Statistical Methods for the Social Sciences

Alan Agresti, Barbara Finlay

Chapter 5 Statistical Inference: Estimation - all with Video Answers

Educators

Chapter Questions

Problem 1

The numbers of children in sis families selected at random froun I 870 census records in a southern town are 1.4.0,0,6,7. Calculate and interpret point estimates of the population mean and standard Jeviation.

Kari Hasz

Numerade Educator

08:51

Problem 2

Find and interpret the $95 \%$ confidence interval for $\mu$, if $\bar{Y}=70$ and $s=10$. based on a sample size of (a) $n=25$. (b) $n=100$.

Willis James

Numerade Educator

01:08

Problem 3

In the General Social Survey respondents are asked to rate their poliucal vicws on a sel enpoint scile, where $1=$ extremely likeral. $4=$ moderale. and $7=$ extremels conservative. For the 1994 survey: a computer printout reports.
a) Construct and interpret a $95 \%$ confidence interval for the mean political ideology.
b) Show the effect of increasing the confidence level, by constructing a $99 \%$ confidence interval
c) What assumption are you malang about the scale of measurement for political ideology. uhen you use the sample neall and standard deviation?

Trinity Steen

Numerade Educator

02:33

Problem 4

Refer to Example $$\$ 1$$ and the responses about the number of male sex partiors. Of the 1431 subjects responding with a posituve numbcr, the mean 1 as 4.8 and the standard deviation was 6.2 .
a) Verify that the $95 \%$ confidence interval for that population mean equals (4.5, 5.1).
b) What statistical factons cause the interval in (a) to be narrower than the $95 \%$ confidence interval reported in Example 51 for the mean number of femalc partors?

Sheryl Ezze

Numerade Educator

03:28

Problem 5

Refer to the data in Table 3.13 regarding the subjects" nuinber of sex partners in the last 12 months A computer printout shuws the following inlommation.
a) Let $\mu$ denote the population mean of this distribution. Report a point ectimate of $\mu$ and a point estimate of the population standard der iation.
b) Based on the repotted sample size and standard deviation, verify the repoited value fot the standard error.
c) Construct a $95 \%$ confidence interval for $\mu$. Interpres.
d) Construcı a $99 \%$ confidence interval for $\mu$. Compare to parl (c)

Sheryl Ezze

Numerade Educator

04:18

Problem 6

A survey is taken to estimate the mean annual family income for families living in public housing in Chicaso. Fos a random sample of 30 families. the annuat incomes (in hundreds of dollars) are as follows.
$$
\begin{array}{rrrrrrrrrr}
83 & 90 & 77 & 100 & 83 & 64 & 78 & 92 & 73 & 122 \\
96 & 60 & 85 & 86 & 106 & 70 & 139 & 56 & 94 & 84 \\
111 & 93 & 120 & 70 & 92 & 100 & 124 & 59 & 112 & 79
\end{array}
$$
a) Construct a stem and leal plot of the incomes. What do you predict abour the shape of the population distnbution?
b) Constuct and interpret point estinaates of $\mu$ and $\sigma$, the means and standaid deviation of the famuly incouncs of all fanulies living in public housing in Chicago.
c) Construct and interpret a $95 \%$ confidence interval for $\mu$
d) Construct a $99 \%$ confidence interval for $\mu$. Interpret the inter-al and compare it to the one in part (c)

Raymond Matshanda

Numerade Educator

02:21

Problem 7

A hospital administrator wants to estimate the mean length of stay for all inpaticuts who usc that hospital. Based on a systematic a andom samnjle of 100 records of patients for the previouc year, she reports that "The sample mean was 5.3 In rcpeated random samples of this size, the cample mean could be expceted to fall within 1.0 of the truc mean abou: 95\% of the time "
a) Construct and interpect a $95^{\circ}$ confidence interval for the inean.
b) The adininistrator decides that this interval is too widc. and she prefers one of only half this width. How large a sample size is nceded?

Aimal Hassan

Numerade Educator

01:12

Problem 8

Find the $\bar{z}$-score used in the formula for a confidence inter val uith confidence cocfficient a) .98 b) .90. c) .50. d) .9973 .

Sheryl Ezze

Numerade Educator

05:07

Problem 9

For the WWW data on number of times a week reading a newspaper. ıefeıred to in Problem 1.7, the observations for the 60 University of Flonda social science students have a mean of 4.1 and standard deviation of 3.0
a) For the population that this sample represents, construct and interpret a $95 \%$ confidence interval for the inean number of times a week reading a newspaper Would a $99 \%$ contidence interval for $\mu$ be wides. narrower, or the sarne width ${ }^2$ Explain.
b) Suppose that the sample slandard dcviation had been 6.0 . Find a $95 \%$ confidence intorval fol the mean, and compare it to the one obtained in part (a).
c) Suppose that the sample sizc had bern 240 , with $\bar{Y}=4.1$ and $s=3.0$. Again find a 05\% confidence interval 1or $\mu$ and compare it to the one obtained in part (a)

Sheryl Ezze

Numerade Educator

01:00

Problem 10

Recent General Social Surveys asked subjects. "How long have you lived in the city town or community where you live nou? The possible ıesponses were (less than one year. 1. 2. 3, 4. ) The responses of 1415 cubjects had a inode of "less than one year," a median of 16 years. a inean of 20.3 and a standard deviation of 18.2 .
a) De you think that the pepulation distulbution is nornal? Why or why not?
bj Based on you answer in (a), is it valid to construct a $99 \%$ confidence interval for the truc mean for the population represented by this sample? If not. explain why not If it is valid. do so and interpret.

Dominador Tan

Numerade Educator

01:17

Problem 11

In the 1994 General Social Survey: responses of 1964 subjects to the question. "On the average day. about how many houis do sou personally watch television ${ }^{\text {'mx }}$ had a mean of 2.8 The standard enmt of the mean was .05 .
a) Calculatc a $98 \%$ confidence interval for the unean daily time spent watching tclevision. Interpret
b) Find the standard deviation of the time spent watching television Do you think that the disunbution was bell-shaped? Why or why not?

Tyler Moulton

Numerade Educator

01:13

Problem 12

A study is conducted of the geographic disinbution of the residences of the employees at a large fastory. in order to detennive the suitability of initiating busing to that factory. One valiable considered is the distance the employee lives from the factory For a randoml sample of 100 employecs , the mean distance is 6.3 miles and the standard deviation is 4.0 .
a) Find and interpret a 90 \% confidence interval for the mcan residential distance fiom the factor of all employes.
b) About how largc a sumple would have been adequale if we merely needed to estimate the incan to within 1.0 . with $90 \%$ confidence?

Eleanor Archer

Numerade Educator

04:07

Problem 13

The 1991 General Social Surney asked. "Dunng the last year. did anyone take something from you by using force-such as a stickup. mugging. or threat? of 48 : subjects, 17 answered yes and 970 answcred no
a) Find the point estimate of the proportion of the population who were rictims of this tipe.
b) Construct and interpres a $95 \%$ confidence interval for the population proportion. Can you conclude that fewer than $5 \%$ were victims of this type?

Andrew Kim

Numerade Educator

02:59

Problem 14

In the 1991 General Social Survey. respondents were asked whether people convicted of murder should receive the death penalty. 1078 responded yes and 336 responded no. Construct a 995 confidence interval for the proporlion of American adulis who would answer yes. Inteıpret. Can you conclude that more than half of all American adulıs would answer yes? Why?

Gaurav Kalra

Numerade Educator

Problem 15

The 1994 General Social Sutvey ashed subjects "Do you believe these is a life aften dcath , Of 1958 respondents. 1425 answered ves. A report based on these data stated that $773^{\circ} \mathrm{C}$ of Americans believe in life aflal death. The margin of error for this result is plus or minus 2\%." Explain how the rescarchers obtained this 1esult.

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03:13

Problem 16

An anthropologist wants to estimate the proportion of childred in a tribe is the Philippines who die before reaching adulthood. Fot families she kneu who had children botn between 1970 and 1975,3 of 30 children died before reaching adulthood.
a) Construct a $90 \%$ confidence interval for the corresponding population proportion.
b) Explain why this interval is highly approximate and perhaps indppropriate. (Hinf. Check the sample size and sampling assumptions.)

Harsh Gadhiya

Numerade Educator

02:51

Problem 17

A study in 1991 by the U S National Center for Health Statistics provided a point estimatc of $255 \%$ for the percentage of adult Americans who were currently smokers. The sample sizc wias 42.000 . Assuning that this sample has the charactenstics of a random sample. construct and interpret a $99 \%$ confidence interval for the proportion of the population who were smohers. (Note: 'hen the sample size is very large. even confidence inter als with large confidence coefficients are narrow)

Nick Johnson

Numerade Educator

01:14

Problem 18

Refer to the previous problem. The same study provided a point estimate of $5.5 \%$ for the percentage of Americans whe averaged two or more alcoholic drinks per day
a) Construct and interpret a $99 \%$ confidence interval for the proportion of the population who drink that much
b) Compare the width of this inter al to the one in the previons prohlem
c) By computing values of the stanklard erron of $\hat{\pi}$ for $n=42.000$ when $\pi=1,3,5,7, .9$, explain why it is more difficult to estimale Ir when it is near 5 than when it is near 0 or 1 .

Maxime Rossetti

Numerade Educator

03:16

Problem 19

A national telcvision networh samples 1400 volers aiter each has cast a vote in a state gubernalorial election OS these 1400 roters, 742 claim to have voled for the Democratic candidate and 658 for the Republican candidate. There are only two candidares in the elaction.
a) Assuming that each sonapled voter actually voted as claimed and that the sample is a random cample from the population of all voters, is there enough evidence to predict the uinner of the election? Base you decision on a $95 \%$ confidence interval.
b) Base jour decision on a $99 \%$ confidence interval Explann why it requires greater evidence to make a prediction when we require greater confidence of heing correct.

Sheryl Ezze

Numerade Educator

Problem 20

Neursueek magazine (March 27. 1989) reponed results of a poll about religious beliefs, conducted by the Gallup organication. Of 750 American adults, $24 \%$ believed in reincarnation Treating this as a random sample, construct and interpres a $95 \%$ confidence interval for the true proportuon of Amencan adults believing in reincarnation.

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01:35

Problem 21

Or 577 .006 pcople involved in motor vehicle accidents in Florida in a secent ycar. 412,878 were wearing seat belts (Flonda Department of Highway Safety and Motor Vehicles). a) Find al point estimate of the population proportion of Florida motorists wearing seat belis.
b) Construct a $99 \%$ confidence inter al for the population proportion wearing seat belts at that time, Can we conclude that a majonty wore seat belts? Explain
c) For the inferences in (a) and (b) to be sensible, explan what you nust assumc about the sample data

Tony Wilson

Numerade Educator

02:45

Problem 22

A random sample is taken of students at a large university to determine the ptoportion who own automobiles. If $66 \%$ of the students in the sample own automobiles. form a 99\% confidence interval for the propotion of all students owning automobiles under the following conditions:
a) Sample sizc is 50 .
b) Sample size is 100 .
c) Sample size is 400 .

Nick Johnson

Numerade Educator

02:06

Problem 23

Out of an election day sample of 400 individuals who voted in a gubernatorial election. 160 voted for Jones and 240 for Smuth.
a) Assuming this is a random sample of all voters, construct a $99 \sigma^{\circ}$ confidence interval for the proportion of votes that Jones will receive Do you think that Jones will lose the election? Hily?
b) Suppose, instead, that the sample size had been 40 . of whom 16 voted Sor Jones Again. fiud the $99 \%$ confidence interval and. if possible. prediet the winuer How does the ıesult compare to (a)?

Sheryl Ezze

Numerade Educator

00:54

Problem 24

A city council voles to appropriate funds for a new civic auditorium. The mayor of the city threatens to veto this decision uniess it call be shoun that a majority of citizens would use it at least twice a yedr. The council commissions a poll of cily residents. For a random sample of 400 Iesidcuts, 230 sas they would use the facility at least tw ice a yea Find a $98 \%$ confidence interval for the propertion of all residents of the tuwn who would say they would use the proposed auditorium at least rwice a year. Intcrpret the inten al and advise the may or.

Hast Aggarwal

Numerade Educator

01:12

Problem 25

Refel to Problem 5.16 Before obldining the samplc. the anthropologist wanted to determine the size of sample nerded to estimale $\pi$ to within 07 with $95 \%$ confidence. Assuming nothing about the salue of $\pi$, calculate the sample size that ensures at leass this degree of accuracy

Sheryl Ezze

Numerade Educator

01:51

Problem 26

To estimase the proportion of traffic deaths in Flot ida last ycat that were alcohol related. determinc the necessary sample size for the esumate io be accurate to within 06 with probability .90 . Based on results of a previous study. we expect the proportion to be about 30

Sheryl Ezze

Numerade Educator

02:22

Problem 27

A study is being planned for estimating the proportion of mamned women with Jiving parents in the United States who live in the same statc as theil parents Approximatel) how large a cample size is needed to guarantce estinating this proportion correct to within a) .10 with probability 95
b) 05 with pıobabizity .95
c) 05 witla probability 99
d) .01 with probability 99
Compare sainple stres for parts (a) and (b), (b) and (c), and (c) and (d)

Sheryl Ezze

Numerade Educator

08:10

Problem 28

A public health unit wants to sample death records for the past year in New York City to estimate the proportion of the deaths that were due to accidents. They want the estimate to be accurate to within 02 uith piobability 95
a) Find the necessary sample size if, based on previous studies they betieve that this proportion docs not exceed 10 .
b) Suppose that in determining the necessary sample sizc they use the sale approach that sels $\vec{z}=5$ in the appropriate formula Then bow many recurds nced to be sampled? Compare the result to the answer in part (a). and note the reduction in sample size that occurs by making an educated ouess for $\pi$

Robin Corrigan

Numerade Educator

02:51

Problem 29

A tax assessor wants to estimate the mean propert tax bill for all homeouners living in Madison. Wisconsin. A survey complered three years ago indicated that the mean and standard deviation were $$\$ 1400$$ and $$\$ 1000$$. Assunning that the standard deviation is still about $$\$ 1000$$. how large a sample of tax records should the tax assessor take in order to extimale the mean correct to within $$\$ 100$$ with probability $.90^{\circ}$

Calin Lupas

Numerade Educator

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Problem 30

An estimate is necded of the meun acreage of farms in Canadi. The cstimate must be correct to within 25 acres with probability 95 . A prcliminary study suggests that 200 acres is a rcasonable guess lor the standard deviation of farm size. Hou large a sample of farms is required?

Danielle Fairburn

Numerade Educator

Problem 31

Refer to the previous exercise. A sample is selected of the size believed to be needed to esumate the mean correct to wilhin 25 acres with probability .95 Suppose, however, than the sample bas a standard deviation of 400 acres, rathes than 200 . Then, how close can we expect the sample mean to be to the true mean? In other words. whar amoumt is added to and subtracted fiom the sainple mean to get a $95 \%$ confidence inler al fot the mean acrcagc of farme?

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02:51

Problem 32

How large a sample size is tiecded to estimate the mean annual income of Native Amcricans correct to within $$\$ 1000$$ with probability $.99^{\circ}$ Suppose there is no prior information about the standard deviation of annual incoine of Native Americans, but we guess that aboun $95 \%$ of their incomes are between $$\$ 6000$$ and $$\$ 50,000$$ and that this disrribution of incomes is approximately mound shaped.

Calin Lupas

Numerade Educator

00:55

Problem 33

A random sample of $n=20$ pnvate compames having at Jcast 100 employees each is chosen to form the basis of a study of political contributions by industry. In particular, a $95 \%$. confidence interval is wanred for the median ycarly total concribution to political campaigns by all companics with at least 100 employees. The interest in the median instead of the mean is duc to the belief that thas distnbution may be very skewed to the right The reported total contributions for the past year for the 20 companies sampled arc, in thousands of dollars
$\begin{array}{rrrrrrrrrr}10 & 17 & 1 & 15 & 14 & 0 & 5 & 2 & 8 & 10 \\ 0 & 3 & 4 & 100 & 3 & 0 & 25 & 12 & 9 & 36\end{array}$
Show that a 95 \% confidence interval for the medidn contrabution is ( $\$ 3000 . \$ 14,000$ )

Maxime Rossetti

Numerade Educator

Problem 34

Refer to Problem 56 Construct a $95 \%$ confidence interval for the inedian annual income of the public housing residents. Interpret

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06:22

Problem 35

Refer to Example 5.5. Constuct a (a) $90 \%$. (b) $99 \%$ confidence interval for the median time since a book was last checked out. Interpret.

Willis James

Numerade Educator

01:17

Problem 36

Refer to the WWW data set (Problem 1.7). Using computeı softwale. construct and interpret a $95 \%$ confidence interval for the mean weekl) number of hotrs spent watching TV

Tyler Moulton

Numerade Educator

02:02

Problem 37

For the WWW data set, find a $95 \%$ confidence interval for the proportion believing in life after death interpret.

Sanchit Jain

Numerade Educator

Problem 38

Refet to the data file created in Problem 1.7 For variables chosen by your instructor, conduct inferential statistical analyses using basic methods of point and inter al cstination. Summarize and interplet your findings

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Problem 39

Based on dara from the National Survey of Families and Households, a recent study (by S South and G Spitze, Ameriran Sociological Review: Vol. 59, 1994. pp 327-347) reported the descriptive stalistics in Table 5.4 for the bours spent in housework and in cmployment per week. Analyze these data. and summarize your conclusions

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06:55

Problem 40

The 1994 General Social Survey asked subjecıs about government spending on the cnvironment. Table 5.5 shows the responses. Analyze and interpret these data
(TABLE CAN'T COPY)

Jeremiah Mbaria

Numerade Educator

02:58

Problem 41

The 1988 General Social Survey asked respondents, "Did you live with your husband/wife belore you got manicd?" The responses were 176 yes, 566 no, 725 nol applicuble, and 14 no answer. Analyze these data.

Sheryl Ezze

Numerade Educator

07:53

Problem 42

Explain what it means for an estimator to be (a) unbiased, (b) efficient.

Tyler Tebbs

Numerade Educator

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Problem 43

Explain the distinction berween point estimation and unrenal estimation. Why is an inlerval estimare more infornative than a point estimate?

Ana Carolina Da Cruz

Numerade Educator

00:51

Problem 44

What is the purpose of forming a confidence interval for a population mean? What does one leam that one could not otheruise learn by calculating a point essimate. such as the cample mean?

Sheryl Ezze

Numerade Educator

01:01

Problem 45

Explain why confidence intervalk are widel when we use (a) larger confidence coeffcients. (b) smaller sample sizes.

Sanchit Jain

Numerade Educator

01:33

Problem 46

Why would it be unusual to sce a $99.9999 \%$ confidence inter al?

Harsh Gadhiya

Numerade Educator

01:07

Problem 47

Give an examplc of a stady in which it would be important to have
a) A high degree of confidence
b) A high degree of precision
c) Both a high degrce of confidence and a high degrec of precision

Tyler Moulton

Numerade Educator

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Problem 48

How does population heterogeneit! affect the sample size required to estimate a population mean? Illustrate wth an example.

Rashmi Sinha

Numerade Educator

03:04

Problem 49

Increasing the confidence coefficient causes the width of $d$ confidence interval to (a) increase, (b) docrease. (c) stay the samc

Harsh Gadhiya

Numerade Educator

01:09

Problem 50

Other things being equal, quadrupling the samplc sice causes the width of a confidence interval to (a) double, (b) halve. (c) be one quarles as wide. (d) stay the same

Jerrah Biggerstaff

Numerade Educator

01:14

Problem 51

Based on responses of 1467 subjects in General Social Surveys in the mid-1980s, a $95 \%$ confidence interval for the mean number of close fnends equals ( 6.8 .80 ). Which of the following interpretations is (are) correct?
a) We can be $95 \%$ confident that $\bar{Y}$ is between 68 and 8.0
b) We can be $95 \%$ confident that $\mu$ is between 6.8 and 8.0 .
c) Ninery-five percent ol the values of $Y=$ number of close friecuds (for this cample) ate betwcen 6.8 and 8.0
d) If random samples of sice 1467 werc repeatedly selected. then $95 \%$ of the lime $\tilde{P}$ would be between 6.8 and 8.0 .
e) If random samiples of size 1467 u ere repeatedly selected, then in the long run $95 \%$ of the coutidence incervals formed uould contain the true value of $\mu$.

Tyler Moulton

Numerade Educator

04:03

Problem 52

A random sample of 50 records yields $395 \%$ confidence interval for the mean age at firm marnage ol women in a certain counly of 21.5 to 23.0 years. Explaill uhat is wrong with each of the following interpretations of this interval.
a) If random samples of 50 records were repeatedly selected. then $95 \%$ of the time the sample inean age at first marriage for women would be between 215 and 23.0 years.
b) Ninet! -five percent of the agcs at first maniage for uomen in the county are betweell 21.5 and 23.0 years
c) We can be $95 \%$ confident that $\bar{Y}$ is between 21.5 and 23.0 years
d) If ue repeatedly sampled the entre population, then $95 \%$ of the time the population mean would be between 21.5 and 235 ycars

Beth Stone

Numerade Educator

Problem 53

Refer to the previous cxcrcisc Provide the pioper interpretation,

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Problem 54

For a random samplc of $n$ subjects. explam why it is about $95 \%$ likely that the sample proportion has an earor of no inore than $1 / \sqrt{n}$ as an estimatc of the popalation proportion.

Nick Johnson

Numerade Educator

00:02

Problem 55

An article reports that ( 4.0 .5 .6 ) is a $95 \%$ confidence interval for the mean length of stav, in days. of parients in hospital fot a particular operation. The artucle reports the sanıple sice of 50 , but not the sample mean or standard deviatiou Find them

Sheryl Ezze

Numerade Educator

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Problem 56

To encourage subjects to make responses on sensitive questions. the method of randomized response is oflen used. The subject is asked to flip a coin. in secret If it is a head, the subject tosses the coin once more and reports the outcome, head or tails. If. instead, the fiust ftip is a tail. the subject icports instead the response to the sensitive question, for instance reporting the response head if the true response is wes and reporting the response tail if the truc response is no. Lel $\pi$ denote the true probability of the yes response on the sensitive question.
a) Explain why the numbers in Table 56 are the probabilities of the four possible outcomes
b) Let $p$ denote the sample proportion of subjects who report head for the second responsc Explain why $\hat{\pi}=2 p-.5$ estimates . .
c) Using this approach, 200 subjecrs are asked whether they have ever knowingly cheated on their income tax. Reporf the estimate of $\pi$ if the nuinber of reported heads equals (i) 50 , (ii) 70 . (iii) 100 . (iv) 150 .
(TABLE CAN'T COPY)

Shu Naito

Numerade Educator

07:14

Problem 57

Tu cunsiruct a large-sampple confidence interval for a proportion $\pi$, it is not necessary to substitule $\vec{\pi}$ for the unh nown value of $\pi$ in the formula for the standard error of $\hat{\pi}$. A less approximate method for constructing a $95 \%$ confidence interval finds the endpoints by delermining the $\pi$ values that dre 1.96 standard errors from the sample proportion. That is, one colves for $\pi$ it the equation
$$
|\hat{\pi}-\pi|=1.96 \sqrt{\frac{\pi(1-\pi)}{n}}
$$

One can solve this by trial and esror. using the endpoints of the usual confidence interval as initial guesses. Oi one can equare both sides of the equation and solve the resulting quadratic equation.
a) Use this method for the datd un Problem 5.16, and compare the result to that ubtained with the usual method.
b) Explan what happens with the usual melhod when the sample proportion equals 0 or 1 . (The two methods give similar results for very large samples. but otherwise can be quite different if the sample proportion is near 0 or 1. )

Jeremiah Mbaria

Numerade Educator

01:25

Problem 58

Derive the formula for determining low large a sample size is needed to estimate a population mean to within $B$ units with probability $.9 \overline{5}$.

Rashmi Sinha

Numerade Educator