Chapter Nine Inferences Based on Two Samples

1
Chapter NineInferences Based on Two Samples
2
Hypothesis Test 2-Sample Means (Known ?) Both
Normal pdf (known ?)Null Hypothesis H0 u1
u2 ?0Test Statistic z x y ?0
? ?21/m ?22/nAlternative Hypothesis
Reject RegionHa u1 u2 gt ?0 (Upper
Tailed) z ? z?Ha u1 u2 lt ?0 (Lower
Tailed) z ? -z?Ha u1 u2 ? ?0
(Two-Tailed) either z ? z?/2 or z ?
-z?/2 P-Value computed the same as 1-Sample
Mean.
3
Example HT 2-Sample Means (Known ?)During a
total solar eclipse the temperature drops quickly
as the moon passes between the earth and the sun.
During the June 2001 eclipse in Africa, data was
collected on the drop in temperature in degrees F
at two types of locations. The average drop in
temperature for 9 samples taken in Mountainous
terrain was 15.0. The average drop in temperature
for 12 samples taken in River-level terrain was
17.5. Assume the variance in temperature drop is
known to be 9 for this type of terrain-temperature
drop experiment and that experiments of this
type follow a Normal pdf. Is there evidence at
the ? .10 level to conclude that there is a
difference in temperature drop between the two
types of terrain in this experiment?P-value?

4
Determining ? (? known)Alternative Type
IIHypothesis Error ?(??)Ha u1 - u2 gt
?0 ? z?- ?? - ?0
? Ha u1 - u2 lt ?0 1 - ? -z?- ?? - ?0
?Ha u1 - u2 ?
?0 ? z?/2 - ??- ?0 - ? -z?/2- ??- ?0
? ? Where ? ?X-Y
?(?21/m) (?22/n)
5
Example HT 2-Sample Means (Known ?)A study of
report writing by student engineers was conducted
at Watson School. A scale that measures the
intelligibility of student engineers English is
devised. This scale, called an index of
confusion, is devised, to the delight of the
students, so that low scores indicate high
readability. Data are obtained on articles
randomly selected from engineering journals and
from unpublished reports. A sample of 16
engineering journals yielded an average score of
1.75 while a sample of 25 unpublished reports
yielded an average score of 2.5. Variance for
this type of scale is known to be 0.48 and the
scores are known to follow a Normal pdf. At a
significance level of .05, does there appear to
be a difference between the average scores of the
two types of reports? What is the Beta error when
the true averages differ by as much as 0.5?
6
Hypothesis Test 2-Sample Means (Large n) Null
Hypothesis H0 u1 u2 ?0Test Statistic z
x y ?0 ? s21/m
s22/nAlternative Hypothesis Reject
RegionHa u1 u2 gt ?0 (Upper Tailed) z
? z?Ha u1 u2 lt ?0 (Lower Tailed)
z ? -z?Ha u1 u2 ? ?0 (Two-Tailed) either
z ? z?/2 or z ? -z?/2 Both m gt 40
n gt 40.
7
Example HT 2-Sample Means (Large n)Aseptic
packaging of juices is a method of packaging that
entails rapid heating followed by quick cooling
to room temperature in an air-free container.
Such packaging allows the juices to be stored
un-refrigerated. A new old machine used to fill
aseptic packages is being compared. The mean
number of containers filled per minute on the new
machine was 114.1 for 50 observations with a
standard deviation of 5.0. The mean number of
containers filled per minute on the old machine
was 112.7 for 72 observations with a standard
deviation of 3.0. Is there evidence that the new
machine is faster than the old machine? Use a
test with ? .01. (Include the P-value).
8
Hypothesis Test 2-Sample Means (Small n) Both
Normal pdf (unknown ?)Null Hypothesis H0 u1
u2 ?0Test Statistic t x y ?0
? s21/m s22/nAlternative Hypothesis
Reject RegionHa u1 u2 gt ?0 (Upper
Tailed) t ? t?, ?Ha u1 u2 lt ?0
(Lower Tailed) t ? -t?, ?Ha u1 u2
? ?0 (Two-Tailed) either t ? t?/2, ?
or t ? -t?/2, ?
9
Estimating the Degrees of Freedom
s21 s22 2 ? m n
(s21/m)2 (s22/n)2 m 1 n -1
10
Example HT 2-Sample Means (Small n)The slant
shear test is used for evaluating the bond of
resinous repair materials to concrete. The test
utilizes cylinder specimens made of 2 identical
halves bonded at 300C. Twelve specimens were
prepared using wire brushing. The sample mean
shear strength (N/mm2) and sample standard
deviation were 19.20 1.58, respectively. Twelve
specimens were prepared using hand-chiseled
specimens the corresponding values were 23.13
4.01.Does the true average strength appear to be
different for the two methods of surface
preparation? Use a significance level of .05 to
test the relevant Hypothesis assume the shear
strength distributions to be Normal. (Include the
P-value).Parameters of interest
R. R.Null Hypothesis
CalculationAlternative
Decision Test Statistic
P-value

11
Example HT 2-Sample Means (Small n)A
manufacturer of power-steering components buys
hydraulic seals from two sources. Samples are
selected randomly from among the seals obtained
from these two suppliers, and each seal is tested
to determine the amount of pressure that it can
withstand. These data resultSupplier I
Supplier II x 1342 lb/in2 y
1338 lb/in2 s2 100 s2 33 m
10 n 11Is there evidence at the ?? .05
level to suggest that the seals from supplier I
can withstand higher pressures than those from
supplier II? (P-value)Assume measurements of
this type are Normal.
12
HT 2-Sample Means-Paired Data (Small n) Both
Normal X and Y (unknown ?)Null Hypothesis H0
uD ?0Test Statistic t d ?0
sD / ? nAlternative Hypothesis
Reject RegionHa uD gt ?0 (Upper Tailed)
t ? t?, ?Ha uD lt ?0 (Lower Tailed)
t ? -t?, ?Ha uD ? ?0
(Two-Tailed) either t ? t?/2, ?
or t ? -t?/2, ? D X Y within each
paired observation.
13
Example HT 2-Sample Means-Paired Data (Small
n)One important aspect of computing is the CPU
time required by an algorithm to solve a problem.
A new algorithm is developed to solve zero-one
multiple objective problems in linear
programming. It is thought that a new algorithm
will solve problems faster than the algorithm
currently used. To obtain statistical evidence to
support this research hypothesis, a number of
problems will be selected at random. Each problem
will be solved twice once using the current
algorithm and once using the newly developed one.
These CPU times are not independent they are
based on the same problems solved by two
different methods and so are paired by design.
The mean difference between the (16) paired data
points was 2.7 seconds with a standard deviation
calculated at 6.0 seconds. Does the data support
this hypothesis at a ? .025 level of
significance? Assume measurements of this type
are known to be Normal. (Give the P-value) Let X
old Y new.
14
Example HT 2-Sample Means-Paired Data (Large
n)Highway engineers studying the effects of wear
on dual-lane highways suspect that more cracking
occurs in the travel lane of the highway than in
the passing lane. To verify this contention, 64
one-hundred-feet-long test strips are selected,
paved, and studied over a period of time. It is
found that the mean difference in the number of
major cracks is 3.3 with a sample deviation of
8.8. Does this data support the research
hypothesis at a significance level of .05?
(Include P-value). Let RV X Travel lane RV Y
Passing lane
15
The F Distribution? W/? ? ?? ? ?/2
x(?/2)-1 Y/? 2 ? ? ? ? ? ? x
1 (??)/2 2 2 ? 0
lt x lt ? W Y are independent Chi-Square RVs
with ? ? degrees of freedom.
16
Hypothesis Test on 2 Population Variances Both
Normal (unknown u1 u2)Null Hypothesis H0
?21 ?22Test Statistic ? S21 / S22
Alternative Hypothesis
Reject RegionHa ?21 gt ?22 (Upper Tail) ? ?
F?, m-1, n-1Ha ?21 lt ?22 (Lower Tail) ? ?
F1- ?, m-1, n-1Ha ?21 ? ?22 (Two-Tail)
either ? ? F?/2, m-1, n-1 or ? ? F1-
?/2, m-1, n-1 F1- ?, m-1, n-1 1 / F?, n-1,
m-1 F-Tables pg. 730-735
17
Example HT 2-Population VariancesThe cost of
repairing a fiberoptic component may depend of
the stage of production at which it fails. The
following data are obtained on the cost of
repairing parts that fail when installed in the
system and on the cost of repairing parts that
fail after the system is installed in the field
System failure Field failure
Sample Size 21 Sample Size 25 Mean 65
Mean 120 s2 25 s2 100 It is
thought that the variance in cost of repairs made
in the field is larger than the variance in cost
of repairs made when the component is placed into
the system. Test at the ?? .10 level to see if
there is statistical evidence to support this
contention.
18
Example HT 2-Population VariancesOxide layers on
semiconductor wafers are etched in a mixture of
gases to achieve the proper thickness. The
variability in the thickness of these oxide
layers is a critical characteristic of the wafer,
and low variability is desirable for subsequent
processing steps. Two different mixtures of gases
are being studied to determine whether one is
superior in reducing the variability of the oxide
thickness. Twenty wafers are etched in each gas.
The sample standard deviation of oxide thickness
are s1 1.96 angstroms and s2 2.13 angstroms,
respectively. Is there any evidence to indicate
that either gas is preferable? Use ? 0.10
assume measurements of this type are Normal.
19
Example HT 2-Population VariancesTwo companies
supply raw materials to the manufacturer of paper
products. The concentration of hardwood in these
materials is important for the tensile strength
of the products. The mean concentration of
hardwood for both suppliers is the same however,
the variability in concentration may differ
between the two companies. The standard deviation
of concentration in a random sample of 26 batches
produced by company X is 4.7 g/l, while for
company Y a random sample of 21 batches yields
6.1 g/l. Is there sufficient evidence to conclude
that the two population variances differ? Use ?
.05 list any assumptions that you make.