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Statistical Methods for the Social Sciences

Alan Agresti, Barbara Finlay

Chapter 7

Comparison of Two Groups - all with Video Answers

Educators

Chapter Questions

01:17

Problem 1

In the 1982 Gencral Social Survey. 350 subjects reported the time spent every day walching television. The sample mean was 41 hours, with standard deviation 3.3. In the 1994 General Social Survey. 1965 subjects reported a mean time spent watching television of 2.8 bours, with standard deviation 2.0 .
a) Construct the $95 \%$ confidence interval fol the difference between the means in 1982 and 1994 Interpret. Is it plausible that the mean was the same in both years?
b) Set up the hypotheses of a significance resi to analyze whether the population means differ in 1982 and 1994.
c) For this test, consuruct the test statistic. Report and interpret the $P$-value.
d) Do you thunk the distribution of the response variable is approximately normal? Why? Does this alfect the validity of your inference? Explain.

Tyler Moulton
Tyler Moulton
Numerade Educator

Problem 2

Refer to Table 6.2 and Example 67 . Table 7.9 is a computer printout that summanzes the responses on the seven-point poliucal ideology scale for the surveys in 1978 and in 1994.
a) Arc these samples independent or dependent?
b) Construct a $95 \%$ confidencc interval for the difference between the inean political ideology in 1994 and 1978 Interpret.
c) Show how to test the hypothesis that the population mean political ideology was equal in 1978 and 1994. against the alternative of a difference. Report the $P$-yalue. and interpret.
d) Show the correspondence between the results of the inferences in (b) and (c) Wuch inference is more informative? Explain.

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02:22

Problem 3

Reter to Problem 5.39 and Table 54. Construct a $99 \%$ confidence interval for the difference between the mean nunber of hours cmployed per week for inen and for women. Interpret Based on this interval, indicate what result would occur in a two-sample test comparing the means. Explain.

Maxime Rossetti
Maxime Rossetti
Numerade Educator
04:12

Problem 4

A recent study compared substance use, delinquency psychological well-being, and social support ainong various faniily types. Ior a samplc of urban Afncan-American adolescent males (M. Zimmerman et al., Child Development. Vol 66. 1995, pp. 1598-1613) The sample contained 108 subjects from single-mother households and 44 from households with both biological parents. Delinquency was measured with four dichotomous items asking the youths if they had ever (i) been in trouble with the police. (ii) been arrested, (iii) appeared before a juvemje court. or (iv) been incarcerared in the last ycar, the delinquency score was the number of items to which a youth answered yes. This score had a mean of 19 and standard deviation of I .5 for single-mother houselsolds and a inean of 1.4 and standard deviation of 1.5 for houscholds with both biological pasents.
a) Construct a $99 \%$ confidence interval for the difference be(ween the true means for the populations represented by this sample. Interpret
b) Test the hypothesis of equal means against the alternative of unequal means. Report the $P$-taluc and interpret.
c) Explain the correspondence betueen the results of (a) and (b)
d) Do you think that delinquency is nonnally distributed for each houseliold type? Explain. and indicate whethee this has any eflect oul your inferential analyses.

James Kiss
James Kiss
Numerade Educator
01:12

Problem 5

Refer to the preceding exercise. The youths also responded to a battery of questions that provides a measure of perceived parental support. This measure had sample means of 46 $(s=9)$ for the single-mother households and $42(s=10)$ for the houscholds with both bio-Iogical parents. Consider the conclusion, "The mean parental support was 4 units higher for the single-mother huuseholds. If the true mcans were equal. a difference of this size could be expected only $2 \%$ of the time For samples of this size. $95 \%$ of the time onc would expect this difference to be within 34 of the true value "Explain how this conclusion refers to the results of a) a confidence interval. b) a test.

Tyler Moulton
Tyler Moulton
Numerade Educator
05:28

Problem 6

For 30 female assistant professors selected at random from arts and science departments in a large state university system, the mean academic year salary is $$\$ 41,000$$ The mean salary for a random sample of 50 male assistant professors is $$\$ 41,800$$. The sainple standard deviations are $$\$ 3000$$ for females and $$\$ 3400$$ for males.
a) Consiruct a $95 \%$ confidence interval for the difference in mean saldry berween male and feınale assistant protessors in this system. Interpret.
b) Perfonn a significance test of whether the truc means are identical. Interpuet the $P$ value
c) What decision would you make in the test, using $\alpha=05$ ? Explain how this decision relates to information displayed by the confidence interval.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
11:26

Problem 7

Seegmiller et al. (J. Psycholog); Vol. 102, 1979, pp. 215-224) sudied the effect of tester sex on sex-role differentiation scores of a sample of preschool chuldren. For the Occupational Preference Test, children were asked to give three choices of what they wanted to be when they grew up. Each occupation was rated on a scale from 1 (traditionally feminine) to 5 (traditionally masculne), and a child's scorc was the mean of the threc selections. a) The mean score was 4.7 for the 152 boys $(s=.7)$, and the mean scorc was 3.1 for the 140 giris $(s=1.3)$. Construct a $95 \%$ confidence inter al to summarize how much more masculine the boys' choices tended to be than the girls' choices. Interpret
b) When the lester was male, the 50 girls had $\bar{Y}=2.9$ and $s=14$. whereas when the tester was fentale, the $90 \mathrm{gr} / \mathrm{s}$ had $\bar{Y}=32$ and $s=1.2$. Test the hypothesis that the tue mean is the same for each testes sex, against the alternalive that they differ. Repont the $P$-valuc and interpret
c) For the test in (b), what is your conclusion. using $\alpha=.05$ ? If you have made an error. is il a Type I error or a Type 11 error?

Erin Moser
Erin Moser
Numerade Educator
04:05

Problem 8

Families headed by low-wage workers may be more likely to be living in poverty now than two decades ago. According to a report by the Center on Budget and Policy Priorities, in $19777.7 \%$ of famalies with children in which the head of the household worked were below the poverty level: in 1993 , this poverty rate was $11.4 \%$. Report a point estimate to compare the statistics from 1977 and 1993, and interpres.

KM
Kanishk Mishra
Numerade Educator
03:05

Problem 9

Chatierjee et al (1995.p 132) described two studies about the effect of condoms in reducing the spread of AIDS. One two-year lalian study followed heterosexual couples where one partner was infecied with the HIV virus. Of 171 couples who always used condoms, 3 partners became infected with HIV, while of 55 couples who did not always use condoins, 8 partners became infected.
a) Estimate the infection rates for the two groups. and construct a $95 \%$ confidence interval to compare them. InterpreL
b) Test whether the rales are significantly dillerent Report the $P$-value and interpret.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:49

Problem 10

For a randem sample of 1600 Canadians taken in January, 880 people indicate approval of the prime minister's performance. A similar poll a month later of a separate random sample of 1600 Canadians has a favorable rating by 944 people. Let $\pi_1$ denote the true proportion in January that approve of the prime munister 's perfonnance, and let $\pi_2$ represent the true proportion in February.
a) Calculate point estimales of these proportions. Calculate the difference. and iuterpret
b) Construct a $99 \%$ confidence interval for $\pi_2-\pi_1$. Interpiet
c) Based on the interval in (b), explain why (i) therc may have been no change in suppott (ii) if an increase in supporl occurred, it may have been faill important. (iii) if a decrease in support occurred, it was probably so small as to be substantively unimportant.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:17

Problem 11

The National Health Interview Survey conducted of 42.000 adults by the U.S. National Center fot Health Statistics in 1991 indicated that $25.6 \%$ of adults weie current smokers A sinilar study conducted in 1965 indicated that $42.4 \%$ were cunent smokers
a) Construct a $95 \%$ confidence interval for the ditlerence bciween the proportion of current sinokers in 1965 and the proportion of current smokers in 1991 (Assume that the sample size was the saine in 1965.) Interpret.
b) L'sing the standard error from the confidence interval. test that there w as no change in the proportion of current smokes Report the test statistic and $P$-ialue, and intcrprel.
c) What statisucal factors cause the test statistic in (b) to be so large?

Neel Faucher
Neel Faucher
Numerade Educator

Problem 12

Refer to Example 7.3.
a) Find point estimates of the proportions of Anericans who disagree with the statement abour women's roles in 1982 and in 1994.
b) Find a $95 \%$ confidence unterval for the change in the disagrec proportion, and compare it to the interval found following Example 7.3 for the change in the agree proportion. Explain the correspondence
c) Conduct a test for equality of the disagıee proportions in 1982 and 1994 Compare to the result of the test for the agree proportions, and explain the correspondence.

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01:20

Problem 13

The General Social Survey asks, "Do you favor or oppose the death penalty for persons convicted of nurder?" In 1982, 154 were in fav or and 165 opposed In 1994. 22 is were in favor and $$\$ 80$$ were opposed. Construct a 95 confidence interval for the difference between the proportions in favor of the death penalty in 1994 and in 1982. Interpuet

Narayan Hari
Narayan Hari
Numerade Educator
02:19

Problem 14

In a survey conducted in 1992 by the Wight State Unirersity School of Medicine. scnior high school students werc asked if they had ever used nuaijuana. Of the females sampled. 445 said yes and 675 said no: of the males sampled. 515 said yes and 641 said no. Compare and interpret these tesponses,
a) Usiug a $95 \%$ confidence interval.
b) Using a signilicance test.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
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Problem 15

A study using a probability sample from the 1984 National Health Interview: Study (R. Coward et al. The Gerontologist. Vol. 35, 1995. pp. 24-34) reported that the percent of lespondents with income below the poverty level was $11.3 \%$ for residepts of metropolitan areats ( $n=7331$ ) and $20.6 \%$ for residents of nonmetrojol itan areas $(n=4235)$ Construct and interpret a $99 \%$ confidence interval for the difference in poverty rates benveen nonmetropolitan and metropolitan areas What is the effect of the very large sample sizes on the interval?

Rashmi Sinha
Rashmi Sinha
Numerade Educator

Problem 16

Table $\overline{.} 10$ shous results from a recent General Social Survey on two variables. gender and whether onc helieves in an afterife.
a) Which of the two variables is more nanurally treated as the response variable?"
b) Let $\pi_1$ denote the population proportion of feunales who would respond "ycs" to whelher they believe in an afterlife, and let $\pi_2$ denore the population piopontion for inales Construct a $95 \%$ confidence inlerval for $\pi_1-\pi_2$, and interpret.
c) Is it plausible that $\pi_1=\pi_2$ ? Explain.
d) Suppose, unknown to us, that $\pi_1=.78$ and $\pi_2=.71$. Does the confidence inter 1 al in
(b) contain the valuc of the paramcter it is designed to estimate? Explain
e) Conduct a test of $H_0: \pi_1=\pi_2$ Make $\alpha$ decision using $\alpha=.05$ If you have made an error, what type of error is it. Type I or Type 11?

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04:08

Problem 17

The 1974-1975 Gieneral Social Survey asked "Do you helicve that women should take care of tunning their homes and leave ruuning the country up to nen ${ }^{2 \prime}$ For 1305 malc respondents. 465 replied yes and 840 replicd no For 1566 feinale Iespondents. 555 replied yes and 1.011 replied no.
a) Find $295 \%$ confidence interval for the difference berween the proportion of males and the proportion of fernales who would respond yes Interpret the intel val
b) Of 487 respondents having 8 years or less education, 330 replied yes. Of 902 respondents having at least 13 years of education. 153 replied jcs Find a $95 \%$ confidence interval tor the difference belween the proportion at the low education level and the proportion at the high education level who would respond yes. Interpret the interval
c) Which vanable. gender or educational level. seems to have had thc greater influence on opinion ${ }^7$ In other words. did opinion tend to differ more between men and women, or between the most and least educated?

James Kiss
James Kiss
Numerade Educator
03:30

Problem 18

Nen'sweek magazine (September 19. 1983) rcported results of a poll to investigate whcther there was a differcnce berw een the perocntage of wonen and the percentage of men who approved of the way Ronald Reagan was bandling his job as president Of 511 women interviewed, $40 \%$ indicated appiotal. OI 507 men, $49 \%$ incicated approval
a) Find a $90 \%$ confidence interval tor the difference between the proportion of men and the proportion of women who approved of Reagan's performance. What assumption do you make to nsc this merhod?
b) interpret the interval obtaincd in part (a). Can you conclude that these is a gender gap?

Marc Lauzon
Marc Lauzon
Numerade Educator
02:19

Problem 19

A study of bulinia dmong college nomen (J. Kern and T. Hastings, Journal of Clinical Psychology; Vol. 51, 1995. p. 499) considered the effect of childhood sexual abuse on vanous components of a Fanuly Environment Scalc. For a measure of fanily cohesion. the sample mcan for the bulimic students was 2.0 for 13 sexually abused students $(s=21)$ and 4.8 for 17 nonabused students $(s=3.2)$
a) Use small-sample methods to construct a $95 \%$ confidence inter al for the difference in mean faunily cohesion tor sexually abused students and nonabused students. Interprct What assumptions have you made?
b) Find the $P$-value for testing whether the popularion medns are equal, against a twosided altemative. Interprct

Jameson Kuper
Jameson Kuper
Numerade Educator
03:58

Problem 20

The property values for three homes selected at random in the Forest Ridge subdivision are (in thousands of dollars) 110.120.130 The property values for two homes selectod al randon in the Hennitage subdivision arc 180.200.
a) Use small-sample methods to construct a $90 \%$ confidence interval for the difference in mean pioperty values between the IEermitage and Forest Radge subdivisions Interpret. What assumptions have you made?
b) Find the $P$-value for testing $H_n: \mu_1=\mu_2$ ayainst $H_a \cdot \mu_1 \neq \mu_2$ InterpreL. Explain the correspondence between results of the confidence interval and results of the lest c) Using connputer software, contpale results of the $t$ test that assumes equal population variances to that of the approximate $f$ test that does not require that assumption. Intccpref

Jameson Kuper
Jameson Kuper
Numerade Educator
06:09

Problem 21

Fol the WWW data for Universiry of Florida soctal science students. described in Problem 1.7, the responses on political ideology had a mean of 318 and standard deviation of 1.72 for the 51 nonvegetarian students and a mean of 2.22 and standard deviation of 67 for the 9 vegelarian students.
a) Show how to compare the means using a $90 \%$ confidence interval. Interpret.
b) When we use software to compare the ineans with a significance test. we obtain the printout
Explain why the results of the two tests differ so nuch, and give your conclusion about whethes the true ineans are equal.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:34

Problem 22

Three editors of large-city newspapers are selected at random and asked to rate on a score of 0 -100 the fainuess of the news media in political reporting ( 0 corresponds to very unfair). The same question is posed to three editors of small-town paspers. The large-city editors gave scores of $55.90,95$, and the small-rown editors gave scores of $40,60,80$. Table 7.11 shows a computer prontout for tesults of a two-sample comparison of means Explain how to interpret results from this printout

Jameson Kuper
Jameson Kuper
Numerade Educator

Problem 23

Refer to Exanple 7.5. The young adults wcre also asked whether they had ever had a same-gender sexuel relationship. Table 7.12 shows results. Use software to test whethor the probability of this is higher for thuse raised by lesbian mothers. Interpret.

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Problem 24

Refer to the previous problein. Subjects were also asked whether they had eves had a same-gender scxual attraction. Table 7.13 shows a compute printout with icsults of the analy sis. Interpiec.

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03:37

Problem 25

Refer to Problem 6.12. Compare the proportions of biacks and whites who think that affirnative action has been good Ior the country
a) Using a $95 \%$ confidence interval Juterpret
b) Using a significance test. Report the $P$-value, and interpret

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:33

Problem 26

Reler to Examples 74 and 76. Construct and interpre! a $90 \%$ confidence interval for $\mu_2-\mu_1$ :
a) In the independent samples case
b) In the dependent samples case.
c) Conpare results of the two analyses.

Jameson Kuper
Jameson Kuper
Numerade Educator
12:40

Problem 27

A recent study compared ficrsonality chatacuenstics between 49 adult children of alcoholics and a conurol group inatclued on age and gender (D. Baker and L. Stepbenson. .Iournal of Clinical Psichology, Vol 51. 1995. p. 694). On a measure of well-heing, the 49 children of alcoholics had a mean of $26.1(s=7.2)$ and the 49 subjocts in the control group had a mean of $28.8(s=6.4)$ The difference soores between the marched subjects from the two groups had a mean of 2.7 ( $s=9.7$ ).
a) Construct a $95 \%$ confidence interval for the difference betu oen means (i) ueating the samples as independent, (ij) treating the samples ds dependent. Incespret.
b) Conduct a test of cquality of the iwo means (i) treating the samples as independent, (ii) treating the sainples as dependent. Report the $P$-t alucs and interpiet.
c) Compart results of the independent-samples and dependent-samples analy ses. (Note. Matching on age and gender does not nomnully result in a "correlation" between responses in the two sainples, so the results are siunilar.)

Robin Corrigan
Robin Corrigan
Numerade Educator

Problem 28

Reter to the previous problem. For the 29 pairs of women. the authors reported a mean of 24.8 on the well-being mcasurc for the children of alcoholics, and a nean of 29.0 for the control group They reported a $t$-score of 2.67 for the test comparing the means. Assuming that this is the result of a dependent-samples analysis, report the $P$-value, and interprel.

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01:51

Problem 29

Refer to Problem 6.26. The changes in heart rate for 15 males had a mean of -1.22 with standard deviation 19.7. Show how to compare the inean changes for females and males. a) using a $95 \%$ confidence interval. b) using a significancc test Interpret in each case.

Akhil Choudhary
Akhil Choudhary
Numerade Educator
03:15

Problem 30

Each of a random sample of ten college freshmen takes a mathematics aptitude lest hoth before and after undergoing an intensive training counse designed to improve such test scores Then. the scores for each studcnt are parred, as shown in Table 7.14.
$$
\begin{aligned}
&\text { TABLE } 7.14\\
&\begin{array}{ccc}
\hline \text { Student } & \text { Bcfore } & \text { After } \\
\hline 1 & 60 & 70 \\
2 & 73 & 80 \\
3 & 42 & 40 \\
4 & 88 & 94 \\
5 & 66 & 79 \\
6 & 77 & 86 \\
7 & 90 & 93 \\
8 & 63 & 71 \\
9 & 55 & 70 \\
10 & 96 & 97 \\
\hline
\end{array}
\end{aligned}
$$
a) Comparc the mean scores after and before the training course by (i) finding the differcnce of the sample means, (ii) finding the mean of the difference scores. Compare
b) Calculate and interpret the $P$-value for testing whether the nean change equals 0 .
c) Coinpare the mean scores afler and before the training coursc by construcung and interpreting a $90 \%$ confidence interval for the population mean difference.
d) Explain the correspondence between the result of the significance test and the result of the confidence interval. What assuinptions does each infescuce make?

Karen Song
Karen Song
Numerade Educator
03:27

Problem 31

A parred-difference experiment dealing with response latencies for noise detection under two conditions used a sample of twelve 9 -month-old children and reported a sample inean difference of 70.1 and standard deviation of 49.4 for the differences ( 1 Morgan and J . Saffran. Child Development, Vol 66, 1995, pp. 911-936). In theur discussion, the authors reported at statistic of 4.9 having $P<01$ for a two- sided alternative. Show how dhey constructed the $t$ statistic, and confirm the $P$-value.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
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Problem 32

Refer to Examplc 7.7. Table 7.15 show's opinions about spending on the environment and on citics.
a) Find the sample proportion favoring increased spending, for each item
b) Test whether the population proportions are equal. Rcport the $P$-value, and interpret.
c) Construct a $95 \%$ confidence interval for the difference of proportions. Interpret

Rashmi Sinha
Rashmi Sinha
Numerade Educator

Problem 33

Table 716 refers to a sample of juveniles convicled of a felony in Florida in 1987. Matched pars were forned using critenia such as age and the number of prior oflenses Fot each pair, one subject was handled in the juvenile court and the other was transferred to the adull court The response of interest was whether the juvenile was rearrested by the end of 1988 .
a) Test the hypothesis that the ure pruportions rearrested were identical for the adult and juvenile court assignments. Use a lwo-sided alternative.
b) Find a 90 \% confidence inter val for the difference between the true proportions of icanest for the adult and juvenile cour assignments Incrpres.

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Problem 34

A recent sludy used data fiom the Longitudinal Study of Aging to insestigate how older people's health and social characteristics influence how far they live from their children (M. Silv erstein, Demography: Vol 32. 1995, p. 35) Consideı Table 7.17, which showe whether an older subject lives with a child al a given time and then again four years lates. a) Test the hypolhesis that the truc proportion living with a child is the same af the two times. Report and interpref the $P$-value. Silverstein cxpected that as people aged and their health delenorated, they would be more likely to live with children Do these data support this belief?
b) Construct a $95 \%$ confidence interval fos the change in the proportino living with a child Interpret.

Victor Salazar
Victor Salazar
Numerade Educator
11:26

Problem 35

Refer to the WWW data set (Prollem 1.7). Using softwase. use a confidence interval and a test to compare males and females in terms of
a) The mean number of umes reading a newspaper, Interpret
b) Opinions about legalized abortion. Interpret.

Erin Moser
Erin Moser
Numerade Educator
02:40

Problem 36

For the WWW data sel (P1oblem 1.7), comparc Democrat and Republican students in ıerms of polifical ideology.
a) Using back-to-back stem and leaf plots.
b) Using numencal summaries
c) Using inferential statistical melhode. Iuterpret

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:39

Problem 37

For the 60 students in the WWW data set (Problem 1 7). compare the mean weekly time spent watching TV to the mean weekly tinue in sports and other physical exercise. using both descriptive and inferential methods.

Akhil Choudhary
Akhil Choudhary
Numerade Educator

Problem 38

Refer to the data file created in Problem 1.7. For vanables chosen by you instructor, conduct infercntial statistical analyses, using methods of this chaper. Prepare a report that summanzes and interpres your finding In this report. also use graphical and numerical methods presented earlicr in this text to describe the data and. if necessary, to check assumptions you make for your analysis.

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01:25

Problem 39

Two nen drugs have been proposed for treating subjects who suffer from high blood pressure. A sample of ten such subjects were randomly allocated to two groups for an experiment comparing the dugs. The five subjects in one group took drug A, and the five subjects in the other group took drug B. Following three months of treatment. the drop in blood pressure was measured. The sample values were
$$
\begin{array}{ll}
\text { Drug A. } & 0,2,2,3,3 \\
\text { Drug B: } & 3,6,6.78
\end{array}
$$
Analyze these data, and interpret your results.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
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Problem 40

The results in Table 7.18 are from a recent study of physical attractiveness and subjective well-being (E Diener el al.. Journal of Personalin and Social Psvchology, Vol 69, 1995. pp. 120-129). As part of the study, a sample of college stmdents were rated bj a panel on their physicad attractiveness. The table presents the number of dates in the past diree monhhs for students rated in the top or hottom quartile of attractiveness. Analy'ze thesc data. and interpret

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:48

Problem 41

A study of the effects of race on professional football players compensation $(\mathrm{L}$ Kahn, Indusrrial \& Labor Relations Revieu, Vol. 45, 1992, p. 295) reported mean salanes, in thousands of dolldess, in the National Football League in 1989 of 328 for 544 whites ( $s=$ 277) and 315 for 819 blacks $(s=248)$. The mean number of tumes the player had been to the pro bowl was .24 for u hites $(s=.87)$ and 26 fur blacks $(s=.82$ ). The proportions who had received college dcgrees were .39 for whites and 20 for blachs. Andlyze thesc data. Prepare 4 short report. summarizing your analy ses and cunclusions

Nick Johnson
Nick Johnson
Numerade Educator

Problem 42

For a sample of high school students in South Carolina, the percentage who reported they had had sexual intercourse was $65 \%$ for white males $(n=1147), 54 \%$ for white females $(n=1314), 89 \%$ for black males $(n=612)$, and $71 \%$ for black females $(n=820)$ (D.
Richter et al., Journal of School Health, Vol. 63. 1993. p 91) Analyze these data. and interpret.

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03:01

Problem 43

Refer to Problem 7.1. In 1982, 11 of the 350 subjects reported watching no television. In 1994. 74 of the 1965 subjects reported watching no television Analyze these data, and interpret

Sheryl Ezze
Sheryl Ezze
Numerade Educator
07:08

Problem 44

According to Newsweek magazine (July 31. 1995), the mean houschold income in 1994 was $$\$ 40.708$$ for whutes and $$\$ 25.409$$ for blacks The article did not provide sample sizes ol standard deviations. Suppose it had reported a standard error of $$\$ 200$$ for the inean for whites and $$\$ 300$$ for the mean fon blacks. Is this sufficient information to construcl a $99 \%$ confidence interval cumpariug the true difference in nicans? If so, construct the interval and interpret; if nol. explain why not.

Foster Wisusik
Foster Wisusik
Numerade Educator
01:39

Problem 45

In the United Slates, the extimated annual probability that a woman over the age of 35 dies of lung cancer equals 00130 for current smokers and .00012 Ior nonsmokers (M. Pagano and K Gauvreat,, Prnciples of Biostatistics, Belmont, CA: Duxbury Press. 1993. p. 134). a) Calculate and interpret the difference between smokers and nonsmokes in the proportons who die of lung cancer in a given year. Interpret. Are the proportions similar. or quite different, for the two groups?
b) Another way to compare the two propontions is through their ratio. The ralio equals 10 when the proportions are equal, and ratios farthes from 1.0 represent larger differences between the groups. Calculate the ratio of the proportions for smokers and nonsmokers, and interpret. According to this ıneasure, are the proportions similar. or quite dilferent. for the two groups?

Nick Johnson
Nick Johnson
Numerade Educator

Problem 46

Rcfer to the previous exetcise. According to the U.S National Center for Healuh Statistics. in 1991 the prohability that a male between the dges of 20 and 24 was a bonicide viction was .00164 for blacks and 00015 for whites. Compare thesc rates using the Jiffesence of proportions and the rclative risk, and interpret

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Problem 47

A poll by Louis Harris and Associates of 1249 Americans in July 1994 indicaled that $36 \%$ believe in ghosts and $37 \%$ belicve in astrology.
a) Is it valid to compare the proportions using inferential methods for independent samples? Explain.
b) Do you have enough inlormation to compare them using methods for dependent samplcs? Explain

James Kiss
James Kiss
Numerade Educator
00:55

Problem 48

Table 7.19 compares two hospitals on the outcomes of patient admissions for severe pneumonia. Alchotgh patient status is an ordinal ranable. two researchers who analyze the data ureat it as an interval variable. The first researcher assigns the scores $(0,5.10)$ to the three categones. The second rescarcher. believing that the middle category is much closer to the third categor $y$ than to the first, usca the scores $(0,9,10)$ Each researcher calculates the means for the two institutions and identifies the institution with the higher mean as the one having inore success in treating its patients.
a) Find the two means for the scoring system used by (i) the first researcher, (ii) the second researcher. Interprel.
b) Norice that the conclusion depends on the scoring syslem, hence, it one uses methods for quantitdive variables with ordinal data, one should tale care in selecting scores.

Maxime Rossetti
Maxime Rossetti
Numerade Educator
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Problem 49

A recent General Social Survey askcd respondents whether piemarital sex was always wrong. and whether extramarital sex was always wrong (ycs. no). Of the respondents. 144 said jes on both. 86 sarid no on both, 243 said no on presnantal and yes on extramarital. and 2 said yes on premarital and no on extramarital. Analyze thesc dala, and interpre.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
07:16

Problem 50

A short questionnaire measung anxiely is given to a random sample of two students taken from a large introductory stadistics class. Andeely is mcasured both hefore and after the students take a midterm exam in stahstics The scores are 91 before and 80 after for student I and 69 before and 60 after for sludent 2 . Analyze these data.

Jerelyn Nevil
Jerelyn Nevil
Numerade Educator
01:04

Problem 51

A pool of six candidates for three managenal positions includes three finales and three males Table 7.20 show's the resulis.
a) Denote the three females by $F_1, F_2, F_3$ add the three males by $M_1, M_2, M_3$. Identify the 20 distinct samples of size three that can be chosen from these six individuals
b) Let $\hat{\pi}_1$ denote the sample proportion of males sclected and $\hat{\pi}_2$ the sample proportion of femalcs. For Table $7.20, \hat{\pi}_1-\hat{\pi}_{\hat{n}}=(2 / 3)-(1 / 3)=1 / 3$. Or the 20 possible samples, show that 10 have $\hat{\pi}_1-\hat{\pi}_2 \geq 1 / 3$. Thus. if the three managers were randomly selecied. the probability would equal $10 / 20=.5$ of obtaining $\hat{\pi}_1-\hat{x}_2 \geq 1 / 3$ lo fact, this is the reasoning that provides the one-sided $P$-value fot Fisher's exact test.
c) Find the $P$-value if all three selected are male. Interpret

Maxime Rossetti
Maxime Rossetti
Numerade Educator
03:30

Problem 52

Refer to l'roblen 710 Suppose the same 1600 people who were interview'ed in January were interviewed again in February, with the results shown in Table 7.21
a) Test for a difference between the proportions who approve in February and in January.
Interpret
b) Find a $95 \%$ confidence interval for the difference berween the proportions $u$ ho approve in Februaly and in January. Interprer. Can you concludc that the approval rating cbanged from Janidry to Fcbruary? Explain.
c) Compare results to those in Problem 710 for tiec independent samples. In particular. compare the standard error for the difference of proportions for the iwo analyses. How does this affoct the widdh of the confidence interval? Which analysis provides more precise results and why does this happen?

Marc Lauzon
Marc Lauzon
Numerade Educator
00:13

Problem 53

Provide an example of a situation in which it is more sensible to compare means using dependent saniples than independent sainples.

Maxime Rossetti
Maxime Rossetti
Numerade Educator

Problem 54

A $99 \%$ confidence interval fol the diflerence $\pi_2-\pi_1$ berwecn the proportions of men and women in Florida who are alcoholics equals (.02, .09).
a) We are $99 \%$ confident that the proportion of dlcoholics is between .02 and .09 .
b) We are $99 \%$ confident that the proportion of men in Flonda who are alcoholics is between 02 and .09 larger than the proportion of wonnen in Florida who are.
c) At this confidence level. therc is insufficient cvidence 10 infer that the population proportions are different.
d) We are $99 \%$ confident that a minonty of Flondd residents are alcobolics
e) Since the confidence inter val does not contain 0 , it is impossible that $\pi_1=\pi_2$.

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03:02

Problem 55

To compare the populatuon mean annual incomes for Iispanics $\left(\mu_1\right)$ and for whites ( $\mu_2$ ) having jobs in construction, we construct a 959 confidence interval lor $\mu_2-\mu_1$.
a) If the confidence interval is $(3000,6000)$, then at this confidence level we conclude that the inean income for whites is higher than for Hispanics.
b) If the contidence interval is $(-1000.3000)$. then the corresponding $\alpha=.05 \mathrm{lcl} \mathrm{cl}$ test of $H_0, \mu_1=\mu_2$ against $H_a: \mu_1 \neq \mu_2$ rejects $H_a$.
c) If the confidence inlerval is $(-1000,3000)$, then it is plausible that $\mu_1=\mu_2$.
d) If the confidence interval is $(-1000.3000$ ), then we are $95 \%$ confident that the mcan annual income for whites is letween $$\$ 1000$$ less and $$\$ 3000$$ more than the mean annual income for Hispanics

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:49

Problem 56

For small samples, the Wilcoxon Icst differs from parametric procedures (for means) in Whe sence that
a) The alternative hypothesis cannot be one-sided
b) It is unnecessary to assume that the population distribution is normal
c) Random sampling is not assumed
d) It applies dircctly to ordinal as well as interval response variables.

Maxime Rossetti
Maxime Rossetti
Numerade Educator
02:56

Problem 57

A test consists of 100 true-false questions Joe did not study, so on each queslion, he randomly guesses the correct response.
a) Find the probability that he scores at least 70. thus passing the exam. (Hink: Usc the sampling distribution for the proportion of correct responses.)
b) Janc studied a liule and has a . 60 chance of a correct response for each question Find the probability that her score is nonetheless lower than Joe"s (Hunt. Use the distibution of the difference of sainple proportions.)
c) How do the answers to (a) and (b) depend on the number of questions? Explain

James Kiss
James Kiss
Numerade Educator

Problem 58

Using their formulas, for independemt samples show that $\hat{\sigma}_{\hat{\gamma}_2}-\hat{\gamma}_1$ is langer than $\hat{\sigma}_{\dot{Y}_2}$ or $\hat{\sigma}_{\gamma_2}$ for either mean separately Provide sonue intuition for why this happens.

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Problem 59

Let $X_i$ denote the observation for the $i$ th subject at time 1. Let $Y_i$ denote the observation for the $i$ th subject at time 2 . Let $D_i=Y_i-X_i$.
a) Letting $\bar{X}, \bar{Y}$. and $\tilde{D}$ denote the means of these observations, show that $\bar{D}=\bar{Y}-\bar{X}$.
b) Is the median difference (i.e.. the median of the $D_i$ values) equal to the difference between the medians of the $Y_r$ and $X_i$ valucs? Show this is true, or give a counterexample to shou that it is false.

Victor Salazar
Victor Salazar
Numerade Educator