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Statistical Methods for the Social Sciences

Alan Agresti, Barbara Finlay

Chapter 6

Statistical Inference: Significance Tests - all with Video Answers

Educators

Chapter Questions

01:29

Problem 1

Fol a large-simple lest oi $H_0: \mu=0$ againct $H_o, \mu \neq 0$. the 2 test stalistic equals 104 .
a) Find the $P$-value. and inlerpret
b) Suppose $z=-2.50$ ralhen than 1.04 Find the $P$-七alue. Does this provide strunger. or wcaker, evidence against the null hypothesis? Explain.

Tyler Moulton
Tyler Moulton
Numerade Educator
02:17

Problem 2

Complete the previous exercise for the alternarives (i) $H_\gamma, \mu>0$. (ii) $H_a \mu<0$.

Natalie Anderson
Natalie Anderson
Numerade Educator
01:26

Problem 3

The $P$-value for a large-sample test aboul a mean is $P=.10$.
a) If the alternative hypothesis ic $H_u . \mu \neq \mu_0$, report the value of the test statistic
b) Does this $P$-value provide stronger. or weaker. evidence against the null hy pothesis than $P=.010$ ? Explain.

Tyler Moulton
Tyler Moulton
Numerade Educator
04:56

Problem 4

Repeal (a) of the previous exercise for (i) $H_a . \mu>\mu_0$, (a) $H_a: \mu<\mu_0$.

Jameson Kuper
Jameson Kuper
Numerade Educator
02:25

Problem 5

Find and interpret the $P$-value for testing $H_0: \mu=100$ agdinst $H_0: \mu \neq 100$, if a cample of 400 obsen alions has $s=40$ and a) $\bar{Y}=103$, b) $Y=97$.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:10

Problem 6

Refer to the previous exercise. Repeat part a) if a) $n=1600$ instead of 400, b) $s=20$ instead of 40 . Comment on the effects of $n$ and $s$ on the results of a significance test.

Maxime Rossetti
Maxime Rossetti
Numerade Educator
00:38

Problem 7

In response to the statement "A preschool child is likely to suffer if his or her mother works," the response calegories (Strongly agree, Agree. Disagree, Strongly disagree) had counts $(91,385,421.99)$ for the 996 responses in the 1991 General Social Survey. With scoles $(2.1,-1,-2)$ for the four categories, computer software reported the following resulls.
a) Set up null and alternative hypotheses to test whedher the population mean response differs Irum the neutral value. 0 .
b) Find the test statistic and $P$-value. Interpret.
c) Explain what this choice of sconng assumes about relative distances between levels of the scale.

Hossam Mohamed
Hossam Mohamed
Numerade Educator
01:38

Problem 8

The mean score for all hgh school seniors taking a college entrance exam equals 500 . A study is conducled to sec whether a different mean applies to those students bom m a foseign country. For a random sample of 100 of sucl students. the neean and standard deviation on this exam equal 508 and 100 .
a) Set up hypothcses for a significance test.
b) Compute the test stausuc.
c) Report the $P$-valute, aud interpret.
d) Based on (c). can you conclude that the population mean for students born in a foreign countiy equals 500 ? Why or why not?
e) Make a decision about $H_0$, using $\alpha=05$
f) Construct a $95 \%$ confidence interval for $\mu$. Show the correspondence betueen whether 500 falls in the interval and whether $H_0 \cdot \mu=500$ is rejected in favor of $H_c: \mu \div 500$

Dominador Tan
Dominador Tan
Numerade Educator
06:51

Problem 9

A social psychologisi plans to conduct un experiment with a random sample of 49 children from some school distrel Before conducting the experinent. the psychologist checks how this sample compares to national norms on several variables that could affect the results of the experiment. The IQ scores for the 49 children have $\bar{Y}=103$ and $s=14$. The national population) ınean IQ equals 100 . Is it plausible that the mean $\mu$ of the population of children in the school district from which these students were sampled equals 100 ?
a) Conduct a test of $H_0: \mu=100$ against $H_s: \mu \neq 100$. Report the $P$-value, and interpiel.
b) Make a decision about $H_0$ using $\alpha=.05$.
c) Construct a $95 \%$ confidence interval for $\mu$, and contrast the types of information obtained with the two luference methods. tests and confidence intervals.
d) Does it inake sense to "accept $H_0{ }^"$ in part (b)? Why or why not"
e) What conclusion applies for cach of the lollowing $\alpha$-levels:
(i) $\alpha=.20$,
(ii) $\alpha=.10$
(iii) $\alpha=.01$.
Why is $\alpha=20$ not typically used in pracuce?

Shu Naito
Shu Naito
Numerade Educator
01:13

Problem 10

Refer to Example 6.2. Suppose we instead use the scores $(-3,-2,-1,0,1,2,3)$, subtracting 4 from each of the original scorcs. We then test $H_0: \mu=0$. Explain the effect of the change in scores on a) the samplc inzan and standard detiation. b) the test statistic, c) the $P$-Value and interpretation

Breanna Ollech
Breanna Ollech
Numerade Educator
03:22

Problem 11

The mean agc at first mamage for unarricd men in a New England community was 280 in 1790 . For a randorn sample of 40 married men in that community in 1990, the sample mean and standard deviation of age at firsi marriage werc $\bar{Y}=26.0$ and $s=9.0$.
a) State the bypotheses, and find the lest slalislic and $P$-valuc for testing whether the mean has changed Interprei.
b) Make a decision, using $\alpha=05$.
c) If the decision in (b) is an earor, what type of error is it, Type I or T) pe II?
d) Report the $P$-value for the alternatuve $H_a \cdot \mu<28$ Interpret.
e) Report the $P$-value for the alternative $H_a \mu>28$. Interpret

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:37

Problem 12

A joint ('SA Teadaw/CNN/Gal lup Pol] in July 1995 indicated that of 832 white adults. $53 \%$ thought affirmarive action has heen good for the country and $37 \%$ thought it had not been good: the remaining $10 \%$ were undecided. Let $\pi$ denote the population proportion of white aduls who believe that affirmative action has hcen good for the conntry
a) Tcst $H_0: \pi=.50$ against $H_0 \cdot \pi \neq .50$. Report the $P$-value. and interpret.
b) Can jou scjecı $H_0$ using $\alpha=0$ ? ? Indicale uhether you can conclude tlaat a majority or minovity of the population think affirinative action has bees good.
c) Construct a $95 \%$ confidence interval for $\pi$. Explain hou this provides greater informacion than the conclusion of the lest in (b).
d) The same poll also interviewed 299 black adults. Of them. $66 \%$ nlough that affirmative action had been good and $23 \%$ thought it had not been good Conduct statistical inlerence for black adults.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
18:27

Problem 13

An experunent consists of giving everjone in soune group a lottery ticket, and then later ashing il they would be willing to exchange their tickel for another one, plus a small ınonctary incentive. Let $\pi$ denote the population proportion who would agree to the exchange When this expenunent was conducted with 26 students in an Israeli classroom recently, only 7 students agreed to the exchange (M. Bar-Hillel and E. Ncter, J. Persnndity and Social Psychology, Vol. 70. 1996. pp 17-27).
a) lising these dala. test $H_0, \pi=.50$ against $H_e{ }^{\prime} \rightarrow \frac{1}{\tau} .50$, and interpret the $P$-value. b) In a reluied experínent, 31 students in an Isracli statisrics class were given a new pen and then later asked to exchange it for another pen and a small monetary incentive All 3I agreed. Rcpcat the lest for these data, and inlerpret.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
01:01

Problem 14

Lel $\pi$ denote the proportion of Floridians who think that government environmental regulations are too strict Test $H_0: \pi=.5$ against $H_n: \pi \neq .5$ using dala from a telephonc poll of 834 people conducted in June 1995 by the Institute for Public Opinion Research al Florida International Universiry. in $u$ bich $26.6 \%$ said ıcgulations were too strict
a) Calculate the test stalistic.
b) Find the $P$-value. and interpret
c) (1sing $\alpha=.01$, can you delennine whether a majority or nunonty thank that envuonmental regulations are too strict. or is it platsible that $\pi=50$ ?
d) Constuct a $99 \%$ conlidence inter val Explain the adv antage of the confidence interval over the lest.

Maxime Rossetti
Maxime Rossetti
Numerade Educator
04:31

Problem 15

The 1994 Genetal Social Survey asked, "Do you rhink ir should be possible for a pregnant woman to obtain a legal abortion if the family has a very los income and cannol afford any more children ${ }^m$ Lel $\pi$ denote the population proportion who would answer yes to this question. An anti-abortion activis claims that a minority of Amencans support legaluzed abortion in such cases a) Set up hypotheses for a significance test that can andlyze the claim.
b) In this survey. 971 answered yes and 954 answered no. Calculate the test statisuc and $P$-value for the hypotheses presented in part (a). Interpret.
c) Using $\alpha=05$, what is your decision? Interpret

Sheryl Ezze
Sheryl Ezze
Numerade Educator
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Problem 16

A mayoral election in Madison. Wisconsin, has two candidates. Exactly half the residents of the cily currently prefer each candidate.
a) Find the probability that, for a random sample of 400 residents. at least 230 iudicate a preference for one of the candidates or the other ${ }^2$ This is also the $P$-value for testing $H_0$. $\pi=.5$ against $H_a: \tau \neq .5$, where $\pi$ denotes the probability that a randomly selected voter prefers a particular candidate
b) For a random sample of 400 voters, 230 voted for a particular candidate. Are you willing to predict the outcome of the election? Why?
c) For a random sainplc of 40 voters, 23 voted for a particular candidate. Hould you be willing to predict the outcomne of the election? Why?

Danielle Fairburn
Danielle Fairburn
Numerade Educator
01:02

Problem 17

The authorslip, of an old documen is in doubt A bistonan hypothesizes that the author was a joumalist named Jacalyn Levme, Upon a thorough investigation of Levine's known works, it is observed that onc unusual feature of her wnting was that she consistently began 65 of her sentences with the word whereas. To test the histonan's hypothesis, it is decided to count the number of sentences in the disputed document that begin with the word whereas Out of the 300 sentences in the documeut, none begin with that word Let $\pi$ denote the probability that any one sentence written by the unkHown author of the docuntent begins with the word whereas. Conduct a test of the hypothesis $H_0: \pi=06$ against $H_0 \pi \neq .06$. What conclusion can you ınake? What assumptoons are needed lor that conclusion to be valad? (Mosteller and Wallace (1964) conducted an invesugation similar to this to determine whether Alexander Hamilton or James Madison was the author of 12 of the Federalist Papers )

Tyler Moulton
Tyler Moulton
Numerade Educator
01:58

Problem 18

Refer to the ordinal scale in Table 6.2. Treat this as a qualitativc variable, as follows. Ignore the subjects responding in the moderate category, Of the remainder not responding in that calegory, let $\pi$ denote the population proportion who make one of the three libelal responses and let $1-\pi$ denote the population proportion making one of the three conser ative respouses. Test the hypothesis that these proportions are identical. Find the $P$-value and mierpret.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
11:31

Problem 19

The ou ncr of a department store in Rochester, New York, initiares a w eek-long neu spaper advertising cainpaign to increase dwareness of the store. Before inicsting any additional money in advertising, the owner takes a phone survey to check whether potential customers are more inclined or less inclined to shop in this store after seeing the ads ertising. Consider the population of residents who have seen the ad and are either more inclined or less inclined to shop there than before seeing the ad. Of these people. let $\pi$ denote the proporlion who are inore inclined. For a random sample of 100 numes selected from the teleplone book, 25 express an opinion Of these 25 subjects, 18 say they are more inclined to shop at the store, and 7 say they are less inclined.
a) Report the $P$-value for testing $H_0 . \pi=.5$ against $H_a \pi \neq .5$. Interpret
b) Aake a decision using $\alpha=05$. Can the owner conclude that, of those with an opinion, a najority are more likely to shop at the store?
c) If you made an error with the decision in (b), would it be a Type I error or a Type II crror?
d) Consider the proportion of residents who have not seen the ad or who are neither more nor less likely to shop at the store as a result of the ad. Can you determine whether this is a majority or minority of the population? (Answer by performing a test or constructing a confideuce interval that uses all 100 observations.)

Robin Corrigan
Robin Corrigan
Numerade Educator
02:00

Problem 20

A multiple-choice lest question has four possible responses. The question is designed to be very difficult, with none of the four responses being obviously wrong, yet with only one correct answer It first occurs on an exain taken by 400 students. The designers test whether more people anstwer the questuon correctly than would be expecied just due to chance (i.e.. if everyonc randomly guessed the correct answer).
a) Set up the hypotheses for the lest
b) Of the 400 students, 125 correctly answer the question. Find the $P$-value, and inter piet.
c) Make a decision about $H_0$, using $\alpha=.05$ Based on this decision. what can you conclude about the parameter?
d) As an alternative inference, constuct a $95 \%$ confidence interval for the parameter Contrast whal you learn using these Iwo methods.

Nick Johnson
Nick Johnson
Numerade Educator
02:13

Problem 21

Two lesearchers conduct separate studies to test $H_0: \mu=500$ against $H_0: \mu \neq 500$. The first researcher gets $\bar{Y}=519.5$, with a standard ecror of 10.0 . The second researcher gets $\bar{Y}=519.7$, with a standard error of 10.0 .
a) Using $\alpha=.05$, test in each case whether the result is "staristically significant"
b) Explain the dangers of rcporting the result of a test as " $P \leq .05^{\circ}$ versus $" P>.05$." or as "reject $H_0$ " versus "Do not reject $H_0$ "

Joshua Argo
Joshua Argo
Numerade Educator
01:38

Problem 22

A study considers whether the mean scoie on a college entrancc exam for students in 1996 is any different froin the mean score of 500 for students who took the saune cxam in 1966. Let $\mu$. represent the mean score for all students who took the exam in 1996. Test $I_0{ }^{\circ} \mu=$ 500 against $H_r . \mu \neq \frac{1}{\mathcal{}} 500$, if for a nationwide random sample of 10.000 students who took the cxam in $1996, \bar{Y}=497$ and $s=100$. Show that the result is highly signibcant statistically, but not praclically significant

Dominador Tan
Dominador Tan
Numerade Educator
01:03

Problem 23

Report the $t$-score that multiplics by the standard error to forin a
a) $95 \%$ confidence interval uith 5 observations
b) $95 \%$ confidence interval with 15 observations
c) $95 \%$ confidence inter ral with 25 observations.
d) $95 \%$ confidence interval with $d f=25$.
e) $99 \%$ confidence interval with $d f=25$.

Samuel Coplin
Samuel Coplin
Numerade Educator
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Problem 24

A $r$ test for a mean uses a saunple of 15 observations Find the $t$ test statistic value that has a $P$-value of $P=05$ uhen the altemative hypothesis is a) $H_a \mu>0, \quad$ b) $H_a: \mu \neq 0$, c) $H_{d x}: \mu<0$ ?

James Kiss
James Kiss
Numerade Educator
00:43

Problem 25

A school board commissions a study of the absentee rate of students at a pat ticular high school in Vancouver For a random sainple of 20 scudenl records from the previous year, the mumber of days absent in the previous year is recorded, yielding $\bar{Y}=40$ and $s=4$. a) Stating the necessary assumiptions, calculate $a 95 \%$ confidence interval for the incan number of day's abseat for all students at that high school during the previous year.
b) Does the normality assumption scen plausible for these ddta? Explain, and discuss the implications of the term "robustness" for you analysis.

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
01:03

Problem 26

Refer to Example 6.11. For the 15 feunale infants, Table 6.9 shows an SPSS pi intout for the data on the changc in beart rale.
a) Explain how to interpret all the results in Ulis table.
b) Explain how to rest whether the true mean is 0 . Report the $P$-valuc for (i) a one-sided alternative of a positive meun change, (ii) two-sided alternative, interpretung in each casc.

Akhil Choudhary
Akhil Choudhary
Numerade Educator
01:01

Problem 27

Explain how the sample size affects the width of a small-sample confidcnce interval through its effect on the a) standard error, b) $t$-scoie that inultiplies the standard error.

Sanchit Jain
Sanchit Jain
Numerade Educator
01:50

Problem 28

According to a union agrcement, the mean income for all senior-level assembly-line workas in a large company equals $$\$ 500$$ per weeh. A representative of a woinen's group decides to analyze whether the nean income for female employees matches this nomn For a randoin samjle of nine female employces, $$\bar{Y}=\$ 410$$ and $s=90$ Conduct a significance lest of whether the mean income of fernale employees difers from $$\$ 560$$ per wcek. Include all assumptions, the hypotheres. test statistic, and $P$-value. and interprel the result

Hossam Mohamed
Hossam Mohamed
Numerade Educator
01:30

Problem 29

Refer to the previous probleml
a) For which $\alpha$-levels in Table B (the $t$ disenbution) can you reject $\mathrm{H}_9$ ?
b) For which confidence coefficients would the confidence interval contain 500 ?
c) Lise (a) and (b) to illustrate the cor respondence between results of tesis and results of conlidence intcrrals

James Kiss
James Kiss
Numerade Educator
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Problem 30

By law. an inductrial plant can dischange no more than 500 gallons of waste waler per hour, on the average, into a neighbonng lake. Baved on othet infractions they have noticed, an environrnental action group believes this limit is being exceeded. Monitoring the plant is expensive. and only a small sample is possible. A random sample of four hours are selected over a penod of a woek A compuier prntout shows the results:
$$
\begin{array}{ccccc}
& \text { Number } & & \\
\text { Variable } & \text { of Cases } & \text { Mean } & \text { SD } & \text { SB of Nean } \\
\text { HASTE } & 4 & 1000.0 & 400.0 & 200.0
\end{array}
$$
a) Test the null hypothesis that the mean discharge equals 500 gallons per hour against the alternative that the limit is being exceeded. Find the $P$-value, and interpret
b) Explain why the Iesult of this test may be invalid, or at least very highly approximate. if the population distribution of discharge is far from normal.
c) Explain how your one-sided analy sis implicitly tests the broader null hypothesis that $\mu \leq 500$
d) Makic a decision. using $\alpha=05$ Interprel.
e) If the decision in (d) was incorrcet, what type of error was made? What could you do to reduce the chance of that type of error?

Victor Salazar
Victor Salazar
Numerade Educator
04:26

Problem 31

A study was conducted of the edfects of a special class designed to improve children's vethal skills. Each cluild took a verbal shills test twice. both before and altel a three-week period in the class. Lel $Y$ be the second exam score minus the first exam score. Ilence, if $\mu$ (the population mean for $Y$ ) is equal to 0 , the class has no effect on the average. Test the null hypothcsis of no effect against the alternarive hypothesis that the effect is positive, if the scores on $Y$ for a random sample of four cliildren having learning problems were 3 . 7.3.3. Interpiet the $P$-value, (Note. The scores could tend to improve simply from the srudents feeling more comfortable with the resting process. A more appropriate design nould also administel the exam twice to a control group that does not take the special class. companng the changes for the experintental and control groups.)

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:25

Problem 32

A jur. list contains the names of all individuals who may be called for jury duty. The proportion of the available jurors on the hst who are women is .5. If a jurz of size 12 is selected at random from the list of available jurors.
a) Find the probability that so woinen are selected.
b) Find the probability that exactly one woman is selected.
c) Find the expecred value and standard deviation of the nuinber of women selected.

Nick Johnson
Nick Johnson
Numerade Educator
02:15

Problem 33

Refer to the ptevious problem. Test the hypothesis that the selections are random, if no woman is selected out of a sample of size 12. Report the $P$-value, and interpret.

Kari Hasz
Kari Hasz
Numerade Educator
03:30

Problem 34

At current rates, the proportion of deaths of Amencan females that are duc to suicide is 01 (Siatistical Abstract of the Linited Stures, 1995). For a random sample of ten deaths. find the probability that
a) None of them is due to sticide.
b) At most oue is due to suicide.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
02:58

Problem 35

Refer to the previous problem The proportion of deaths due to suicide for Amencan males is 02 .
a) For 1000 American male deaths, find the mean and the standard deviation of the piobability distribution of the number that were due to suicide.
b) Would it be surprising if uone were due to suicidc? Explain.

Trent Speier
Trent Speier
Numerade Educator
04:13

Problem 36

For each free throw. a basketball player has probability .80 of making the shot and 20 of missing it.
a) Find the probahility that le makes ten fiee throus in a row
b) He is fouled and takes two free Unows. Find the probabilities in the distrbution of the number of free throws that he inales
c) What assumptions must you make for the calculations in (a) and (b) to be ralid?

Nick Johnson
Nick Johnson
Numerade Educator
01:40

Problem 37

A football team has probability $\pi=50$ of winning anj particular game.
a) Find the plobability that the team wins all or none of its six conference games. What assumptions does this calculation require?
b) If the teum wins all six games, can you be quite confident that its actual probability of sinning any single game diffess fioun .50 ? Explain.

Christopher Stanley
Christopher Stanley
Numerade Educator
00:53

Problem 38

A perzon claiming to possess extrasensory perteption (ESP) says she can guese more often than not the outcome of a fip of a balanced coin in another room. not visible to her
a) Introduce appropnate notation, and state hypotheses for lesting her claim
b) Of ten coin flips. she guesses the correct result seven tomes Find the $P$-valuc and inlenuret

Nick Johnson
Nick Johnson
Numerade Educator
01:30

Problem 39

A weather lorecusier states "The chance of rain is $50 \%$ on Saluday and $50 \%$ again on Sunday So, there's a $100 \%$ chance of rain sometime over the neekend." If whether it Idins on Saturday is independent of whether it rains on Sunday. find the actual probability of rain at least once during the weekend.

Hailey Tomashek
Hailey Tomashek
Numerade Educator
04:31

Problem 40

${ }^3$ Refer to Table 6.2. Of those subjects responding in one of the two extrenc categories. let $\pi$ denote the population proportion who make the extrcinely liberal response, with $1-\pi$ being the population proportion making the extremel) conservative response. Using small-sainple nethods, show that the $P$-value for testing that these probabilities are identical equals 10 . Interpret

Lucas Finney
Lucas Finney
Numerade Educator
06:55

Problem 41

In a given year: the probability that an Aıncrican female adult dies in a motor vehicle accident equals . 0001 (Statistical Abstract of the linited Stares, 1995).
a) In a city having 1 mullion Aınerican female adults, find the mean and standard detiation of the number of dealhs from notor vchicle accidents.
b) Based on the normal approximation to the binomial, find an interval within which the number of deaths has probability .95 of occurring
c) The rate for Amencan ınale adults is twice as latge Repeat (a) and (b) using the rate .0002 , and compare results to those for females

Jacquelinne S. Mejia Sandoval
Jacquelinne S. Mejia Sandoval
Numerade Educator
01:54

Problem 42

Refer to Problem 6.16. Find the $P$-value for testing $H_0: \pi=5$ against $H_a, \pi \neq 5$ when five people are randomly selected and all of them prefer a particular candidate Intes prct For these hypotheses. show that a sanple of at least six people is needed to oblain $P$-value below . 05 .

Lucas Finney
Lucas Finney
Numerade Educator
02:54

Problem 43

A fraternal organization admits $80 \%$ of all applicants who satisf) certain requirements. Of four members of a ninority group who recently applicd for admission, all met the requirements but none was accepted.
a) Find the probability that none would be accepred if the same admissions slandards werc applied to the minority goup, other things being equal.
b) Find the $P$-valuc for testing that the probability of admission for a minont) group member equals 80 . against the altenuative that it is less than 80 . and interpret

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:49

Problem 44

A decision is planned in a rest of $H_0, \mu=0$ against $H_0: \mu>0$, using $\alpha=.05$. If $\mu=5$, the probability of a Type JI error equals . 17 .
a) Explain the meaning of this last sentence.
b) If the test were conducted at the $\alpha=0 \mathrm{I}$ level, would the probahility of a Type II enor be less than. cqual to. or greater than 17 . Explain
c) If $\mu=10$, would the probability of a Type Il crror be less than. equal to, ur grcater than .177 Explain.

Nick Johnson
Nick Johnson
Numerade Educator
03:49

Problem 45

Refer to Exainple 611.
a) Show that the probability of Type 11 error is less than . 001 at $\mu=15$.
b) Show that the probability of Typc 11 error equals .49 at $\mu=5$
c) Find the probability of Type $\Pi$ error at $\mu=8$.
d) If $\alpha=.01$ instead of 05 . venfy thal the probability of Type It error al $\mu=10$ cquals 16 .
e) If $\alpha=.0 \mathrm{~J}$ hut $n=81$, vel ify that the jrobability of Type Il el rot at $\mu=10$ equals . 004

Nick Johnson
Nick Johnson
Numerade Educator

Problem 46

Refer to the hypotheses in Example 6.11. Suppose a sample of size 25 wele taken (instead of 36) and suppose the estimated standard error of $\hat{\gamma}$ is $\hat{\sigma}_{\hat{\beta}}=3.6$. When $a=.05$, compute the probabilities of a Type II error for the altemative $\mu$ values of (a) 5 , (b) 10 , (c) 15. Compare these to the probabilities of Type II errol when $\hat{\sigma}_j=3.0$, and usc them to illustrate the effect of the sample size on $P$ (Type II errot).

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10:44

Problem 47

Let $\pi$ denote the proportion of schizophrenies who respond posituvely to a particular treatment A tcst is conducted of $U_n \pi=5$ against $H_n: \pi>.5$. buscd on a sample of size 25. using $\alpha=.05$.
a) Find the region of sample proportion values for which $H_d$ is scjecred. Ior a large-sample $z$ lest
b) Suppose that, unknown to the rescarcher, the true value of $\pi$ is 60 . Find the probability thal a Type II error occurs.
c) Explain why the probability of Type $I I$ error increases toward .95 as the true value of $\pi$ moves down toward 5. (Assuinc $n$ and $\alpha$ stay fixed.)

Gaurav Kalra
Gaurav Kalra
Numerade Educator
03:11

Problem 48

Refer to the WWW data set (Problem 1.7).
a) Test whethes the mean political ideology diffes from 4.0 Report the $P$-value, and interpret
b) Tect whether the proportion lav onng Jegalized ahortion equals. or diffors, from . 50 . Report the $P$-value, and interpret

Sheryl Ezze
Sheryl Ezze
Numerade Educator

Problem 49

Refer to the data file you created in Problein 1.7. For variables chosen by your instructor, conduct inferential statistical analyses Prepare d report. summarizing and interpreting your findings In this rcport, also use graphical and numerical incthods piesented earlier in this text to describe the data and. it necessary, to check assumptions that you make for your analysis.

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Problem 50

In response to the question "Do you think it should or should not be the government's responsibility to provide a job for everyonc who wants one." the categories "Definitely should be," "Proliably should be," "Probably should not be." "Definitely should not be" had response coints $242,3,30,333,362$ in the 1991 General Social Survey, Applying appropriate inferential methods, analyze these data.

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01:38

Problem 51

An article in a sociology joumal that deals with changes in religious beliefs ovet time states. "For these subjects. the difference in their responses on the scale of religiosity betwcen age 16 and the current survey was significant $(P<.05$ )."
a) Explain what it means for the result to be "significant."
b) Explain why it would have been more informative if the authors provided the actual $P$-value rather than simply indicating that it is below 05 . What orber information might they have piovided?
c) Presumably the statement refers to a population thean $\mu$ of difterence scoses betwcen the curremt survey and dge 16 Based on the authors' statcunent, what can you conclude about $\mu$ ?
d) Can you conclude that an important change in religiosity las occurred hetu een age 16 and the time of the current surccy Why or why not"

Qudsiya Anis
Qudsiya Anis
Numerade Educator
07:45

Problem 52

A new'ly created randon number generator is supposed to gencrate a sequence of digits such that each digit is equally likely to be any of $0,1,2, \ldots 9$. The lirst 20 numbers geuerated are
$$
7,7.3,0.5 .6,3,2,6.1,0.9 .9 .4 .0 .8,5,0.6 .2
$$
As a check of whether the process works correctly. test whether the mean differs significautly fiom the value expected. Report the $P$-value and interpret.

Ryan Mcalister
Ryan Mcalister
Numerade Educator

Problem 53

In making a decisiou in a significance test, a researcher worrics about the possibility of rejecting the null hypothesis when it is actually true Explain how to control the prohability of this type of error.

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04:16

Problem 54

Consiter the analogy between making a decision about a hy pothesis in a significance test and making a dccision about the innocence or guilt of a defendant in a criminal trial. Identify, "Ho true." "Holse" with "Defendant innocent. "Defcndant guilty." and "Reject $H_0$," "Do not reject $H_0$ " with "Convict defendant," "Acquit defendant."
a) Explain the difference between Type I and Type II cnors m this tial setting. Which t) pe of error would you consider more serious and hope to set at a very small Jevel? Why" b) In this setting, explain why decreasing the chance of Type l error increases the chance of Type II error.

Sherrie Fenner
Sherrie Fenner
Numerade Educator

Problem 55

Medical tests for diagnosing conditions such as cancer or HIN'+ are fallible. just like decisions in significance tests identily ( $H_0$ true. $H_0$ false) with (Disease absent. Disease precent). and (Reject $H_0$. Do nol reject $H_0$ ) with (Diagnostic test is positive, Diagnostic test is negative), whcre a positive diagnosis means that the test predicts that the disease is present Explain the difference between Ty pe 1 and Type 11 enors in this setting. Explain u liy decreasing the chance of Type I error increases the chance of Type II enor.

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01:18

Problem 56

An article in a political science journal states that "no significant diffes ence was found between med and woinen in their voting rates ( $P=.63$ ).- Can we conclude that the population roting ratcs are identical for men and women? Explain.

Sheryl Ezze
Sheryl Ezze
Numerade Educator

Problem 57

A research sludy conducts 60 significance tests. Of these. 3 are significant at the 05 level. The duthois white a report stressing only the three cases in which Uhey found "significant" results, not mentuoning the other 57 tests they conducted that were "not significant." Explain what is misleading about this report.

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02:00

Problem 58

Somc medical joumals have a policy of publishing results only of rescarch thar is statistically significant at some level
a) Explain the dangers of this by descrihing u hat could happen if researche1s at 20 different institutions independently and without communication anong each other, conducted similar studies and tesled the same null trypothesis using $\alpha=.05$, w hen that hypothesis is actually true.
b) When medical stones in the mass media report supposed large dangers or benefits of certain agents (c.g., coffee drinking, fiber in cereal). larer research often suggests that the effects are not ncarly as large as first believed. Explain why.
Select the courect response(s) in Problems 6 . 59-6 62

James Kiss
James Kiss
Numerade Educator
01:41

Problem 59

A $95 \%$ condidence interval for $\mu$ is (96.110) Which of the following starements about significance tests for the same dana is larel correct?
a) In testing $H_0 \quad \mu=100$ aqainst $H_n: \mu \neq 100, P>.05$.
b) In lesting $H_C: \mu=100$ against $H_a \cdot \mu \neq 100 . P<.05$. the confidence interval.
d) In testing $H_0: \mu=\mu_0$ aganst $H_a: \mu \neq \mu_{0 .} P>.05$ if $\mu_0$ is any of the numbers outside the confidence interval.

R M
R M
Numerade Educator
03:08

Problem 60

The $P$-talue for testing $H_{0^*} \mu=100$ against $H_0 \mu \neq 100$ is $P=, 001$. This indicates that
a) There is strong evidence that $\mu=100$
b) There is strong evidence that $\mu \neq 100$.
c) There is strong cvidence that $\mu>100$
d) There is strong evidence that $\mu<100$
e) If $\mu$ were equal to 100 it would be unusual to obtain data such as those ohsern ed

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator

Problem 61

Refer to the previons problem. Suppose the value of the test statistic was $z=329$ Then
a) There is stong evidence that $\mu=100$.
b) There is stong cvidence that $\mu>100$.
c) There is strong evidence that $\beta<100$

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01:34

Problem 62

Refer to Problem 6.30. When we inake a decision using $\alpha=.05$. this means that a) If the plant is not exceediug the limit, but actually $\mu=500$. there is only a $5 \%$ chance that we will conclude that they are exceeding the limit.
b) If the plant is exceeding the limit. there is only a $5 \%$ chance that we will conclude that they ane not exceeding the limit.
c) The probability of getting the sample mean we observed would equal .05 if $I_0$ were true,
d) If we reject $H_0$. the probabilin that it is actually true is . 05
e) All of the above

Tyler Moulton
Tyler Moulton
Numerade Educator
01:27

Problem 63

True or False $P$ (Type II errori) $=1-P$ (Type I error). Explain.

Tyler Moulton
Tyler Moulton
Numerade Educator
01:07

Problem 64

True on False. If we reject $H_0$ using $\alpha=.01$, then we also reject it using $\alpha=.05$. Explain.

Tyler Moulton
Tyler Moulton
Numerade Educator
03:08

Problem 65

An article in an anthropology joun nal reports $P=.043$ for testing $H_0 . \mu=0$ against $H_{G^*} \mu \neq 0$. Thue or False- If the authors had instcad reported a $95 \%$ confidence inter val for 12 . then the inter al would have contained zero. Explain, and discusc what one could learn from the confidence interval but not fiom the lest.

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
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Problem 66

a) Definc (i) $P$-valuc. (ii) $\alpha$-level. (tii) Type 11 error, (iv) rejection region. (v) power.
b) Explain the difference between one-sided and two-sided hypotheses. and explain fow this affects calculation of the $P$-value

Victor Salazar
Victor Salazar
Numerade Educator

Problem 67

Explain why the tes minology "do nol reject $H_{10}{ }^{\text {" }}$ is preferable to "accepi $H_0$."

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02:25

Problem 68

A randorn sample of size 40 has $\bar{Y}=120$ The $P$-value for lesting $H_0{ }^* \mu=100$ against $H_a \mu \stackrel{1}{\bar{T}} 100$ is $P=.057$ Explain what is inconect about each of the following interprelations of this $P$-value, and provide a proper interpretation.
a) The probability that the null hypothesis is correct equals . 0.57 .
b) The probabulity that $\bar{Y}=120$ if $H_c$ is truc cquals .057 .
c) If in fact $\mu \neq 100$. the probability equals . 057 that the data would be at least as contradictory to $H_0$ as the observed data
d) The probability of Type 1 ctror equals 057.
e) We can accept $H_0$ at the $\alpha=05$ level.
f) W'c can reject $H_0$ al the $\alpha=$. (S level.
g) The valte of the $z$ rest statistic is $＝ 158$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:15

Problem 69

${ }^T$ The $P$-value for testing $H_{\mathrm{L}} . \mu=100$ agains $H_0 \mu>100$ is $P=043$ Does a $95 \%$ confidence inten al for $\mu$ contain 100 ? Explain.

Lucas Finney
Lucas Finney
Numerade Educator
03:26

Problem 70

Explain why one cim interpret the $P$-value as the sinallest $\alpha$-level at which one can reject $H_0$, that is, $P$ equals the smallest level at which the dala are siguificant. Illustrate using the $P$-value of 057 fiom Problem 668

Cerys Evans
Cerys Evans
Numerade Educator
01:52

Problem 71

Refer to Example 6.2 and to the corrcspondence between results of confidence intervals and two-sided rests Since $P=.52$ in that exauple, and since $1-.52=48$, explain why the 48 s. confidence interval is the nanowest confidence interval centered about $\bar{\gamma}$ that contains $\mu_0=4.0$

Manik Pulyani
Manik Pulyani
Numerade Educator
01:48

Problem 72

We know the sample incan $Y$ of $n$ ineasurements. Show that if we know ( $n-1)$ of those measurements, then we can detennine the remaining observation. In othet words. given the value of $\bar{Y}$. the values of $(n-1)$ obser ations delermine the remaining one. In sununauring scores on a quantitatic vaiable. thene are $(n-1)$ degrees of treedom. since ouly that many observations asc hdependent.

Manik Pulyani
Manik Pulyani
Numerade Educator
00:42

Problem 73

A researcher conducts a significance test even time she anal zes a new data set. Over time. she conducts 100 tests
a) Suppose the null hypothesis is tue in every case. What is the distroution of the number of times she rejects the null hypothesis at the 05 level?
b) Suppose slee rejects the null bypothesis in seven of the lests. Is it plausible that the null hypothesis is correct in every casc? Explain

Lucas Finney
Lucas Finney
Numerade Educator