This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 9,75. Consider the test of H0: 0'2 : 5 against H1: 0'2 < 5.
Approximate the P—value for each of the following test statistics.
(3) xﬁ = 25.2 andn = 20
(b) 1%, =15.2andn =12 = 4.2 and r1 = 15 Consider the hypothesis test of H0: CI"2 = 10 against
1: 02 > 10. Approximate the Pvalue for each of the follow—
ing test statistics. (a) xﬁ = 25.2 andn = 20 ('3) x3 =15.2andn =12 (c) 1% = 4.23am =15 977. The data from Medicine and Science in Sports and
Exercise described in Exercise 848 considered ice hockey
player performance aﬁer electrostirnulation training. In
summary, there were 17 players and the sample standard devi— ation of performance was 0.09 seconds.
(a) Is there strong evidence to conclude that the standard devi ation of performance time exceeds the historical value of
0.75 seconds? Use or = 0.05. Find the P—value for this test. (b) Discuss how part (a) could be answered by constructing a
95% onesided conﬁdence interval for 0. 978 The data from Technometrics described in Exercise
8 1 considered the variability in repeated measurements of
the weight of a sheet of paper. In summary, the sample stan C ' (lard deviation from 15 measurements was 0.0083 grams. (a) Does the measurement standard deviation differ from 0.01
grams at ct = 0.05? Find the P—value for this test. (b) Discuss how part (a) could be answered by constructing a
conﬁdence interval for 0'. 979. Reconsider the percentage of titanium in an alloy used in aerospace castings from Exercise 847. Recall thats = 0.37 andn = 51. . (a) Test the hypothesis Ho: 0 = 0.25 versus H1: 0‘ sh 0.25 using or = 0.05. State any necessary assumptions about
the underlying distribution of the data. Find the P—value.
(b) Explain how you could answer the question in part (a) by
constructing a 95% twosided conﬁdence interval for 6. 980. Data from an Izod impact test was described in
Exercise 828. The sample standard deviation was 0.25 and H = 20 specimens were tested. 95 TESTS ON A POPULATION PROPORTION 32.3 conclusion. State any necessary assumptions about the
underlying distribution of the data. (b) What is the Pvalue for this test? (c) Could the question in part (a) have been answered by
constructing a 99% tWOsided conﬁdence interval for 0'2? 981. Data for the life was described in Exercise 827. The sample standard deviation was 3645.94 kilometers and n = 16. (a) Can you conclude, using or = 0.05, that the standard devia
tion of tire life is less than 4000 kilometers? State any nec~
essary assumptions about the underlying distribution of the
data. Find the Pvalue for this test. (b) Explain how you could answer the question in part (a) by
constructing a 95% one—sided conﬁdence interval for (r. 9—82. It the standard deviation of hole diameter exceeds
0.01 millimeters , there is an unacceptably high probability that
the rivet will not ﬁt. Suppose that n = 15 and s = 0.008 mil:
limeter. (a) Is there snong evidence to indicate that the standard devi
ation of hole diameter exceeds 0.01 millimeter? Use or =
0.01. State any necessary assumptions about the underly
ing distribution of the data. Find the Pvalue for this test. (b) Suppose that the actual. standard deviation of hole diame—
ter exceeds the hypothesized value by 50%. What is the
probability that this difference will be detected by the test
described in part (a)? (c) If U is really as large as 0.0125 millimeters, what sample size
will be required to detect this with power of at least 0.8? 983. Recall the sugar content of the syrup in canned
peaches from Exercise 846. Suppose that the variance is
thought to be 02 = 18 (milligramsf. Recall that a random
sample of n = 10 cans yields a sample standard deviation of
s = 4.8 milligrams. (a) Test the hypothesis H0: cr2 = 18 versus H1: 02 at 18 using
(1 = 0.05. Find the P—value for this test. (b) Suppose that the actual standard deviation is twice as
large as the hypothesized value. What is the probability
that this diﬁerence will be detected by the test described in
part (a)? ' ' (c) Suppose that the true 'variance is o2 = 40. How large a
sample wouldbe required to detect this difference with probability at least 0.90? (a) Test the hypothesis that 0' = 0.10 against an alternative
specifying that (I at 0.10, using or = 0.01, and draw a 9—5 TESTS ON A POPULATION PROPORTION It is often necessary to test hypotheses on a population proportion. For example, suppose that
a random sample of size n has been taken from a large (possibly inﬁnite) population and that
X(S n) observations in this sample belong to a class of interest. Then P = X/n is a point
estimator of the proportion of the population p that belongs to this class. Note that n. and p are
the parameters of a binomial distribution. Furthermore, from Chapter 7 we know that the
sampling distribution of P is approximately normal with mean p and variance p(l — p) / n, if are. 96 SUMMARY TABLE OF INFERENCE PROCEDURES FOR A SINGLE SAMPLE 3 29 9.86. Suppose that 1000 customers are surveyed and 850 are satisﬁed or very satisﬁed with a corporation’s products and services. (a) Test the hypothesis Ho: p —— 0.9 against Hi: p at 0.9 at
or = 0.05. Find the Pvalue. ([3) Explain how the question in part (a) could be answered by
constructing a 95% two—sided conﬁdence interval for p. Suppose that 500 parts are tested in manufacturing
an 10 are rejected.
(a) Test the hypothesis H0: p = 0.03 against H1: p < 0.03 at
or = 0.05. Find the P—value.
(b) Explain how the question in part (a) could be answered by constructing a 95% onesided conﬁdence interval for p.
988. A random sample of 300 circuits generated 13
defectives. (a) Use the data to test H0: p = 0.05 versus H ,1 p =# 0.05. Use
or = 0.05. Find the P—value for the test. (b) Explain how the question in part (a) could be answered
with a conﬁdence interval. 9'89. An article in the British Medical Journal [“Comparison of Treatment of Renal Calculi by Operative Surgery, Percutaneous Nephrolithotomy, and ExtraCorporeal Shock Wave Lithotrips,” (1986, Vol. 292, pp. 879—882)] found that percutaneous nephrolithotomy (PN) had a success rate in removing kidney stones'of 289 out of 350 patients. The tradi tional method was 78% effective. (a) Is there evidence that the success rate for PN is greater
than the historical success rate? Find the Pvalue. (b) Explain how the question in part (a) could be answered
with a conﬁdence interval. 9—90. A manufacturer of interocular lenses is qualifying a new grinding machine and will qualify the machine if there is evidence that the percentage of polished lenses that contain Surface defects does not exceed 2%. A random sample of 250 lenses contains six defective lenses. (3) Formulate and test an appropriate set of hypotheses to de
termine if the machine can be qualiﬁed Use or = 0.05.
Find the Pvalue. 0)) Explain how the question in part (a) could be answered
with a conﬁdence interval. 991. A researcher claims that at least 10% of all football helmets have manufacturing flaws that could potentially cause
injury to the wearer. A sample of 200 helmets revealed that 16 helmets contained such defects. (a) Does this ﬁnding support the researcher’s claim? Use
at = 0.01. Find the Pvalue. (b) Explain how the question in part (a) could he answered
with a conﬁdence interval. 9—92. An article in Fortune (September 21, 1992) claimed
that nearly onehalf of all engineers continue academic studies
beyond the BS. degree, ultimately receiving either an MS. or a PhD. degree. Data from an article in Engineering Horizons (Spring 1990) indicated that 117 of 484 new engineering graduates were planning graduate study. (a) Are the data from Engineering Horizons consistent with
the claim reported by Fortune? Use or = 0.05 in reaching
your conclusions. Find the Pvalue for this test. 03) Discuss how you could have answered the question in part
(a) by constructing a two—sided conﬁdence interval on p. 993. The advertised claim for batteries for cell phones is
set at 48 operating hours, with proper charging procedures. A
study of 5000 batteries is carried out and 15 stop operating
prior to 48 hours. Do these experimental results support the
claim that less than 0.2 percent of the company’s batteries
will fail during the advertised time period, with proper
charging procedures? Use a hypothesistesting procedure
with or = 0.01. 994. A random sample of 500 registered voters in Phoenix is asked if they favor the use of oxygenated fuels yearround to reduce air pollution. If more than 315 voters respond posi tively, we will conclude that at least 60% of the voters favor the use of these fuels. (a) Find the probability of type I error if exactly 60% of the
voters favor the use of these fuels. (b) What is the type II error probability B if 75% of the voters
favor this action? ' In a random sample of 85 automobile engine crank
s: . ' bearings, 10 have a surface ﬁnish roughness that exceeds the speciﬁcations. Does this data present strong evidence that the proportion of crankshaft bearings exhibiting excess surface roughness exceeds 0.10? (a) State and test the appropriate hypotheses using or = 0.05. (b) If it is really the situation that p = 0.15, how likely is it
that the test procedure in part (a) will not reject the null
hypothesis? (c) If p = 0.15, how large would the sample size have to be
for us to have a probability of correctly rejecting the null
hypothesis of 0.9‘? 96 SUMMARY TABLE OF INFERENCE PROCEDURES FOR A SINGLE SAMPLE The table in the end papers of this book (inside back cover) presents a summary of all the
singlesample inference procedures from Chapters 8 and 9. The table contains the null
hypothesis statement, the test statistic, the various alternative hypotheses and the criteria “1.4 Lina 360 CHAPTER 10 STATISTICAL INFERENCE FOR Two SAMPLES 10%. Consider the hypothesis test Ho‘: 11.1 = pt; against
H] : u; > n2 with known variances 9‘1 = 10 and o; = 5.
Suppose that sample sizes n1 = 10 and it; = 15 and that
i, = 24.5 and E2 = 21.3. Use or = 0.01. (a) Test the hypothesis and ﬁnd the P—value. (b) Explain how the test could be conducted with a conﬁ
dence interval. (c) What is the power of the test in part (a) if ul is 2 units
greater than M? (d) Assuming equal sample sizes, what sample size shouldbe
used to obtain B = 0.05 if u] is 2 units greater than 1.1.2?
Assume that or = 0.05. 104. Two machines are used for ﬁlling plastic bottles with a net volume of 16.0 ounces. The ﬁll volume can be assumed normal, with standard deviation 0', = 0.020 and oz = 0.025 ounces. A member of the quality engineering staff suspects that both machines ﬁll to the same mean net volume, whether
or not this volume is 16.0 ounces. A random sample of 10 hot—
tles is taken from the output of each machine. 7, Machlnez
16:03 7 , . . '136102 _. 116.03.
_16.04 15.96 _ _ i_5.97_ 16.04
116.051 " ,;.1,5:9s '_ 15396; 16.02 '
16.05 16.02 7 16.01 16.01
.:1:6_L'02'_""1715.907 r...'1__5f99._’2_f 16.00 (a) Do you think the engineer is correct? Use or = 0.05. What
is the P—value for this test? (b) Calculate a 95% conﬁdence interval 011 the diﬁ'erence in
means. Provide a practical interpretation of this interval. (0) What is the power of the test in part (a) for a true diifer
ence in means of 0.04? (d) Assuming equal sample sizes, what sample size should be
used to assure that B = 0.05 if the true difference in cans is 0.04? Assume that Gt = 0.05.
Two types of plastic are suitable for use by an elec4 tronics component manufacturer. The breaking strength of this plastic is important. It is known that 01 = 0'; = 1.0 psi. From a random sample of size r11 = 10 and r22 = 12, we obtain 31 = 162.5 and 32 = 155.0. The company will not adopt plas— tic 1 unless its mean breaking strength exceeds that of plastic 2 by at least 10 psi. (a) Based on the sample information, should it use plastic 1?
Use or = 0.05 in reaching a decision. Find the P—value. (b) Calculate a 95% conﬁdence interval on the difference in
means. Suppose that the true difference in means is really
12 psi. (c) Find the power of the test assuming that or = 0.05. (d) If it is really important to detect a difference of 12 psi, are
the sample sizes employed in part (a) adequate, in your
opinion? 106. The burning rates of two different solidfuel propel ' lants used in aircrew escape systems are being studied. It is
known that both propellants have approximately the same
standard deviation of burning rate; that is 0'] = 0'2 =3
centimeters per second. Two random samples of n] = 20 and
a2 = 20 specimens are tested; the sample mean burning rates
are 3?, = 18 centimeters per second and $2 = 24 centimeters
per second.
(a) Test the hypothesis that both propellants have the same
mean burning rate. Use (1'. = 0.05. What is the P—Value'?
(b) Construct a 95% conﬁdence interval on the difference in‘
means 3)., — [4.2. What is the practical meaning of this
interval? (c) What is the B—error of the test in part (a) if the true differ
ence in mean burning rate is 2.5 centimeters per second? (CD Assuming equal sample sizes, what sample size is needed to
obtain power of 0.9 at a true diiTerence in means of 14 cm/S? 10—7. Two different formulations of an oxygenated motor fuel are being tested to study their road octane numbers. The variance of road octane number for formulation 1 is of = 1.5, and for formulation 2 it is 0% = 1.2. Two random satu ples of size HI = 15 and n; = 20 are tested, and the mean road octane numbers observed are 31 = 89.6 and E2 = 92.5. Assume normality. (a) If formulation 2 produces a higher road octane number
than formulation 1, the manufacturer would like to detect
it. Formulate and test an appropriate hypothesis, using
or = 0.05. What is the P—value? (b) Explain how the question in part (a) could be answered
with a 95% conﬁdence interval on the difference in mean
road octane number. (c) What sample size would be required in each population
if we wanted to be 95% conﬁdent that the error in esti mating the difference in mean road octane number is less
than 1? 108. A polymer is manufactured in a hatch chemical
process. Viscosity measurements are normally made on each batch, and long experience with the process has indicated that — the Variability in the process is fairly stable with [T = 20.
Fifteen batch viscosity measurements are given as follows: 724, 718, 776, 760, 745, 759, 795, 756, 742, 740, 761, 749,
739, 747, 742 A process change is made which involves switching the type
of catalyst used the process. Following the process change,
eight batch viscosity measurements are taken: 735, 775, 729, 755, 783, 760, 738, 780 Assume that process variability is unaffected by the catalyst change. If the difference in mean batch viscosity is 10 or less, the manufacturer would like to detect it with a high probability. (a) Formulate and test an appropriate hypothesis using or =
0.10. What are your conclusions? Find the Pvalue. 10,2 INFERENCE ON THE DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES UNKNOWN 369 ; Therefore, the pooled standard deviation estimate is Sp = V1952 = 4.4. The 95% conﬁdence interval is found using Equation 1019: 3. 1 1 7“ * — :‘c t s — + — s — i I; 2 0025,23 p HI R: M Pz ‘ < _ _ I l
— x! — x2 + r0,025,233,» :1 + IT: or upon substimfing the sample values and using raw, = 2.069, ' l 1 1 i — . — . 4.4 — + — S — 90.0 87 0 2 069( ) 10 15 a, p,
' s 90.0 — 37.0 + 2.069(4.4)1ii + i ‘ 10 15 Case 2.: 0% at: 0% which reduces to
—0.72 5 ll] — pa 5 6.72 Practical Interpretation: Notice that the 95% conﬁdence
interval includes zero; therefore, at this level of conﬁdence we
cannot conclude that there is a difference in the means. Put
another way, there is no evidence that doping the cement with
lead affected the mean weight percent of calcium; therefore,
we cannot claim that the presence of lead affects this aspect of
the hydration mechanism at the 95% level of conﬁdence. In many situations it is not reasonable to assume that 011 = 6%. When this assumption is unwarrantediwe niay still ﬁnd a 10
that T3 = [X] — X; * (pd — u2)]/ Shi/n] + Syn; is distributed approximately as twith 0(1 — 0L % conﬁdence interval on m i 51.2 using the fact \/_)__ degrees of freedom v given by Equation 1016. The CI expression follows. Case 2:
Approximate
Conﬁdence
Interval on the
Difference in :
Means, I
Variances
Unknown Are
Not Assumed I
Equal ' gm'esssp, .:— p;ng ' _ EXERCISES FOR SECTION 102 _.'If_iﬁ,’5‘c;,_s%, failsare the insane andivariances aim ia‘ndoiri assessorssee a, "an .
_ _' a2, respectiveljg ﬁ‘Om two independent normal populations with unknown and unequal ' _
'_ Variances; an_approxirnate._100(_1._—. 00% 'conﬁdenc.e::iriterva1_on the difference in; .  ‘ _' wheie {is given 10— 1 Grand theupper (it/2 percentagepoint of .i.
'  idem115.11% with? désfeés offreédomt _. ' l " ' i '= '    .0949?  ( 1010.i Consider the computer output below. (c) This test was done assuming that the two population vari—
. ances were equal. Does this seem reasonable? TwoSample T—Test and CI Sample N Mean StDev SE Mean
1 12 10.94 1.26 0.36
2 16 12.15 1.99 0.50
Difference = mu (1) H mu (2) Estimate for difference: —l.210 95% CI for difference: (—2.560, 0.140) TTest of difference r“: 0 (vs not =) : T—Value
== ? PValue = ? DF 5 ? Both use Pooled StDev 5 '? (61) Fill in the missing values. Is this a onesided or a two—sided
test? Use lower and upper bounds for the Pvalue.
(b) What are your conclusions ifOL = 0.05? What ifs: = 0.01? (0) Suppose that the hypothesis had been H0: in = pa versus
H0: is] < pa. What would your conclusions be if OL = 0.05? 101 1. Consider the computer output below.
TwoSample T—Test and Cl Sample N Mean StDev SE Mean
1 15 54.73 2.13 0.55
2 20 58.64 5.28 1.2
Difference = mu (1)  mu (2) Estimate for difference: —3.91 95% upper bound for difference: '9 T—Test of difference = 0(vs <): TValue =
—3.00 P~Value = ? DF = ? i l
vi
2
l
l
.1
'1
l
f
0
l 370 (a) Fill in the missing values. Is this a onesided or a twosided
test? Use lower and upper bounds for the Pvalue. (b) What are your conclusions if 0t = 0.05? What if or = 0.01? (c) This test was done assuming that the two population vari
ances were different. Does this seem reasonable? (d) Suppose that the hypotheses had been H0: J..1 = 11.2 versus
H0: u, 7": 114. What would your conclusions be if or = 0.05? @ Consider the hypothesis test H0 : u! = u; against
1 : it, is uz.‘ Suppose that sample sizes are 721 = 15 and
n, 2 15, that r, 2 4.7 and r, 2 7.8, and that s? 2 4 and
3% = 6.25. Assume that 621 = 0% and that the data are drawn
ﬁ'om normal distributions. Use or = 0.05. (a) Test the hypothesis and ﬁnd the Pvalue. (b) Explain how the test could be conducted with a conﬁ
dence interval. (c) What is the power of the test in part (a) for a true differ
ence in means of 3? ' (d) Assuming equal sample sizes, what sample size should
be used to obtain B = 0.05 if the true difference in means
is —2? Assume that or = 0.05. 1013. Consider the hypothesis test H0 : J..1 = it; against H] : “.1 < 11.2. Suppose that sample sizes n] = 15 and n; = 15, that r, 2 6.2 and s, 2 7.8, and that 3% 2 4 and 33 2 6.25. Assume that 0'] : 0% and that the data are drawn from normal distributions. Useor = 0.05. (a) Test the hypothesis and ﬁnd the P—value. (b) Explain how the test could be conducted with a conﬁ
dence interval. (c) Whatis the power of the test in part (a) if it] is 3 units less
than 11.2? (d) Assuming equal sample sizes, what sample size should be
used to obtain [3 = 0.05 if 11.1 is 2.5 units less than 1L2?
Assume that at = 0.05. 1014. Consider the hypothesis test H0 : u1 = it; against H1 : '11 > [1.2. Suppose that sample sizes 121 = 10 and 722 = 10, that rt, 2 7.3 and i, 2 5.6, and that 52, 2 4 and SE 2 9. Assume that 0% 2 0% and that the data are drawn ﬁom normal distributions. Use or = 0.05. 2 (a) Test the hypothesis and find the P—value. (b) Explain how the test could be conducted with a conﬁ
dence interval. (c) What is the power of the test in part (a) if it] is 3 units
greater than 13.2? (d) Assuming equal sample sizes, what sample size should be
used to obtain [3 = 0.05 if M is 3 units greater than 112?
Assume that at = 0.05. 1015. The diameter of steel rods manufactured on two dif
ferent extrusion machines is being investigated. Two random
samples of sizes 11] = 15 and h2 = 17 are selected, and the
sample means and sample variances are it] = 8.73, s, = 0.35,
22 = 8.68, and s; = 0.40, respectively. Assume that o] = 6%
and that the data are drawn from a normal distribution.
(a) Is there evidence to support the claim that the two ma
chines produce rods with different mean diameters? CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES Use or = 0.05 in arriving at this conclusion. Find the
Pvalue. (b) Construct a 95% conﬁdence interval for the diﬁerence in
mean rod diameter. Interpret this interval. 1016. An article in Fire Technology investigated two dif.
ferent foam expanding agents that can be used in the nozzles
of ﬁreﬁghting spray equipment. A random sample of ﬁve ob
servations with an aqueous ﬁlmforming foam (AFFF) had a
sample mean of 4.7 and a standard deviation of 0.6. A random
sample of ﬁve observations with alcoholtype concentrates
(AT C) had a sample mean of 6.9 and a standard deviation 0.8_
(a) Can you draw any conclusions about differences in mean
foam expansion? Assume that both populations are well
represented by normal distributions with the same stan—
dard deviations.
(b) Find a 95% conﬁdence interval on the difference in mean
foam expansion of these two agents. 1017. Two catalysts may be used in a batch chemical
process. Twelve batches were prepared using catalyst l , resulting
in an average yield of 86 and a sample standard deviation of 3,
Fifteen batches were prepared using catalyst 2, and they
resulted in an average yield of 89 with a standard deviation
of 2. Assume that yield measurements are approximately
normally distributed with the same standard deviation.
(a) Is there evidence to support a claim that catalyst 2 pro
duces a higher mean yield than catalyst 17 Use or = 0.01.
(b) Find a 99% conﬁdence interval on the difference in mean
yields that can be used to test the claim in part (a). 1018. The deﬂection temperature under load for two dif—_
ferent types of plastic pipe is being investigated. Two random
samples of 15 pipe specimens are tested, and the deﬂection
temperatures observed are as follows (in 0E): Type 1: 206, 188, 205, 187, 194, 193, 207, 185, 189, 213,
192, 210, 194, 178, 205 Type 2: 177, 197, 206, 201, 180, 176, 185, 200, 197, 192,
198, 188, 189, 203, 192 (a) Construct box plots and normal probability plots for the
two samples. Do these plots provide support of the asstunp'
tions of normality and equal variances? Write a practical
interpretation for these plots. (b) Do the data support the claim that the deﬂection tempera
ture under load for type 1 pipe exceeds that of type 2? [u
reaching your conclusions, use or = 0.05. Calculate a
P—value. (c) Ifthe mean deﬂection temperature for type 1 pipe exceeds
that of type 2 by as much as 5°F, it is important to detect this
diﬁ'erence with probability at least 0.90. Is the choice of
m = 712 = 15 adequate? Use a = 0.05. 1019. In semiconductor manufacturing, wet chemical etch—
ing is often used toremove silicon from the backs of wafers
prior to metallization. The etch rate is an important characteris
tic in this process and known to follow a normal distribution
Two different etching solutions have been compared, using two 10,2 {NFERENCE ON THE DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS. VARIANCES UNKNOWN 3 7 1 random samples of 10 wafers for each solution. The observed
etch rates are as follows (in mils per minute): minimal. Salaam 2
‘ 9's ' 10.6 r 10.2 . 10.0
9.4 10.3 10.5 10.2
7 9.3' 10.0. I ' 10.7 ' jicj
9.6 10.3 10.4 10.4
7710.2" ' 10.1 : ' 710.5' ' 10.3. (a) Construct normal probability plots for the two samples.
Do these plots provide support for the assmnptions of nor
mality and equal variances? Write a practical interpreta
tion for these plots. (b) Do the data support the claim that the mean etch rate is the
same for both solutions? In reaching your conclusions, use
01 = 0.05 and assume that both population variances are
equal. Calculate a Pvalue. (c) Find a 95% conﬁdence interval on the difference in mean
etch rates. _ Two suppliers manufacture a plastic gear used in a
ser printer. The impact strength of these gears measured in
footpounds is an important characteristic. A random sample
of 10 gears from supplier 1 reSults in i1 = 290 and s] = 12=
while another random sample of 16 gears from the second
supplier results min = 321 and s2 = 22. '(a) Is there evidence to support the claim that supplier 2 ' provides gears with'higher mean impact strength? Use
or : 0.05, and assume that both populations are normally
distributed but the variances are not equal. What is the
P—value for this test? (1)) Do the data support the claim that the mean impact
strength of gears from supplier 2 is at least 25 foot—pounds
higher than that of supplier 1? Make the same assump—
tions as in part (a). (C) Construct a conﬁdence interval estimate for the difference
in mean impact strength, and explain how this interval could
be used to answer the question posed regarding supplier
tosupplier differences. 1021. The melting points of two alloys used in formulating
SOlder were investigated by melting 21 samples of each mate
rial. The sample mean and standard deviation for alloy 1 was
31 = 420°F and S1 = 4°12, while for alloy 2 they were
7:2 = 426°F anus2 = 31°F.
(3) Do the sample data support the claim that both alloys have
M the same melting point? Use or = 0.05 and assume that
both populations are normally distributed and have the
same standard deviation. Find the Pvalue for the test.
(b) Suppose that the true mean difference in melting points
is 3°F. How large a sample would be required to detect this
. difference using an or = 0.05 level test with probability at
least 0.9? Use 0'1 = 02 = 4 as an initial estimate of the
common standard deviation. 1042. A photoconductor ﬁlm is manufactured at a nominal
thickness of 25 mils. The product engineer wishes to increase
the mean speed of the ﬁlm, and believes that this can be
achieved by reducing the thickness of the ﬁlm to 20 mils.
Eight samples of each ﬁlm thickness are manufactured in a pi
lot production process, and the ﬁlm speed (in microj oules per
square inch) is measured For the 25—mil ﬁlm the sample data
result is £1 = 1.15 and s] = 0.11, while for the 20mil ﬁlm
the data yield 5:; = 1.06 and s2 = 0.09. Note that an increase
in ﬁlm speed would lower the value of the observation in
microjoules per square inch. (a) Do the data support the claim that reducing the ﬁlm thick
ness increases the mean speed of the ﬁlm? Use or = 0.10
and assume that the two population variances are equal
and the underlying population of ﬁlm speed is normally
distributed. What is the Pvalue for this test? (b) Find a 95% conﬁdence interval on the difference in the
two means that can be used to test the claim in part (a). 1043. Two companies manufacture a rubber material in
tended for use in an automotive application. The part will be
subjected to abrasive wear in the ﬁeld application, so we decide
to compare the material produced by each company in a test.
Twentyﬁve samples of material from each company are tested
in an abrasion test, and the amount of wear after 1000 cycles is
observed. For company 1, the sample mean and standard de
viation of wear are i1 : 20 miliigramsflOOO cycles and
s1 = 2 milligrams/ 1000 cycles, while for company 2 we obtain
E2 = 15 milligrams/ 1000 cycles and .92 = 8 milligrams! 1000
cycles. (a) Do the data support the claim that the two companies pro
duce material with different mean wear? Use or = 0.05,
and assume each population is normally distributed but
that their variances are not equal. What is the Pvalue for
this test? (b) Do the data support a claim that the material from com
pany 1 has higher mean wear than the material from com
pany 2? Use the same assumptions as in part (a). (c) Construct conﬁdence intervals that will address the ques
tions in parts (a) and (b) above. 10—24. The thickness of a plastic ﬁlm (in mils) on a sub
strate material is thought to be inﬂuenced by the temperature
at which the coating is applied. A completely randomized
experiment is carried out. Eleven substrates are coated at
125°F, resulting in a sample mean coating thickness of it] = 103.5 and a sample standard deviation of s; t 10.2. Another 13 Substrates are coated at 150°F, for which it; = 99.7 and 32 = 20.1 are observed. It was originally suspected that raising the process temperature would reduce mean coating
thickness. (a) Do the data support this claim? Use or = 0.01 and assume
that the two population standard deviations are not equal.
Calculate an approximate P—value for this test. (b) How could you have answered the question posed regard
ing the effect of temperature on coating thickness by using
a conﬁdence interval? Explain your answer. Hale
Design Design
Language Language
Programmer ' . l ' 2 (a) Is the assumption that the difference in coding time is nor—
mally distributed reasonable? 0:) Find a 95% conﬁdence interval on the difference in mean
coding times. ls there any indication that one design lan— guage is preferable? 1041. Fifteen adult males between the ages of 35 and 50
participated in a study to evaluate the effect of diet and exercise
on blood cholesterol levels. The total cholesterol was meas
ured in each subject initially and then three months aﬁer
participating in an aerobic exercise program and switching
to a lowfat diet. The data are shown in the accompanying
table _______—1—«— Blood cugtesteror Level 104 PAIRED t’TEST 381 (a) Do the data support the claim that lowfat diet and aerobic
exercise are of value in producing a mean reduction in
blood cholesterol levels? Use on = 0.05. Find the Pvalue. (b) Calculate a onesided conﬁdence limit that can be used c answer the question in part (a). An article in the Journal ofAircmﬁ (Vol. 23, 1986,
p. 859—864) described a new equivalent plate analysis
method formulation that is capable of modeling aircraﬁ struc
tures such as cranked wing boxes= and that produces results
similar to the more computationally intensive ﬁnite element
analysis method. Natural vibration frequencies for the cranked
wing box structure are calculated using both methods, and
results for the ﬁrst seven natural frequencies follow: ' Finite, 7 Equivalent
' . ' Element ' ' Plate,
Freq._ 'Cycler's ' 7' ejs 2 48.52 49.11) 4 113.99 117.53 6 " 212.72 “7220.14 (a) Do the data suggest that the two methods provide the same
mean value for natural vibration frequency? Use or = 0.05. Find the Pvalue.
(b) Find a 95% conﬁdence interval on the mean difference between the two methods.
10—43. Ten individuals have participated in a dietmodiﬁcation
program to stimulate weight loss. Their weight both before and
after participation in the program is shown in the following list. .ﬁybieot ' ' 1710'" ' ' 3'10 285 (a) Is there evidence to support the claim that this particular
dietmodiﬁcation program is effective in producing a
mean weight reduction? Use OL = 0.05. (b) Is there evidence to support the claim that this particular
dietmodiﬁcation program will result in a mean weight
loss of at least 10 pounds? Use or = 0.05. ...
View
Full
Document
This note was uploaded on 03/13/2012 for the course ISE 130 taught by Professor Patel,n during the Spring '08 term at San Jose State.
 Spring '08
 Patel,N

Click to edit the document details