GERTRUDE COX: SPREADING THE GOSPEL ACCORDING TO ST. GERTRUDE
Step 2 Decide on the significance level, α . Step 3 Compute the value of the test statistic
z= pˆ− p0 p0(1− p0)/n and denote that valuez0.
CRITICAL-VALUE APPROACH OR P-VALUE APPROACH
Step 4 The critical value(s) are
±zα/2 −zα zα
or or
(Two tailed) (Left tailed) (Right tailed) Use Table II to find the critical value(s).
/2
0 z
Left tailed
−z
Right tailed
0 z
z
0 z
Two tailed
−z/2 z/2
Do not rejectH0 Reject
H0
Reject H0
Do not rejectH0 Reject
H0
Do not rejectH0 Reject H0
/2
Step 5 If the value of the test statistic falls in the rejection region, reject H0; otherwise, do not rejectH0.
Step 4 Use Table II to obtain the P-value.
0
P- value 0
0 z z z
P- value
Two tailed Left tailed Right tailed
z0 z0
−|z0| |z0| P- value
Step 5 If P≤α, reject H0; otherwise, do not reject H0.
Step 6 Interpret the results of the hypothesis test.
Solution Becausen=1053 andp0=0.50 (50%), we have
np0=1053ã0.50=526.5 and n(1−p0)=1053ã(1−0.50)=526.5. Because bothnp0andn(1−p0)are 5 or greater, we can apply Procedure 11.2.
Step 1 State the null and alternative hypotheses.
Letpdenote the proportion of all U.S. adults who favored passage of the economic stimulus package. Then the null and alternative hypotheses are, respectively,
H0: p=0.50 (it is not true that a majority favored passage) Ha: p>0.50 (a majority favored passage).
Note that the hypothesis test is right tailed.
Step 2 Decide on the significance level,α.
We are to perform the hypothesis test at the 5% significance level; so,α=0.05.
Step 3 Compute the value of the test statistic
z= pˆ− p0 p0(1− p0)/n.
We haven=1053 andp0=0.50. The number of U.S. adults surveyed who favored passage was 548. Therefore the proportion of those surveyed who favored passage is pˆ =x/n=548/1053=0.520 (52.0%). So, the value of the test statistic is
z= 0.520−0.50
√(0.50)(1−0.50)/1053 =1.30.
CRITICAL-VALUE APPROACH OR P-VALUE APPROACH Step 4 The critical value for a right-tailed test iszα.
Use Table II to find the critical value.
For α=0.05, the critical value is z0.05=1.645, as shown in Fig. 11.2A.
FIGURE 11.2A
z Do not rejectH0 RejectH0
1.645 0
0.05
Step 5 If the value of the test statistic falls in the rejection region, rejectH0; otherwise, do not rejectH0.
From Step 3, the value of the test statistic isz=1.30, which, as Fig. 11.2A shows, does not fall in the rejection region. Thus we do not rejectH0. The test results are not statistically significant at the 5% level.
Step 4 Use Table II to obtain the P-value.
From Step 3, the value of the test statistic isz=1.30.
The test is right tailed, so the P-value is the probability of observing a value ofz of 1.30 or greater if the null hypothesis is true. That probability equals the shaded area in Fig. 11.2B, which by Table II is 0.0968.
FIGURE 11.2B
z
z= 1.30 0
P-value
Step 5 If P≤α, rejectH0; otherwise, do not rejectH0.
From Step 4, P=0.0968. Because the P-value ex- ceeds the specified significance level of 0.05, we do not reject H0. The test results are not statistically signifi- cant at the 5% level, but (see Table 9.8 on page 360) the data do provide moderate evidence against the null hypothesis.
Step 6 Interpret the results of the hypothesis test.
Interpretation At the 5% significance level, the data do not provide sufficient evidence to conclude that a majority of U.S. adults favored passage of the economic stimulus package.
Report 11.2
Exercise 11.65 on page 459
Note: Example 11.6 illustrates how statistical results are sometimes misstated. The headline on the Web site featuring the survey read, “In U.S., Slim Majority Supports Economic Stimulus Plan.” In fact, the poll results say no such thing. They say only that a slim majority (52%) of thosesampledsupported the economic stimulus plan. As we have demonstrated, at the 5% significance level, the poll does not provide sufficient evidence to conclude that a majority of U.S. adults supported passage of the economic stimulus plan.
THE TECHNOLOGY CENTER
Most statistical technologies have programs that automatically perform the one- proportionz-test. In this subsection, we present output and step-by-step instructions for such programs.
EXAMPLE 11.7 Using Technology to Conduct a One-Proportion z -Test
Economic Stimulus Of 1053 U.S. adults who were asked whether they favored or opposed passage of a new 800 billion dollar economic stimulus package, 548 said that they favored passage. Use Minitab, Excel, or the TI-83/84 Plus to decide, at the 5% significance level, whether the data provide sufficient evidence to conclude that a majority of U.S. adults favored passage.
Solution Let pdenote the proportion of all U.S. adults who favored passage of the economic stimulus package. The task is to perform the hypothesis test
H0: p=0.50 (it is not true that a majority favored passage) Ha: p>0.50 (a majority favored passage)
at the 5% significance level. Note that the hypothesis test is right tailed.
We applied the one-proportion z-test programs to the data, resulting in Out- put 11.3. Steps for generating that output are presented in Instructions 11.2.
OUTPUT 11.3 One-proportionz-test on the data on passage of the economic stimulus package
Test and CI for One Proportion
Test of p = 0.5 vs p > 0.5
95% Lower
Sample X N Sample p Bound Z-Value P-Value 1 548 1053 0.520418 0.495095 1.33 0.093 Using the normal approximation.
MINITAB
EXCEL TI-83/84 PLUS
UsingCalculate
UsingDraw
As shown in Output 11.3, the P-value for the hypothesis test is 0.093. Because theP-value exceeds the specified significance level of 0.05, we do not rejectH0. At the 5% significance level, the data do not provide sufficient evidence to conclude that a majority of U.S. adults favored passage of the economic stimulus package.
INSTRUCTIONS 11.2 Steps for generating Output 11.3
MINITAB EXCEL TI-83/84 PLUS
1 ChooseStat➤Basic Statistics➤ 1 Proportion. . .
2 Select theSummarized data option button
3 Click in theNumber of events text box and type548
4 Click in theNumber of trialstext box and type1053
5 Check thePerform hypothesis testcheck box
6 Click in theHypothesized proportiontext box and type0.50
7 Click theOptions. . . button 8 Click the arrow button at the right
of theAlternativedrop-down list box and selectgreater than 9 Check theUse test and interval
based on normal distribution check box
10 ClickOKtwice
1 Store the sample size, 1053, and the number of successes, 548, in ranges named n and x,
respectively
2 ChooseDDXL➤Hypothesis Tests
3 SelectSumm 1 Var Prop Test from theFunction type drop-down list box
4 Specify x in theNum Successes text box
5 Specify n in theNum Trialstext box
6 ClickOK
7 Click theSet p0button 8 Click in theHypothesized
Population Proportiontext box and type0.50
9 ClickOK
10 Click the.05button 11 Click thep>p0button 12 Click theComputebutton
1 PressSTAT, arrow over to TESTS, and press5 2 Type0.50forp0and press
ENTER
3 Type548forxand pressENTER 4 Type1053fornand press
ENTER
5 Highlight >p0and press ENTER
6 Press the down-arrow key, highlightCalculateorDraw, and pressENTER
Exercises 11.2
Understanding the Concepts and Skills
11.57 Of what procedure is Procedure 11.2 a special case? Why do you think that is so?
11.58 The paragraph immediately following Example 11.6 dis- cusses how statistical results are sometimes misstated. Find an article in a newspaper, magazine, or on the Internet that misstates a statistical result in a similar way.
In each of Exercises11.59–11.64, we have given the number of successes and the sample size for a simple random sample from a population. In each case, do the following.
a. Determine the sample proportion.
b. Decide whether using the one-proportion z-test is appropriate.
c. If appropriate, use the one-proportion z-test to perform the specified hypothesis test.
11.59 x=8,n=40,H0:p=0.3,Ha:p<0.3,α=0.10 11.60 x=10,n=40,H0:p=0.3,Ha:p<0.3,α=0.05 11.61 x=35,n=50,H0:p=0.6,Ha:p>0.6,α=0.05 11.62 x=40,n=50,H0:p=0.6,Ha:p>0.6,α=0.01
11.63 x =16,n=20,H0: p=0.7,Ha: p=0.7,α=0.05 11.64 x =3,n=100,H0: p=0.04,Ha: p=0.04,α=0.10 In Exercises11.65–11.70, use Procedure 11.2 on page 456 to per- form an appropriate hypothesis test. Be sure to check the condi- tions for using that procedure.
11.65 Generation Y Online. People who were born between 1978 and 1983 are sometimes classified by demographers as be- longing to Generation Y. According to aForrester Researchsur- vey published inAmerican Demographics(Vol. 22(1), p. 12), of 850 Generation Y Web users, 459 reported using the Internet to download music.
a. Determine the sample proportion.
b. At the 5% significance level, do the data provide sufficient ev- idence to conclude that a majority of Generation Y Web users use the Internet to download music?
11.66 Christmas Presents. The Arizona Republic conducted a telephone poll of 758 Arizona adults who celebrate Christmas.
The question asked was, “In your family, do you open presents on Christmas Eve or Christmas Day?” Of those surveyed, 394 said they wait until Christmas Day.
a. Determine the sample proportion.
b. At the 5% significance level, do the data provide sufficient evi- dence to conclude that a majority (more than 50%) of Arizona families who celebrate Christmas wait until Christmas Day to open their presents?
11.67 Marijuana and Hashish. The Substance Abuse and Mental Health Services Administrationconducts surveys on drug use by type of drug and age group. Results are published inNa- tional Household Survey on Drug Abuse. According to that pub- lication, 13.6% of 18- to 25-year-olds were current users of mari- juana or hashish in 2000. A recent poll of 1283 randomly selected 18- to 25-year-olds revealed that 205 currently use marijuana or hashish. At the 10% significance level, do the data provide suffi- cient evidence to conclude that the percentage of 18- to 25-year- olds who currently use marijuana or hashish has changed from the 2000 percentage of 13.6%?
11.68 Families in Poverty. In 2006, 9.8% of all U.S. families had incomes below the poverty level, as reported by theU.S. Cen- sus Bureauin American Community Survey. During that same year, of 400 randomly selected Wyoming families, 25 had in- comes below the poverty level. At the 1% significance level, do the data provide sufficient evidence to conclude that, in 2006, the percentage of families with incomes below the poverty level was lower among those living in Wyoming than among all U.S. families?
11.69 Labor Union Support. Labor Day was created by the U.S. labor movement over 100 years ago. It was subsequently adopted by most states as an official holiday. In aGallup Poll, 1003 randomly selected adults were asked whether they approve of labor unions; 65% said yes.
a. In 1936, about 72% of Americans approved of labor unions.
At the 5% significance level, do the data provide sufficient evidence to conclude that the percentage of Americans who approve of labor unions now has decreased since 1936?
b. In 1963, roughly 67% of Americans approved of labor unions.
At the 5% significance level, do the data provide sufficient evidence to conclude that the percentage of Americans who approve of labor unions now has decreased since 1963?
11.70 An Edge in Roulette? Of the 38 numbers on an Amer- ican roulette wheel, 18 are red, 18 are black, and 2 are green. If the wheel is balanced, the probability of the ball landing on red is
18
38 =0.474. A gambler has been studying a roulette wheel. If the wheel is out of balance, he can improve his odds of winning. The gambler observes 200 spins of the wheel and finds that the ball lands on red 93 times. At the 10% significance level, do the data provide sufficient evidence to conclude that the ball is not landing on red the correct percentage of the time for a balanced wheel?
In each of Exercises11.71–11.76, use the technology of your choice to conduct the required hypothesis test.
11.71 Recovering From Katrina. A CNN/USA TODAY/
Gallup Poll, conducted in September, 2005, had the headline
“Most Americans Believe New Orleans Will Never Recover.”
Of 609 adults polled by telephone, 341 said they believe the hurricane devastated the city beyond repair. At the 1% signifi- cance level, do the data provide sufficient evidence to justify the headline? Explain your answer.
11.72 Delayed Perinatal Stroke. In the article “Prothrombotic Factors in Children With Stroke or Porencephaly” (Pediatrics Journal, Vol. 116, Issue 2, pp. 447–453), J. Lynch et al. com- pared differences and similarities in children with arterial is- chemic stroke and porencephaly. Three classification categories were used: perinatal stroke, delayed perinatal stroke, and child- hood stroke. Of 59 children, 25 were diagnosed with delayed perinatal stroke. At the 5% significance level, do the data provide sufficient evidence to conclude that delayed perinatal stroke does not comprise one-third of the cases among the three categories?
11.73 Drowning Deaths. In the article “Drowning Deaths of Zero to Five Year Old Children in Victorian Dams, 1989–2001” (Australian Journal of Rural Health, Vol. 13, Issue 5, pp. 300–308), L. Bugeja and R. Franklin examined drowning deaths of young children in Victorian dams to identify common contributing factors and develop strategies for future prevention. Of 11 young children who drowned in Victorian dams located on farms, 5 were girls. At the 5% significance level, do the data provide sufficient evidence to conclude that, of all young children drowning in Victorian dams located on farms, less than half are girls?
11.74 U.S. Troops in Iraq. In aZogby International Poll, con- ducted in early 2006 in conjunction with Le Moyne College’s Center for Peace and Global Studies, roughly 29% of the 944 mil- itary respondents serving in Iraq in various branches of the armed forces said the United States should leave Iraq immediately. Do the data provide sufficient evidence to conclude that, at the time, more than one-fourth of all U.S. troops in Iraq were in favor of leaving immediately? Useα=0.01.
11.75 Washing Up. A recent Harris Interactivesurvey found that 92.0% of 1001 American adults said they always wash up after using the bathroom.
a. At the 5% significance level, do the data provide sufficient ev- idence to conclude that more than 9 of 10 Americans always wash up after using the bathroom?
b. Repeat part (a), using a 1% level of significance.
11.76 Illegal Immigrants. ANew York Times/CBS Newspoll asked a sample of U.S. adults whether illegal immigrants who have been in the United States for at least 2 years should be allowed to apply for legal status. Of the 1125 people sampled, 62% replied in the affirmative. At the 1% significance level, do the data provide sufficient evidence to conclude that less than two-thirds of all U.S. adults feel that illegal immigrants who have been in the United States for at least 2 years should be allowed to apply for legal status?
11.3 Inferences for Two Population Proportions
In Sections 11.1 and 11.2, you studied inferences for one population proportion. Now we examine inferences for comparing two population proportions. In this case, we have two populations and one specified attribute; the problem is to compare the proportion
of one population that has the specified attribute to the proportion of the other popula- tion that has the specified attribute. We begin by discussing hypothesis testing.
EXAMPLE 11.8 Hypothesis Tests for Two Population Proportions
Eating Out Vegetarian Zogby Internationalsurveyed 1181 U.S. adults to gauge the demand for vegetarian meals in restaurants. The study, commissioned by the Vegetarian Resource Groupand published in theVegetarian Journal, polled inde- pendent random samples of 747 men and 434 women. Of those sampled, 276 men and 195 women said that they sometimes order a dish without meat, fish, or fowl when they eat out.
Suppose we want to use the data to decide whether, in the United States, the percentage of men who sometimes order a dish without meat, fish, or fowl is smaller than the percentage of women who sometimes order a dish without meat, fish, or fowl.
a. Formulate the problem statistically by posing it as a hypothesis test.
b. Explain the basic idea for carrying out the hypothesis test.
c. Discuss the use of the data to make a decision concerning the hypothesis test.
Solution
a. The specified attribute is “sometimes orders a dish without meat, fish, or fowl,” which we abbreviate throughout this section as “sometimes orders veg.”
The two populations are
Population 1: All U.S. men Population 2: All U.S. women.
Let p1and p2denote the population proportions for the two populations:
p1=proportion of all U.S. men who sometimes order veg p2=proportion of all U.S. women who sometimes order veg.
We want to perform the hypothesis test
H0: p1= p2(percentage for men is not less than that for women) Ha: p1< p2(percentage for men is less than that for women).
b. Roughly speaking, we can carry out the hypothesis test as follows:
1. Compute the proportion of the men sampled who sometimes order veg, pˆ1, and compute the proportion of the women sampled who some- times order veg, pˆ2.
2. If pˆ1is too much smaller than pˆ2, rejectH0; otherwise, do not rejectH0. c. To use the data to make a decision concerning the hypothesis test, we apply
the two steps just listed. The first step is easy. Because 276 of the 747 men sampled sometimes order veg and 195 of the 434 women sampled sometimes order veg,x1=276,n1=747,x2 =195, andn2=434. Hence,
ˆ p1= x1
n1 = 276
747 =0.369(36.9%) and
ˆ p2= x2
n2 = 195
434=0.449(44.9%).
For the second step, we must decide whether the sample proportion ˆ
p1=0.369 is less than the sample proportion pˆ2=0.449 by a sufficient amount to warrant rejecting the null hypothesis in favor of the alternative hy- pothesis. To make that decision, we need to know the distribution of the differ- ence between two sample proportions.
The Sampling Distribution of the Difference Between Two Sample Proportions for Large and Independent Samples Let’s begin by summarizing the required notation in Table 11.2.
TABLE 11.2 Notation for parameters and statistics when two population proportions are being considered