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Øving 6
STAT111
Sondre Hølleland
Auditorium π 4
14. Mars 2016
Oppgaver
• Section 10.1: 2, 3, 4, 5
• Section 10.2: 31, 33.
Fasit
Section 10.1:
2. 4.85
3. 1.76
4. a) (1250.67, 2949.33)
b) 6352.95
5. a) -2.90
b) 0.0019
c) 0.8212
d) 65.15, så bruk 66
Section 10.2:
31. p-verdi: 0.048
q
33. a) x − y ± tα/2,m+n−2 · sp m1 + n1 ,
b) (−0.24, 3.64)
c) (−0.34, 3.74)
496
CHAPTER
10
Inferences Based on Two Samples
horizontal axis). Would the shape of the curve
necessarily be the same for sample sizes of
10 batteries of each type? Explain.
2. Let m1 and m2 denote true average tread lives for
two competing brands of size P205/65R15 radial
tires. Test H0: m1 m2 ¼ 0 versus Ha: m1 m2 6¼ 0 at level .05 using the following data:
m ¼ 45, x ¼ 42; 500, s1 ¼ 2200, n ¼ 45,
y ¼ 40; 400, and s2 ¼ 1900.
3. Let m1 denote true average tread life for a premium
brand of P205/65R15 radial tire and let m2 denote
the true average tread life for an economy brand of
the same size. Test H0: m1 m2 ¼ 5000 versus Ha:
m1 m2 > 5000 at level .01 using the following
data: m ¼ 45, x ¼ 42; 500, s1 ¼ 2200, n ¼ 45,
y ¼ 36; 800, and s2 ¼ 1500.
4. a. Use the data of Exercise 2 to compute a 95% CI
for m1 m2. Does the resulting interval suggest
that m1 m2 has been precisely estimated?
b. Use the data of Exercise 3 to compute a 95%
upper confidence bound for m1 m2.
5. Persons having Raynaud’s syndrome are apt to
suffer a sudden impairment of blood circulation
in fingers and toes. In an experiment to study the
extent of this impairment, each subject immersed
a forefinger in water and the resulting heat output
(cal/cm2/min) was measured. For m ¼ 10 subjects
with the syndrome, the average heat output was
x ¼ :64, and for n ¼ 10 nonsufferers, the average
output was 2.05. Let m1 and m2 denote the true
average heat outputs for the two types of subjects.
Assume that the two distributions of heat output
are normal with s1 ¼ .2 and s2 ¼ .4.
a. Consider testing H0: m1 m2 ¼ 1.0 versus
Ha: m1 m2 < 1.0 at level .01. Describe in
words what Ha says, and then carry out the test.
b. Compute the P-value for the value of Z
obtained in part (a).
c. What is the probability of a type II error when
the actual difference between m1 and m2 is m1 m2 ¼ 1.2?
d. Assuming that m ¼ n, what sample sizes are
required to ensure that b ¼ .1 when m1 m2 ¼ 1.2?
6. An experiment to compare the tension bond
strength of polymer latex modified mortar (Portland
cement mortar to which polymer latex emulsions
have been added during mixing) to that of unmodified mortar resulted in x ¼ 18:12 kgf=cm2 for the
modified mortar (m ¼ 40) and y ¼ 16:87 kgf=cm2
for the unmodified mortar (n ¼ 32). Let m1 and
m2 be the true average tension bond strengths for
the modified and unmodified mortars, respectively.
Assume that the bond strength distributions are both
normal.
a. Assuming that s1 ¼ 1.6 and s2 ¼ 1.4, test H0:
m1 m2 ¼ 0 versus Ha: m1 m2 > 0 at level .01.
b. Compute the probability of a type II error for
the test of part (a) when m1 m2 ¼ 1.
c. Suppose the investigator decided to use a level
.05 test and wished b ¼ .10 when m1 m2 ¼ 1.
If m ¼ 40, what value of n is necessary?
d. How would the analysis and conclusion of part
(a) change if s1 and s2 were unknown but
s1 ¼ 1.6 and s2 ¼ 1.4?
7. Are male college students more easily bored than
their female counterparts? This question was examined in the article “Boredom in Young Adults—
Gender and Cultural Comparisons” (J. Cross-Cult.
Psych., 1991: 209–223). The authors administered
a scale called the Boredom Proneness Scale to 97
male and 148 female U.S. college students. Does
the accompanying data support the research hypothesis that the mean Boredom Proneness Rating is
higher for men than for women? Test the appropriate hypotheses using a .05 significance level.
Gender
Sample
Size
Sample
Mean
Sample
SD
Male
Female
97
148
10.40
9.26
4.83
4.68
8. Is touching by a coworker sexual harassment? This
question was included on a survey given to federal
employees, who responded on a scale of 1–5, with
1 meaning a strong negative and 5 indicating a
strong yes. The table summarizes the results.
Gender
Sample
Size
Sample
Mean
Sample
SD
Female
Male
4343
3903
4.6056
4.1709
.8659
1.2157
Of course, with 1–5 being the only possible
values, the normal distribution does not apply
here, but the sample sizes are sufficient that it
does not matter. Obtain a two-sided confidence
interval for the difference in population means.
Does your interval suggest that females are more
likely than males to regard touching as harassment?
Explain your reasoning.
508
CHAPTER
1
1
1
2
2
2
2
2
2
2
2
10
Inferences Based on Two Samples
Kenyon
Oberlin
Franklin and
Marshall
Goucher
Randolph-Macon
Thomas Aquinas
Beloit
Austin
Ursinus
Siena
Juniata
ber of cycles to break were 4358 and 2218, respectively, whereas a sample of 20 polyisoprene
condoms gave a sample mean and sample standard
deviation of 5805 and 3990, respectively. Is there
strong evidence for concluding that the true average number of cycles to break for the polyisoprene
condom exceeds that for the natural latex condom
by more than 1000 cycles? [Note: The article presented the results of hypothesis tests based on the t
distribution; the validity of these depends on
assuming normal population distributions.]
38140
36282
36480
31082
26830
20400
30138
21586
35160
22685
28920
33. Consider the pooled t variable
a. Construct a comparative boxplot of expenses,
and comment on any interesting features.
b. Obtain a 95% confidence interval for the difference of population means. Interpret your
result in terms of the additional cost of attending a more prestigious college. Moving up
from tier 2 to tier 1 raises the cost by roughly
what percentage?
31. The article “Characterization of Bearing Strength
Factors in Pegged Timber Connections” (J. Struct.
Engrg., 1997: 326–332) gave the following summary data on proportional stress limits for specimens constructed using two different types of wood:
Type
of Wood
Red oak
Douglas fir
Sample
Size
Sample
Mean
Sample
SD
14
10
8.48
6.65
.79
1.28
Assuming that both samples were selected from
normal distributions, carry out a test of hypotheses
to decide whether the true average proportional
stress limit for red oak joints exceeds that for
Douglas fir joints by more than 1 MPa.
32. According to the article “Fatigue Testing of
Condoms” (Polym. Test., 2009: 567–571), “tests
currently used for condoms are surrogates for the
challenges they face in use,” including a test for
holes, an inflation test, a package seal test, and
tests of dimensions and lubricant quality (all fertile territory for the use of statistical methodology!). The investigators developed a new test
that adds cyclic strain to a level well below breakage and determines the number of cycles to break.
The cited article reported that for a sample of 20
natural latex condoms of a certain type, the sample
mean and sample standard deviation of the num-
T¼
ðX YÞ ðm1 m2 Þ
rffiffiffiffiffiffiffiffiffiffiffiffi
1 1
þ
Sp
m n
which has a t distribution with m + n 2 df when
both population distributions are normal with
s1 ¼ s2 (see the Pooled t Procedures subsection
for a description of Sp).
a. Use this t variable to obtain a pooled t confidence interval formula for m1 m2.
b. A sample of ultrasonic humidifiers of one particular brand was selected for which the observations on maximum output of moisture (oz)
in a controlled chamber were 14.0, 14.3, 12.2,
and 15.1. A sample of the second brand gave
output values 12.1, 13.6, 11.9, and 11.2
(“Multiple Comparisons of Means Using
Simultaneous Confidence Intervals,” J. Qual.
Techn., 1989: 232–41). Use the pooled t formula from part (a) to estimate the difference
between true average outputs for the two
brands with a 95% confidence interval.
c. Estimate the difference between the two m’s
using the two-sample t interval discussed in this
section, and compare it to the interval of part (b).
34. Refer to Exercise 33. Describe the pooled t test for
testing H0: m1 m2 ¼ 0 when both population
distributions are normal with s1 ¼ s2. Then use
this test procedure to test the hypotheses suggested
in Exercise 32.
35. Exercise 35 from Chapter 9 gave the following
data on amount (oz) of alcohol poured into a short,
wide tumbler glass by a sample of experienced
bartenders: 2.00, 1.78, 2.16, 1.91, 1.70, 1.67,
1.83, 1.48. The cited article also gave summary
data on the amount poured by a different sample
of experienced bartenders into a tall, slender
(highball) glass; the following observations are
consistent with the reported summary data: 1.67,
1.57, 1.64, 1.69, 1.74, 1.75, 1.70, 1.60.