Confidence intervals for population means analysis môn Xác suất thống kê| Trường Đại học Ngoại Thương

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304 ❍CHAPTER 8 LARGE-SAMPLE ESTIMATION
Solution The confidence interval is constructed using a value of zwith tail area
a/2 !.025 to its right; that is, z
# #!1.96. Using the sample standard deviations to ap- proximate the unknown popul .0
at2i5on standard deviations, the 95% confidence interval is ) #1.96" s s (x! "x ! # # $2 2 n1$ % $n2$ 1 2 (756 "762) #1.96" 1 2 35 30 $ 2 2 50$ % $50$ "6 #12.78 or "18.78 &(m
"m ) &6.78. Look at the possible values for (m "m ) in the con-
fidence interval 1. It is 2possible that the difference (m"m ) could 1be neg2ative (indicat-
ing that the average for women exceeds the av1erage 2for men), it could be positive
(indicating that men have the higher average), or it could be 0 (indicating no difference
between the averages). Based on this information, you should not be willing to conclude
that there is a difference in the average daily intakes of dairy products for men and women.
Examples 8.9 and 8.10 deserve further comment with regard to using sample esti-
mates in place of unknown parameters. The sampling distribution of "# # (x! "x ! ) "(m "m ) $ 1 $$ 2 1 2 s s $ 2 2 n1$ % $n2$ 1 2
has a standard normal distribution for all sample sizes when both sampled populations
are normal and an approximate standard normal distribution when the sampled
populations are not normal but the sample sizes are large ('30). When s and s 2 2
are not known and are estimated by the sample estimates s and s , the result 1ing stati 2 s- 2 2
tic will still have an approximate standard normal distrib 1 ution 2 when the sample sizes
are large. The behavior of this statistic when the population variances are unknown
and the sample sizes are small will be discussed in Chapter 10. 8.6 EXERCISES BASIC TECHNIQUES
8.42 Independent random samples were selected from
8.41 Independent random samples were selected from
populations 1 and 2. The sample sizes, means, and
populations 1 and 2. The sample sizes, means, and variances are as follows: variances are as follows: Population Population 12 12 Sample Size 64 64 Sample Size 35 49 Sample Mean 2.9 5.1 Sample Mean 12.7 7.4 Sample Variance 0.83 1.67 Sample Variance 1.38 4.14
a. Find a 90% confidence interval for the difference in
a. Find a 95% confidence interval for estimating the
the population means. What does the phrase “90%
difference in the population means (m "m ). confident” mean?
b. Based on the confidence interval in p1art a, 2 can you
b. Find a 99% confidence interval for the difference in
conclude that there is a difference in the means for
the population means. Can you conclude that there is the two populations? Explain.
a difference in the two population means? Explain.
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8.6 ESTIMATING THE DIFFERENCE BETWEEN TWO POPULATION MEANS ❍305
8.43 Independent random samples of size n
8.47 Teaching Biology An experiment was con- 1 !n 2 !
100 were selected from each of two populations. The
ducted to compare a teacher-developed curriculum,
mean and standard deviations for the two samples
“Biology: A Community Context” (BACC) that was were x!
standards-based, activity-oriented, and inquiry-
1 !125.2, x !2 !123.7, s 1 ! 5.6, and s 2 ! 6.8.
a. Construct a 99% confidence interval for estimating
centered to the traditional presentation using lecture,
the difference in the two population means.
vocabulary, and memorized facts. The perhaps not-so-
startling results when students were tested on biology
b. Does the confidence interval in part a provide suffi-
concepts, published in The American Biology Teacher,
cient evidence to conclude that there is a difference
are shown in the following table.11
in the two population means? Explain. Sample Standard
8.44 Independent random samples of size n 1 !n 2 ! Mean Size Deviation
500 were selected from each of two populations. The Pretest: All BACC Classes 13.38 372 5.59
mean and standard deviations for the two samples Pretest: All Traditional 14.06 368 5.45
were x!1 !125.2, x !2 !123.7, s 1 ! 5.6, and s 2 ! 6.8. Posttest: All BACC Classes 18.5 365 8.03
a. Find a point estimate for the difference in the two Posttest: All Traditional 16.5 298 6.96
population means. Calculate the margin of error.
a. Find a 95% confidence interval for the mean score
b. Based on the results in part a, can you conclude that
for the posttest for all BACC classes.
there is a difference in the two population means?
b. Find a 95% confidence interval for the mean score Explain.
for the posttest for all traditional classes. APPLICATIONS
c. Find a 95% confidence interval for the difference in
mean scores for the posttest BACC classes and the
8.45 Selenium A small amount of the trace element posttest traditional classes.
selenium, 50–200 micrograms (mg) per day, is consid-
ered essential to good health. Suppose that random
d. Does the confidence interval in c provide evidence samples of n
that there is a real difference in the posttest BACC 1 !n
2 !30 adults were selected from
two regions of the United States and that a day’s
and traditional class scores? Explain.
intake of selenium was recorded for each person. The
Source: From “Performance Assessment of a Standards-Based High School Biology
Curriculum,” by W. Leonard, B. Speziale, and J. Pernick in The American Biology
mean and standard deviation of the selenium daily
Teacher, 2001, 63(5), 310–316. Reprinted by permission of National Association of
intakes for the 30 adults from region 1 were x Biology Teachers. !1 !
167.1 and s1 !24.3 mg, respectively. The correspond-
ing statistics for the 30 adults from region 2 were x
8.48 Are You Dieting? To compare two weight ! 2 ! 140.9 and s
reduction diets A and B, 60 dieters were randomly
2 !17.6 mg. Find a 95% confidence inter-
val for the difference in the mean selenium intakes for
selected. One group of 30 dieters was placed on diet
the two regions. Interpret this interval.
A and the other 30 on diet B, and their weight losses
were recorded over a 30-day period. The means and
8.46 9-1-1 A study was conducted to compare the
standard deviations of the weight-loss measurements
mean numbers of police emergency calls per 8-hour
for the two groups are shown in the table. Find a
shift in two districts of a large city. Samples of
95% confidence interval for the difference in mean
100 8-hour shifts were randomly selected from the
weight loss for the two diets. Interpret your confi-
police records for each of the two regions, and the dence interval.
number of emergency calls was recorded for each
shift. The sample statistics are listed here: Diet A Diet B Region x! !21.3 x !13.4 A !B s !2.6 s !1.9 12 A B Sample Size 100 100
8.49 Starting Salaries As a group, students major- Sample Mean 2.4 3.1
ing in the engineering disciplines have the highest salary Sample Variance 1.44 2.64
expectations, followed by those studying the computer
Find a 90% confidence interval for the difference in
science fields, according to results of NACE’s 2010 Stu-
the mean numbers of police emergency calls per shift
dent Survey.12 To compare the starting salaries of col-
between the two districts of the city. Interpret the
lege graduates majoring in engineering and computer interval.
science, random samples of 50 recent college graduates
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306 ❍CHAPTER 8 LARGE-SAMPLE ESTIMATION
in each major were selected and the following informa-
a. Find a 95% confidence interval for the difference in tion obtained:
the average room rates for the Marriott and the Major Mean ($) SD Westin hotel chains.
b. Find a 99% confidence interval for the difference in Engineering 56,202 2225 Computer science 50,657 2375
the average room rates for the Westin and the Dou- bletree hotel chains.
a. Find a point estimate for the difference in the aver-
c. Do the intervals in parts a and b contain the value
age starting salaries of college students majoring in (m
engineering and computer science. What is the mar-
1 "m 2) !0? Why is this of interest to the researcher?
gin of error for your estimate?
d. Do the data indicate a difference in the average
b. Based upon the results in part a, do you think that there
room rates between the Marriott and the Westin
is a significant difference in the average starting
chains? Between the Westin and the Doubletree
salaries for engineers and computer scientists? Explain. chains?
8.50 Biology Skills Refer to Exercise 8.47. In addi-
tion to tests involving biology concepts, students were
8.52 Noise and Stress To compare the effect of
also tested on process skills. The results of pretest and
stress in the form of noise on the ability to perform
posttest scores, published in The American Biology
a simple task, 70 subjects were divided into two Teacher, are given below. 11
groups. The first group of 30 subjects acted as a con-
trol, while the second group of 40 were the experi- Sample Standard
mental group. Although each subject performed the Mean Size Deviation
task, the experimental group subjects had to perform Pretest: All BACC Classes 10.52 395 4.79
the task while loud rock music was played. The time Pretest: All Traditional 11.97 379 5.39
to finish the task was recorded for each subject and the Posttest: All BACC Classes 14.06 376 5.65
following summary was obtained: Posstest: All Traditional 12.96 308 5.93
a. Find a 95% confidence interval for the mean score Control Experimental
on process skills for the posttest for all BACC n30 40 classes. x! 15 minutes 23 minutes s4 minutes 10 minutes
b. Find a 95% confidence interval for the mean score
on process skills for the posttest for all traditional
a. Find a 99% confidence interval for the difference in classes.
mean completion times for these two groups.
c. Find a 95% confidence interval for the difference in
b. Based on the confidence interval in part a, is there
mean scores on process skills for the posttest
sufficient evidence to indicate a difference in the
BACC classes and the posttest traditional classes.
average time to completion for the two groups?
d. Does the confidence interval in c provide evidence Explain.
that there is a real difference in the mean process
skills scores between posttest BACC and traditional
8.53 What’s Normal, continued Of the 130 people class scores? Explain.
in Exercise 8.39, 65 were female and 65 were male.9
The means and standard deviation of their tempera-
Source: From “Peformance Assessment of a Standards-Based High School Biology
Curriculum,” by W. Leonard, B. Speziale, and J. Pernick in The American Biology tures are shown below.
Teacher, 2001, 63(5), 310–316. Reprinted by permission of National Association of Biology Teachers. Men Women
8.51 Hotel Costs Refer to Exercise 8.20. The Sample Mean 98.11 98.39
means and standard deviations for 50 billing state- Standard Deviation 0.70 0.74
ments from each of the computer databases of each of
Find a 95% confidence interval for the difference in
the three hotel chains are given in the table: 4
the average body temperatures for males versus Marriott Westin Doubletree
females. Based on this interval, can you conclude that
there is a difference in the average temperatures for Sample Average ($) 150 165 125 males versus females? Explain. Sample Standard Deviation 17.2 22.5 12.8