Chapter 8 - intro to probability and statistics: large sample estimation concepts môn Xác suất thống kê| Trường Đại học Ngoại Thương

8.7 ESTIMATING THE DIFFERENCE BETWEEN TWO BINOMIAL PROPORTIONS ❍309 .11 !(2.58)(.0770) .11 !.199
or (".089, .309). Since this interval contains the value ( p "p ) #0, it is 1 2
possible that p #p , which implies that there may be no difference in the 1 2
proportions favoring the bond issue in the two sections of the city.
2. If there is no difference in the two proportions, then the two samples are not
really different and might well be combined to obtain an overall estimate of
the proportion of the city residents who will vote for the bond issue. If both
samples are pooled, then n#150 and 103 pˆ # $150$ #.69 Therefore, the
" point" estimate of the overall value of pis .69, with a margin of error given by !1.96! (.69)(.31) $ #!1.96(.0378) #!.074 15 0 $
Notice that .69 !.074 produces the interval .62 to .76, which includes only
proportions greater than .5. Therefore, if voter attitudes do not change adversely
prior to the election, the bond proposal should pass by a reasonable majority. 8.7 EXERCISES BASIC TECHNIQUES
a. Find a 99% confidence interval for the difference
8.54 Independent random samples of n #500 and
(p "p ) in the two population proportions. What 1 2 1
does “99% confidence” mean?
n #500 observations were selected from binomial 2
populations 1 and 2, and x #120 and x #147 suc-
b. Based on the confidence interval in part a, can you 1 2 cesses were observed.
conclude that there is a difference in the two bino-
a. What is the best point estimator for the difference mial proportions? Explain.
(p "p ) in the two binomial proportions? 1 2
b. Calculate the approximate standard error for the APPLICATIONS statistic used in part a.
8.57 M&M’S Does Mars, Incorporated use the same
c. What is the margin of error for this point estimate?
proportion of red candies in its plain and peanut vari-
8.55 Independent random samples of n #800 and
eties? A random sample of 56 plain M&M’S contained 1
n #640 observations were selected from binomial
12 red candies, and another random sample of 32 2
populations 1 and 2, and x #337 and x #374 suc-
peanut M&M’S contained 8 red candies. 1 2 cesses were observed.
a. Construct a 95% confidence interval for the differ-
a. Find a 90% confidence interval for the difference
ence in the proportions of red candies for the plain
(p "p ) in the two population proportions. Inter- and peanut varieties. 1 2 pret the interval.
b. Based on the confidence interval in part a, can you
b. What assumptions must you make for the confidence
conclude that there is a difference in the proportions
interval to be valid? Are these assumptions met?
of red candies for the plain and peanut varieties? Explain.
8.56 Independent random samples of n #1265 and 1
n #1688 observations were selected from binomial
8.58 Different Priorities As we approached the 2
populations 1 and 2, and x #849 and x #910 suc-
midterm elections, in the summer of 2010, Democrats 1 2 cesses were observed.
and Republicans were split about our nation’s top
310 ❍CHAPTER 8 LARGE-SAMPLE ESTIMATION
priorities.13 A sample of n#900 registered voters
people are trying to meet quotas, and talking to a man-
were asked the following question: “Which ONE of
ager rather than a salesperson. Suppose that random
the following items do you think is most important for
samples of 200 men and 200 women are taken, and that
the federal government to be working on right now?”
the men were more likely than the women to say they
The list of options is shown in the table below.
“always or often” bargained (30% compared with 25%).
Options were rotated to reduce bias, and voters were
a. Construct a 95% confidence interval for the differ-
allowed to indicate “All,” “None,” or “Unsure.”
ence in the proportion of men and women who say
they “always or often” negotiate for a better deal.
All Democrats Republicans Independents (%) (%) (%) (%)
b. Do the data indicate that there is a difference in the Economy 47 55 37 48
proportion of men and women who say they “always and Jobs
or often” negotiate for a better deal? Explain. Deficit, 15 8 22 16 Spending
8.61 Catching a Cold Do well-rounded people get Terrorism, 8 6 10 10
fewer colds? A study on the Chronicle of Higher Edu- Security
cation was conducted by scientists at Carnegie Mellon Iraq and 7 9 5 4
University, the University of Pittsburgh, and the Uni- Afghanistan
versity of Virginia. They found that people who have Immigration 5 3 7 6
only a few social outlets get more colds than those All (vol.) 16 18 16 13 None/
who are involved in a variety of social activities.16 Other (vol.) 1 - 1 2
Suppose that of the 276 healthy men and women Unsure 1 - 1 -
tested, n1 #96 had only a few social outlets and
Suppose that there were 400 Democrats, 350 Republi-
n2 #105 were busy with six or more activities. When
cans, and 150 Independents in the sample. Use a large-
these people were exposed to a cold virus, the follow-
sample estimation procedure to compare the propor- ing results were observed:
tions of Republicans and Democrats who mentioned Few Social Outlets Many Social Outlets
the economy and jobs as the most important item for Sample Size 96 105
the federal government to work on. Compare the pro- Percent with Colds 62% 35%
portions of Republicans and Independents who men-
tioned deficit spending as the most important item.
a. Construct a 99% confidence interval for the Explain your conclusions.
difference in the two population proportions.
b. Does there appear to be a difference in the
8.59 Baseball Fans The first day of baseball comes
population proportions for the two groups?
in late March, ending in October with the World
Series. Does fan support grow as the season goes on?
c. You might think that coming into contact with more
Two CNN/USA Today/Gallup polls, one conducted in
people would lead to more colds, but the data show
March and one in November, both involved random
the opposite effect. How can you explain this
samples of 1001 adults aged 18 and older. In the unexpected finding?
March sample, 45% of the adults claimed to be fans of
8.62 Union, Yes! A sampling of political
professional baseball, while 51% of the adults in the
candidates—200 randomly chosen from the West and
November sample claimed to be fans.14
200 from the East—was classified according to
a. Construct a 99% confidence interval for the differ-
whether the candidate received backing by a national
ence in the proportion of adults who claim to be
labor union and whether the candidate won. In the fans in March versus November.
West, 120 winners had union backing, and in the East,
b. Does the data indicate that the proportion of adults
142 winners were backed by a national union. Find a
who claim to be fans increases in November,
95% confidence interval for the difference between the
around the time of the World Series? Explain.
proportions of union-backed winners in the West ver-
sus the East. Interpret this interval.
8.60 When Bargaining Pays Off According to a
national representative survey done by Consumer
8.63 Birth Order and College Success In a study
Reports, you should always try to negotiate for a better
of the relationship between birth order and college suc-
deal when shopping or paying for services.15 Tips
cess, an investigator found that 126 in a sample of 180
include researching prices at other stores and on the
college graduates were firstborn or only children. In a
Internet, timing your visit late in the month when sales-
sample of 100 nongraduates of comparable age and
8.8 ONE-SIDED CONFIDENCE BOUNDS ❍311
socioeconomic background, the number of firstborn or
used the second pain reliever, 96% indicated that it
only children was 54. Estimate the difference between relieved their pain.
the proportions of firstborn or only children in the two
a. Find a 99% confidence interval for the difference in
populations from which these samples were drawn. Use
the proportions experiencing relief from pain for
a 90% confidence interval and interpret your results. these two pain relievers.
8.64 Generation Next Born between 1980
b. Based on the confidence interval in part a, is there
and 1990, Generation Next is engaged with technology,
sufficient evidence to indicate a difference in the
and the vast majority is dependent upon it.17 Suppose
proportions experiencing relief for the two pain
that in a survey of 500 female and 500 male students in relievers? Explain.
Generation Next, 345 of the females and 365 of the
8.66 Auto Accidents Last year’s records of auto
males reported that they decided to attend college in
accidents occurring on a given section of highway order to make more money.
were classified according to whether the resulting
a. Construct a 98% confidence interval for the differ-
damage was $1000 or more and to whether a
ence in the proportions of female and male students
physical injury resulted from the accident. The data
who decided to attend college in order to make follows: more money. Under $1000 $1000 or More
b. What does it mean to say that you are “98% Number of Accidents 32 41 confident”? Number Involving Injuries 10 23
c. Based on the confidence interval in part a, can you
conclude that there is a difference in the proportions
a. Estimate the true proportion of accidents involving
of female and male students who decided to attend
injuries when the damage was $1000 or more for
college in order to make more money?
similar sections of highway and find the margin of error.
8.65 Excedrin or Tylenol? In a study to compare
b. Estimate the true difference in the proportion of
the effects of two pain relievers it was found that of
accidents involving injuries for accidents with dam-
n1 #200 randomly selected individuals who used the
age under $1000 and those with damage of $1000
first pain reliever, 93% indicated that it relieved their
or more. Use a 95% confidence interval.
pain. Of n2 #450 randomly selected individuals who ONE-SIDED CONFIDENCE BOUNDS 8.8
The confidence intervals discussed in Sections 8.5 to 8.7 are sometimes called two-sided
confidence intervals because they produce both an upper (UCL) and a lower (LCL)
bound for the parameter of interest. Sometimes, however, an experimenter is interested
in only one of these limits; that is, he needs only an upper bound (or possibly a lower
bound) for the parameter of interest. In this case, you can construct a one-sided confi-
dence bound for the parameter of interest, such as m, p, m1 "m 2, or p1 "p 2.
When the sampling distribution of a point estimator is approximately normal, an
argument similar to the one in Section 8.5 can be used to show that one-sided confi-
dence bounds, constructed using the following equations when the sample size is large,
will contain the true value of the parameter of interest (1 "a)100% of the time in repeated sampling.
A (1 !a)100% LOWER CONFIDENCE BOUND (LCB)
(Point estimator) "z a %(Standard error of the estimator)
A (1 !a)100% UPPER CONFIDENCE BOUND (UCB)
(Point estimator) &z a %(Standard error of the estimator)