digit, through 9, will occur with the same probability Random sampling has the advantage that each unit of the population has an equal chance of being selected One disadvantage is that the units of the population must be numbered; if the population is large, this could be somewhat time-consuming Systematic sampling has an advantage in that once the first unit is selected, each succeeding unit selected has been determined This saves time A disadvantage would be if the list of units was arranged in some manner so that a bias would occur, such as selecting all men when the population consists of both men and women An advantage of stratified sampling is that it ensures representation for the groups used in stratification; however, it is virtually impossible to stratify the population so that all groups are represented 10 Clusters are easy to use since they already exist, but it is difficult to justify that the clusters actually represent the population 11–20 Answers will vary Exercises 14–2 Flaw—biased; it’s confusing Flaw—the purpose of the question is unclear You could like him personally but not politically Flaw—the question is too broad Flaw—none The question is good if the respondent knows the mayor’s position; otherwise his position needs to be stated Flaw—confusing words How many hours did you study for this exam? Possible order problem—ask first, “Do you use artificial sweetener regularly?” Flaw—confusing words If a plane were to crash on the border of New York and New Jersey, where should the victims be buried? Flaw—none Answers will vary 10 Answers will vary Exercises 14–3 Simulation involves setting up probability experiments that mimic the behavior of real-life events Answers will vary John Von Neumann and Stanislaw Ulam Using the computer to simulate real-life situations can save time, since the computer can generate random numbers and keep track of the outcomes very quickly and easily The steps are as follows: a List all possible outcomes b Determine the probability of each outcome c Set up a correspondence between the outcomes and the random numbers d Conduct the experiment by using random numbers e Repeat the experiment and tally the outcomes f Compute any statistics and state the conclusions Random numbers can be used to ensure the outcomes occur with appropriate probability When the repetitions increase, there is a higher probability that the simulation will yield more precise answers Use a table of random numbers Select 40 random numbers Numbers 01 through 16 mean the person is foreign-born Use three-digit random numbers; numbers 001 through 681 mean that the mother is in the labor force 10 Select two-digit random numbers in groups of For one person, 01 through 70 means a success For the other person, 01 through 75 means a success 11 Select 100 two-digit random numbers Numbers 00 to 34 mean the household has at least one set with premium cable service Numbers 35 to 99 mean the household does not have the service 12 Use the odd digits to represent a match and the even digits to represent a nonmatch 13 Let an odd number represent heads and an even number represent tails Then each person selects a digit at random 14–24 Answers will vary IS–73 blu38582_ans_IS1-IS76.qxd 9/28/10 8:24 PM Page 74 Instructor’s Section Answers Review Exercises 1–8 Answers will vary Use one-digit random numbers through for a strikeout and through and represent anything other than a strikeout 10 Use two-digit random numbers: 01 through 15 represent an overbooked plane, and 16 through 99 and 00 represent a plane that is not overbooked 11 In this case, a one-digit random number is selected Numbers through represent the numbers on the face Ignore 7, 8, 9, and and select another number 12 The first person selects a two-digit random number Any two-digit random number that has a 7, 8, 9, or is ignored, and another random number is selected Player selects a one-digit random number; any random number that is not through is ignored, and another one is selected 13 Let the digits through represent rock, let through represent paper, let through represent scissors, and omit 14–18 Answers will vary 19 Flaw—asking a biased question Have you ever driven through a red light? 20 Flaw—using a double negative Do you think students who are not failing should be given tutoring if they request it? 21 Flaw—asking a double-barreled question Do you think all automobiles should have heavy-duty bumpers? 18 Use two-digit random numbers to represent the spots on the face of the dice Ignore any two-digit random numbers with 7, 8, 9, or For cards, use two-digit random numbers between 01 and 13 19 Use two-digit random numbers The first digit represents the first player, and the second digit represents the second player If both numbers are odd or even, player wins If a digit is odd and the other digit is even, player wins 20–24 Answers will vary Appendix A A–1 362,880 A–2 5040 A–3 120 A–4 A–5 A–6 A–7 1320 A–8 1,814,400 A–9 20 A–10 7920 A–11 126 A–12 120 A–13 70 A–14 455 A–15 A–16 10 A–17 560 A–18 1980 A–19 2520 A–20 90 A–21 121; 2181; 14,641; 716.9 A–22 56; 550; 3136; 158 A–23 32; 258; 1024; 53.2 A–24 150; 4270; 22,500; 1457.5 22 Answers will vary A–25 328; 22,678; 107,584; 1161.2 A–26 829; 123,125; 687,241; 8584.8333 Chapter Quiz True True A–27 693; 50,511; 480,249; 2486.1 False True A–28 409; 40,333; 167,281; 6876.80 a c A–29 318; 20,150; 101,124; 3296 c Larger A–30 Ϫ20; 778; 400; 711.3334 Biased 10 Cluster A–31 y 11–14 Answers will vary 15 Use two-digit random numbers: 01 through 45 means the player wins Any other two-digit random number means the player loses 16 Use two-digit random numbers: 01 through 05 means a cancellation Any other two-digit random number means the person shows up 17 The random numbers 01 through 10 represent the 10 cards in hearts The random numbers 11 through 20 represent the 10 cards in diamonds The random numbers 21 through 30 represent the 10 spades, and 31 through 40 represent the 10 clubs Any number over 40 is ignored IS–74 (1, 6) (3, 2) –6 –5 –4 –3 –2 –1–1 –2 –3 –4 –5 –6 x blu38582_ans_IS1-IS76.qxd 9/28/10 8:24 PM Page 75 Instructor’s Section Answers A–32 A–35 y (0, 5) y 10 10 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1–1 10 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1–1 –2 –3 –4 –5 –6 –7 –8 –9 –10 A–33 A–36 (0, 5) x (–1, 3) A–34 –8 –7 –6 –5 –4 –3 –2 –1–1 x –1 10 –8 –7 –6 –5 –4 –3 –2 –1–1 –2 (–1, –2) –3 –4 –5 –6 –7 –8 –9 –10 10 y = + 2x y (–7, 8) x 10 y (3, 6) –6 –5 –4 –3 –2 –1–1 –2 –3 –4 –5 –6 (10, 3) –2 –3 –4 –5 –6 –7 –8 –9 –10 y (–2, 4) (6, 3) (8, 0) x A–37 x y x –2 –3 –4 –5 –6 –7 –8 –9 –10 y y = –1 + x (1, 0) x –6 –5 –4 –3 –2 –1–1 (0, –1) x y –2 –1 –3 –4 –5 –6 IS–75 blu38582_ans_IS1-IS76.qxd 9/28/10 8:24 PM Page 76 Instructor’s Section Answers A–38 A–39 y (0, 4) (1, 1) (1, 7) y = + 4x (0, 3) x –6 –5 –4 –3 –2–1 –1 –2 –3 –4 –5 –6 –6 –5 –4 –3 –2–1 –1 –2 –3 –4 –5 –6 x y = – 3x y y = –2 – 2x (–2, 2) x y –2 –2 Appendix B–2 B–1 0.65 –6 –5 –4 –3 –2 –1–1 IS–76 A–40 y –2 –3 –4 –5 –6 x (0, –2) B–2 0.579 B–3 0.653 B–4 0.005 B–5 0.379 B–6 0.585 B–7 B–8 B–9 0.64 B–10 0.467 B–11 0.857 B–12 0.33 blu38582_index_I1-I6.qxd 9/25/10 2:14 PM Page I-1 Index A Addition rules, 199–204 Adjusted R2, 579–580 Alpha, 406 Alternate approach to standard normal distribution, 765–768 Alternative hypotheses, 401 Algebra review, 753–757 Analysis of variance (ANOVA), 631–662 assumptions, 631–650 between-group variance, 631 degrees of freedom, 632, 649 F-test, 633 hypotheses, 631, 648–649 one-way, 631–637 summary table, 633, 651 two-way, 647–655 within-group variance, 631 Assumptions for the use of chi-square test, 448, 594, 613 Assumptions for valid predictions in regression, 556 Averages, 105–116 properties and uses, 116 B Bar graph, 69–70 Bayes’ theorem, 761–764 Bell curve, 301 Beta, 406, 459 Between-group variance, 631 Biased sample, 721 Bimodal, 60, 111 Binomial distribution, 271–276 characteristics, 271 mean for, 274 normal approximation, 340–346 notation, 271 standard deviation, 274 variance, 274 Binomial experiment, 271 Binomial probability formula, 271 Boundaries, Boundaries, class, 39 Boxplot, 162 C Categorical frequency distribution, 38–39 Census, Central limit theorem, 331–338 Chebyshev’s theorem, 134–136 Chi-square assumptions, 448, 594, 613 contingency table, 606–607 degrees of freedom, 386 distribution, 386–388 goodness-of-fit test, 593–598 independence test, 606–611 use in H-test, 694 variance test, 447–453 Yates correction for, 613, 617 Class, 37 boundaries, 39 limits, 39 midpoint, 40 width, 39–40 Classical probability, 186–191 Cluster sample, 12, 728 Coefficient of determination, 569 Coefficient of nondetermination, 569 Coefficient of variation, 132–133 Combination, 229–232 Combination rule, 230 Complementary events, 189–190 Complement of an event, 189 Compound event, 186 Conditional probability, 213, 216–218 I–1 blu38582_index_I1-I6.qxd 9/25/10 2:14 PM Page I-2 Index Confidence interval, 358 hypothesis testing, 457–459 mean, 358–373 means, difference of, 478, 486, 499 median, 672 proportion, 377–379 proportions, differences, 508–509 variances and standard deviations, 385–390 Confidence level, 358 Confounding variable, 15 Consistent estimator, 357 Contingency coefficient, 617 Contingency table, 606–607 Continuous variable, 6–7, 253, 300 Control group, 14 Convenience sample, 12–13 Correction factor for continuity, 342 Correlation, 534, 538–547 Correlation coefficient, 539 multiple, 578 Pearson’s product moment, 539 population, 543 Spearman’s rank, 700–702 Critical region, 406 Critical value, 406 Cumulative frequency, 54 Cumulative frequency distribution, 42–43 Cumulative frequency graph, 54–56 Cumulative relative frequency, 57–58 D Data, Data array, 109 Data set, Data value (datum), Deciles, 151 Degrees of freedom, 370 Dependent events, 213 Dependent samples, 492 Dependent variable, 14, 535 Descriptive statistics, Difference between two means, 473–479, 484–487, 492–499 assumptions for the test to determine, 473, 486, 493 proportions, 504–509 Discrete probability distribution, 254 Discrete variable, 6, 253 Disordinal interaction, 653 Distribution-free statistics (nonparametric), 672 I–2 Distributions bell-shaped, 59, 301 bimodal, 60, 111 binomial, 270–276 chi-square, 386–388 F, 513 frequency, 37 hypergeometric, 286–289 multinomial, 283–284 negatively skewed, 60, 117, 301 normal, 302–311 Poisson, 284–286 positively skewed, 60, 117, 301 probability, 253–258 sampling, 331–333 standard normal, 304 symmetrical, 59, 117, 301 Double sampling, 729 E Empirical probability, 191–193 Empirical rule, 136 Equally likely events, 186 Estimation, 356 Estimator, properties of a good, 357 Event, simple, 185 Events complementary, 189–190 compound, 189 dependent, 213 equally likely, 186 independent, 211 mutually exclusive, 199–200 Expectation, 264–266 Expected frequency, 593 Expected value, 264 Experimental study, 14 Explained variation, 566 Explanatory variable, 14 Exploratory data analysis (EDA), 162–165 Extrapolation, 556 F Factorial notation, 227 Factors, 647 F-distribution, characteristics of, 513 Finite population correction factor, 337 Five-number summary, 162 Frequency, 37 blu38582_index_I1-I6.qxd 9/25/10 2:14 PM Page I-3 Index Frequency distribution, 37 categorical, 38–39 grouped, 39–42 reasons for, 45 rules for constructing, 41–42 ungrouped, 43 Frequency polygon, 53–54 F-test, 513–519, 631 comparing three or more means, 633–636 comparing two variances, 513–519 notes for the use of, 516 Fundamental counting rule, 224–227 G Gallup poll, 720 Gaussian distribution, 301 Geometric mean, 122 Goodness-of-fit test, 593–598 Grand mean, 632 Grouped frequency distribution, 39–42 H Harmonic mean, 121 Hawthorne effect, 15 Hinges, 165 Histogram, 51–53 Homogeniety of proportions, 611–614 Homoscedasticity assumption, 568 Hypergeometric distribution, 286–288 Hypothesis, 4, 401 Hypothesis testing, 4, 400–404 alternative, 401 common phrases, 402 critical region, 406 critical value, 406 definitions, 401 level of significance, 406 noncritical region, 406 null, 401 one-tailed test, 406 P-value method, 418–421 research, 402 statistical, 401 statistical test, 404 test value, 404 traditional method, steps in, 411 two-tailed test, 402, 408 types of errors, 404–405 I Independence test (chi-square), 606–611 Independent events, 211 Independent samples, Independent variables, 14, 535, 647 Inferential statistics, 484 Influential observation or point, 557 Interaction effect, 648 Intercept (y), 552–555 Interquartile range (IQR), 151, 162 Interval estimate, 358 Interval level of measurement, K Kruskal-Wallis test, 693–696 L Law of large numbers, 193–194 Left-tailed test, 402, 406 Level of significance, 406 Levels of measurement, 7–8 interval, nominal, ordinal, 7–8 ratio, Limits, class, 39 Line of best fit, 551–552 Lower class boundary, 39 Lower class limit, 39 Lurking variable, 547 M Main effects, 649 Marginal change, 555 Margin of error, 359 Mean, 106–108 binomial variable, 274 definition, 106 population, 106 probability distribution, 259–261 sample, 106 Mean deviation, 141 Mean square, 633 Measurement, levels of, 7–8 Measurement scales, 7–8 Measures of average, uses of, 116 Measures of dispersion, 123–132 I–3 blu38582_index_I1-I6.qxd 9/25/10 2:14 PM Page I-4 Index Measures of position, 142–151 Measures of variation, 123–134 Measures of variation and standard deviation, uses of, 132 Median, 109–111 confidence interval for, 672 defined, 109 for grouped data, 122 Midquartile, 155 Midrange, 115 Misleading graphs, 18, 76–80 Modal class, 112 Mode, 111–114 Modified box plot, 165, 168 Monte Carlo method, 739–744 Multimodal, 111 Multinomial distribution, 283–284 Multiple correlation coefficient, 578 Multiple regression, 535, 575–580 Multiple relationships, 535, 575–580 Multiplication rules probability, 211–216 Multistage sampling, 729 Mutually exclusive events, 199–200 N Negatively skewed distribution, 117, 301 Negative linear relationship, 535, 539 Nielsen television ratings, 720 Nominal level of measurement, Noncritical region, 406 Nonparametric statistics, 672–710 advantages, 673 disadvantages, 673 Nonrejection region, 406 Nonresistant statistic, 165 Normal approximation to binomial distribution, 340–346 Normal distribution, 302–311 applications of, 316–321 approximation to the binomial distribution, 340–346 areas under, 305–307 formula for, 304 probability distribution as a, 307–309 properties of, 303 standard, 304 Normal quantile plot, 324, 328–330 Normally distributed variables, 300–302 Notation for the binomial distribution, 271 Null hypothesis, 401 I–4 O Observational study, 13–14 Observed frequency, 593 Odds, 199 Ogive, 54–56 One-tailed test, 406 left, 406 right, 406 One-way analysis of variance, 631–637 Open-ended distribution, 41 Ordinal interaction, 653 Ordinal level of measurement, 7–8 Outcome, 183 Outcome variable, 14 Outliers, 60, 113, 151–153, 322 P Paired-sample sign test, 677–679 Parameter, 106 Parametric tests, 672 Pareto chart, 70–71 Pearson coefficient of skewness, 141, 322–324 Pearson product moment correlation coefficient, 539 Percentiles, 143–149 Permutation, 227–229 Permutation rule, 228 Pie graph, 73–76 Point estimate, 357 Poisson distribution, 284–286 Pooled estimate of variance, 487 Population, 4, 721 Positively skewed distribution, 117, 301 Positive linear relationship, 535, 539 Power of a test, 459–460 Practical significance, 421 Prediction interval, 572–573 Probability, 4, 182 addition rules, 199–204 at least, 218–219 binomial, 270–276 classical, 186–191 complementary rules, 190 conditional, 213, 216–218 counting rules, 237–239 distribution, 253–258 empirical, 191–193 experiment, 183 multiplication rules, 211–216 subjective, 194 blu38582_index_I1-I6.qxd 9/25/10 2:14 PM Page I-5 Index Properties of the distribution of sample means, 331 Proportion, 377, 437 P-value, 418 for F test, 518 method for hypothesis testing, 418–421 for t test, 430–432 for X2 test, 451–453 Q Quadratic mean, 122 Qualitative variables, Quantitative variables, Quantile plot, 324, 328–330 Quartiles, 149–151 Quasi-experimental study, 14 Questionnaire design, 736–738 R Random numbers, 11, 722–725 Random samples, 10, 721–725 Random sampling, 10–11, 721–725 Random variable, 3, 253 Range, 41, 124–125 Range rule of thumb, 133 Rank correlation, Spearman’s, 700–702 Ranking, 673–674 Ratio level of measurement, Raw data, 37 Regression, 534, 551–558 assumptions for valid prediction, 556 multiple, 535, 575–580 Regression line, 551 equation, 552–556 intercept, 552–554 line of best fit, 551–552 prediction, 535 slope, 552–553 Rejection region, 406 Relationships, 4–5, 535 Relative frequency graphs, 56–58 Relatively efficient estimator, 357 Requirements for a probability distribution, 257 Research hypothesis, 402 Research report, 759 Residual, 567–568 Residual Plot, 568–569 Resistant statistic, 165 Right-tailed test, 402–406 Robust, 357 Run, 703 Runs test, 702–706 S Sample, 4, 721 biased, 721 cluster, 12, 728 convenience, 12–13 random, 10, 721–725 size for estimating means, 363–365 size for estimating proportions, 379–381 stratified, 12, 726–728 systematic, 11–12, 725–726 unbiased, 721 Sample space, 183 Sampling, 10–13, 721–730 distribution of sample means, 331–333 double, 729 error, 331 multistage, 729 random, 10–11, 721–725 sequence, 729 Scatter plot, 535–538 Scheffé test, 642, 643 Sequence sampling, 729 Short-cut formula for variance and standard deviation, 129 Significance, level of, 406 Sign test, 675–677 test value for, 675 Simple event, 185 Simple relationship, 535 Simulation technique, 739 Single sample sign test, 675–677 Skewness, 59–60, 301–302 Slope, 552–553 Spearman rank correlation coefficient, 700–702 Standard deviation, 125–132 binomial distribution, 274 definition, 127 formula, 127 population, 127 sample, 128 uses of, 132 Standard error of difference between means, 474 Standard error of difference between proportions, 505 Standard error of the estimate, 570–572 I–5 blu38582_index_I1-I6.qxd 9/25/10 2:14 PM Page I-6 Index Standard error of the mean, 333 Standard normal distribution, 304 Standard score, 142–143 Statistic, 106 Statistical hypothesis, 401 Statistical test, 406 Statistics, descriptive, inferential, misuses of, 16–19 Stem and leaf plot, 80–83 Stratified sample, 12, 726–728 Student’s t distribution, 370 Subjective probability, 194 Sum of squares, 633 Surveys, 9–10, 736–738 mail, 9–10 personal interviews, 10 telephone, Symmetrical distribution, 59, 117, 301 Systematic sampling, 11–12, 725–726 T t-distribution, characteristics of, 370 Test of normality, 322–324, 328–330, 598–600 Test value, 404 Time series graph, 71–73 Total variation, 566 Treatment groups, 14, 648 Tree diagram, 185, 215, 225–226 t-test, 427 coefficient for correlation, 543–545 for difference of means, 484–487, 492–500 for mean, 427–433 Tukey test, 644–645 Two-tailed test, 402, 408 Two-way analysis of variance, 647–655 Type I error, 405–406, 459–460 Type II error, 405–406, 459–460 U Unbiased estimate of population variance, 128 Unbiased estimator, 357 Unbiased sample, 721 Unexplained variation, 566 Ungrouped frequency distribution, 43–44 Uniform distribution, 60, 310 I–6 Unimodal, 60, 111 Upper class boundary, 39 Upper class limit, 39 V Variable, 3, 253, 535 confounding, 15 continuous, 6–7, 253, 300 dependent, 14, 535 discrete, 6, 253 explanatory, 14 independent, 14, 535 qualitative, quantitative, random, 3, 253 Variance, 125–132 binomial distribution, 274 definition of, 127 formula, 127 population, 127 probability distribution, 262–264 sample, 128 short-cut formula, 129 unbiased estimate, 128 uses of, 132 Variances equal, 513–514 unequal, 513–514 Venn diagram, 190–191, 203, 218 W Weighted estimate of p, 505 Weighted mean, 115 Wilcoxon rank sum test, 683–686 Wilcoxon signed-rank test, 688–692 Within-group variance, 631 Y Yates correction for continuity, 613, 617 y-intercept, 552–555 Z z-score, 142–143 z-test, 413 z-test for means, 413–421, 473–479 z-test for proportions, 437–441, 504–508 z-values (score), 304 blu38582_IBC.qxd 9/13/10 Table F d.f 7:08 PM Page The t Distribution Confidence intervals 80% 90% 95% 98% 99% One tail, A 0.10 0.05 0.025 0.01 0.005 Two tails, A 0.20 0.10 0.05 0.02 0.01 3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.309 1.307 1.306 1.304 1.303 1.301 1.299 1.297 1.296 1.295 1.294 1.293 1.292 1.291 1.290 1.283 1.282 1.282a 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.694 1.691 1.688 1.686 1.684 1.679 1.676 1.673 1.671 1.669 1.667 1.665 1.664 1.662 1.660 1.648 1.646 1.645b 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 2.037 2.032 2.028 2.024 2.021 2.014 2.009 2.004 2.000 1.997 1.994 1.992 1.990 1.987 1.984 1.965 1.962 1.960 31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.449 2.441 2.434 2.429 2.423 2.412 2.403 2.396 2.390 2.385 2.381 2.377 2.374 2.368 2.364 2.334 2.330 2.326c 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.738 2.728 2.719 2.712 2.704 2.690 2.678 2.668 2.660 2.654 2.648 2.643 2.639 2.632 2.626 2.586 2.581 2.576d 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 32 34 36 38 40 45 50 55 60 65 70 75 80 90 100 500 1000 (z) ϱ a This value has been rounded to 1.28 in the textbook This value has been rounded to 1.65 in the textbook c This value has been rounded to 2.33 in the textbook d This value has been rounded to 2.58 in the textbook One tail Two tails b Source: Adapted from W H Beyer, Handbook of Tables for Probability and Statistics, 2nd ed., CRC Press, Boca Raton, Fla., 1986 Reprinted with permission Area ␣ t Area ␣ Ϫt Area ␣ ϩt blu38582_IBC.qxd 9/13/10 7:08 PM Page Glossary of Symbols a y intercept of a line MR Midrange a Probability of a type I error MSB Mean square between groups b Slope of a line MSW Mean square within groups (error) b Probability of a type II error n Sample size C Column frequency N Population size cf Cumulative frequency n(E) Number of ways E can occur nCr Number of combinations of n objects taking r objects at a time n(S) Number of outcomes in the sample space O Observed frequency C.V Critical value P Percentile; probability CVar Coefficient of variation p Probability; population proportion D Difference; decile pˆ Sample proportion ᎐ _ D Mean of the differences p d.f Degrees of freedom P(B͉A) Conditional probability d.f.N Degrees of freedom, numerator P(E) d.f.D Degrees of freedom, denominator E Event; expected frequency; maximum error of estimate ᎐ Weighted estimate of p ᎐ Probability of an event E P(E ) Probability of the complement of E n Pr Number of permutations of n objects taking r objects at a time E Complement of an event p e Euler’s constant Ϸ 2.7183 Pi Ϸ 3.14 Q Quartile E(X) Expected value q f Frequency Ϫ p; test value for Tukey test qˆ F F test value; failure Ϫ pˆ _ FЈ Critical value for the Scheffé test q R Ϫ p– MD Median FS Scheffé test value GM Geometric mean Range; rank sum blu38582_IBC.qxd 9/13/10 7:08 PM Page H Kruskal-Wallis test value rS Spearman rank correlation coefficient H0 Null hypothesis S Sample space; success H1 Alternative hypothesis s Sample standard deviation HM Harmonic mean s k Number of samples s Sample variance Population standard deviation l Number of occurrences for the Poisson distribution s Standard deviation of the differences sX Standard error of the mean sD Standard error of estimate ͚ Summation notation sest SSB Sum of squares between groups ws Smaller sum of signed ranks, Wilcoxon signed-rank test SSW Sum of squares within groups X sB2 Between-group variance Data value; number of successes for a binomial distribution sW2 Within-group variance X Sample mean t t test value x Independent variable in regression ta͞2 Two-tailed t critical value X GM Grand mean m Population mean Xm Midpoint of a class ᎐ ᎐ Population variance mD Mean of the population differences Chi-square mX Mean of the sample means y Dependent variable in regression w Class width; weight yЈ Predicted y value r Sample correlation coefficient z z test value or z score R Multiple correlation coefficient za͞2 Two-tailed critical z value r2 Coefficient of determination ! Factorial r Population correlation coefficient ... hypothesis, since the P-value is greater than 0.05 See Figure 8? ?20 Figure 8? ?20 P-Values and A Values for Example 8–7 Area = 0. 029 4 Area = 0. 029 4 Area = 0. 025 Area = 0. 025 Step 8 .2 Summarize the results... 32 35 25 30 26 .5 26 25 .5 29 .5 32 30 28 .5 30 32 28 31.5 29 29 .5 30 34 29 32 27 28 33 28 27 32 29 29 .5 Source: www.healthepic.com Salaries of Government Employees The mean salary of federal government... Almanac 22 Farm Sizes Ten years ago, the average acreage of farms in a certain geographic region was 65 acres The standard deviation of the population was acres A recent study consisting of 22