1/30/2014
1
Chapter 7:
Inferences for Two Samples
1
Introduction
In Chapters 5 and 6, we saw how to
construct confidence intervals and perform
hypothesis tests concerning a single mean
or proportion.
There are cases in which we have two
populations, and we wish to study the
difference between their means,
proportions, or variance.
2
1/30/2014
2
Motivating Example
Suppose that a metallurgist is interested in estimating the difference in strength between two types of welds.
She conducts an experiment in which a sample of 6 welds of one type has an average ultimate testing strength (in ksi) of 83.2 with a standard deviation of 5.2 and a sample of 8 welds of the other type has an average strength of 71.3 with a standard deviation of 3.1.
It is easy to compute a point estimate for the difference in strengths. The difference between the sample means is 83.2 β 71.3 = 11.9.
To construct a CI, however, we need to know how to find a standard error and a critical value for this point estimate.
To perform a hypothesis test to determine whether we can conclude that the mean strengths differ, we will need to know how to construct a test statistic. 3
Section 7.1: Large-Sample Inferences on
Differences Between 2 Population Means
Set-Up:
Let π and π be independent, with π ~ N(X, ππ2)
and π ~ N(Y, ππ2). Then
π β π ~ N(X β Y , ππ2 + ππ
2).
4
(7.1)
1/30/2014
3
CI
Let π1, β¦ , πππbe a large random sample of size ππ from a
population with mean X and standard deviation X, and let
π1, β¦ , πππbe a large random sample of size ππ from a
population with mean Y and standard deviation Y. If the
two samples are independent, then a level 100(1-)% CI for
X β Y is
( π β π) Β± π§ πΌ 2
ππ2
ππ+
ππ2
ππ
When the values of X and Y are unknown, they can be
replaced with the sample standard deviations π π and π π.
5
(7.2)
Example 7.1
The chemical composition of soil varies with depth. An
article in Communications in Soil Science and Plant
Analysis describes chemical analyses of soil taken from
a farm in Western Australia. Fifty specimens were each
taken at depths 50 and 250 cm. At a depth of 50 cm, the
average NO3 concentration (in mg/L) was 88.5 with a
standard deviation of 49.4. At a depth of 250 cm, the
average concentration was 110.6 with a standard
deviation of 51.5. Find a 95% confidence interval for the
difference in NO3 concentrations at the two depths.
6
1/30/2014
4
Hypothesis Tests on the
Difference Between Two Means
Now, we are interested in determining whether or
not the means of two populations are equal to
some specified value.
The data will consist of two samples, one from
each population.
We will compute the difference of the sample
means. Since each of the sample means follows
an approximate normal distribution, the difference
is approximately normal as well.
7
Hypothesis Test
Let π1, β¦ , πππand π1, β¦ , πππ
be large (e.g., ππ 30 and ππ 30) samples from populations with mean X and Y and standard deviations X and Y, respectively. Assume the samples are drawn independently of each other.
To test a null hypothesis of the form H0: X β Y Ξ0, H0: X
β Y β₯ Ξ0, or H0: X β Y = Ξ0.
Compute the z-score:
π = π β π β Ξ0
ππ2
ππ+
ππ2
ππ
If X and Y are unknown they may be approximated by π πand π π.
8
(7.4)
1/30/2014
5
P-value
Compute the P-value. The P-value is an area
under the normal curve, which depends on the
alternate hypothesis as follows.
If the alternative hypothesis is H1: X - Y > Ξ0,
then the P-value is the area to the right of z.
If the alternative hypothesis is H1: X - Y < Ξ0,
then the P-value is the area to the left of z.
If the alternative hypothesis is H1: X - Y Ξ0,
then the P-value is the sum of the areas in the
tails cut off by z and -z.
9
Example 7.2
An article compares properties of welds made using carbon dioxide as a shielding gas with those of welds made using a mixture of argon and carbon dioxide. One property studied was the diameter of inclusions, which are particles embedded in the weld. A sample of 544 inclusions in welds made using argon shielding averaged 0.37m in diameter, with a standard deviation of 0.25 m. A sample of 581 inclusions in welds made using carbon dioxide shielding averaged 0.40 m in diameter, with a standard deviation of 0.26 m. Can you conclude that the mean diameters of inclusions differ between the two shielding gases?
10
1/30/2014
6
Section 7.2: Inference on the
Difference Between 2 Proportions
11
Set-Up:
Let π be the number of successes in ππ independent
Bernoulli trials with success probability ππ, and let π be
the number of successes in ππ independent Bernoulli
trials with success probability ππ, so that π~π΅ππ ππ, ππ
and π~π΅ππ ππ, ππ .
Define
ππ = ππ + 2 ππ = ππ + 2
ππ =π+1
ππ ππ =
π+1
ππ
CI
12
(7.5)
Given the set-up just described, the 100 1 β πΌ % CI for
the difference (ππ β ππ) is
ππ β ππ Β± π§ πΌ 2
ππ 1 β ππ
ππ+
ππ(1 β ππ)
ππ
If the lower limit of the confidence interval is less than
-1, replace it with -1.
If the upper limit of the confidence interval is greater
than 1, replace it with 1.
There is a traditional confidence interval as well. It is a
generalization of the one for a single proportion.
1/30/2014
7
Example 7.4
Methods for estimating strength and stiffness requirements should be conservative in that they should overestimate rather than underestimate. The success rate of such a method can be measured by a probability of an overestimate. An article in Journal of Structural Engineering presents the results of an experiment that evaluated a standard method for estimating the brace force for a compression web brace. In a sample of 380 short test columns the method overestimated the force for 304 of them, and in a sample of 394 long test columns, the method overestimated the force for 360 of them. Find a 95% confidence interval for the difference between the success rates for long columns and short columns.
13
Hypothesis Tests on the Difference
Between Two Proportions
The procedure for testing the difference between
two populations is similar to the procedure for
testing the difference between two means.
One of the null and alternative hypotheses are
H0: pX β pY β₯ 0 versus H1: pX β pY < 0.
14
1/30/2014
8
Comments
The test is based on the statistic ππ β ππ.
We must determine the null distribution of this
statistic.
By the Central Limit Theorem, since ππ and ππ are
both large, we know that the sample proportions
for π and π have an approximately normal
distribution.
15
More on Proportions
The difference between the proportions is also
normally distributed.
Let π =π+π
ππ+ππ, then
ππ β ππ~π 0, π(1 β π)1
ππ+
1
ππ
16
(7.8)
1/30/2014
9
Hypothesis Test
Let π ~ π΅ππ(ππ, ππ) and π ~ π΅ππ(ππ, ππ). Assume ππ and
ππ are large, and that π and π are independent.
To test a null hypothesis of the form π»0: ππ β ππ 0,
π»0: ππ β ππ β₯ 0, and π»0: ππ β ππ = 0.
Compute
ππ =π
ππ, ππ =
π
ππ, πππ π =
π + π
ππ + ππ
Compute the z-score:
π§ = ππ β ππ
π(1 β π)(1ππ
+1ππ
)
17
(7.9)
P-value
Compute the P-value. The P-value is an area
under the normal curve, which depends on the
alternative hypothesis as follows:
If the alternative hypothesis is H1: pX β pY > 0, then the P-value is the area to the right of z.
If the alternative hypothesis is H1: pX β pY < 0, then the P-value is the area to the left of z.
If the alternative hypothesis is H1: pX β pY 0, then the P-value is the sum of the areas in the tails cut off by z and -z.
18
1/30/2014
10
Example 7.5
Industrial firms often employ methods of βrisk transferβ, such as insurance or indemnity clauses in contracts, as a technique of risk management. An article reports the results of a survey in which managers were asked which methods played a major role in the risk management strategy of their firms. In a sample of 43 oil companies, 22 indicated that risk transfer played a major role, while in a sample of 93 construction companies, 55 reported that risk transfer played a major role. Can we conclude that the proportion of oil companies that employ the method of risk transfer is less than the proportion of construction companies that do?
19
Section 7.3: Small-Sample Inferences
on the Difference Between 2 Means
Let X1,β¦,XnXbe a random sample of size nX from a normal
population with mean X and standard deviation X, and let
Y1,β¦,YnYbe a random sample of size nY from a normal population
with mean Y and standard deviation Y. Assume that the two
samples are independent. If the populations do not necessarily
have the same variance, a level 100(1-)% CI for X β Y is
π β π Β± π‘π, πΌ 2
π π2
ππ+
π π2
ππ
The number of degrees of freedom, v, is given by (rounded down to the
nearest integer)
20
(7.11)
(7.10)π =
π π2
ππ+
π π2
ππ
2
(π π
2
ππ)2
ππ β 1+
(π π
2
ππ)2
ππ β 1
1/30/2014
11
Example 7.6
Resin-based composites are used in restorative dentistry. An article presents a comparison of the surface hardness of specimens cured for 40 seconds with constant power with that of specimens cured for 40 seconds with exponentially increasing power. Fifteen specimens were cured with each method. Those cured with constant power had an average surface hardness of 400.9 with a standard deviation of 10.6. Those cured with an exponentially increasing power had an average surface hardness of 367.2 with a standard deviation of 6.1. Find a 98% confidence interval for the difference in mean hardness between specimens cured by the two methods.
21
Another CI
Suppose we have the same set-up as before, but
the populations are known to have nearly the
same variance. Then a 100(1 β )% CI for
X β Y is
π β π Β± π‘ππ+ππβ2, πΌ 2 π π1
ππ+
1
ππ
The quantity π π2 is the pooled variance, given by
π π2 =
ππ β 1 π π2 + ππ β 1 π π
2
ππ + ππ β 2
22
1/30/2014
12
Hypothesis Test for Unequal
Variance Let π1, β¦ , πππ
and π1, β¦ , πππbe samples from normal
populations with mean X and Y and standard deviations X and Y, respectively. Assume the samples are drawn independently of each other.
Assume that X and Y are not known to be equal. To test a null hypothesis of the form H0: X β Y Ξ0, H0: X
β Y β₯ Ξ0, or H0: X β Y = Ξ0. Compute π, rounded to the nearest integer.
Compute the test statistic π‘ = πβ π ββ0
π π2
ππ+
π π2
ππ23
(7.13)
π =
π π2
ππ+
π π2
ππ
2
(π π
2
ππ)2
ππ β 1+
(π π
2
ππ)2
ππ β 1
P-value
Compute the P-value. The P-value is an area under the
Studentβs t curve with v degrees of freedom, which
depends on the alternate hypothesis as follows.
If the alternative hypothesis is H1: X - Y > Ξ0, then the P-value is the area to the right of t.
If the alternative hypothesis is H1: X - Y < Ξ0, then the P-value is the area to the left of t.
If the alternative hypothesis is H1: X - Y Ξ0, then the P-value is the sum of the areas in the tails cut off by t and -t.
24
1/30/2014
13
Example 7.7
Good website design can make Web navigation
easier. An article presents a comparison of item
recognition between two designs. A sample of 10
users using a conventional Web design averaged 32.3
items identified, with a standard deviation of 8.56. A
sample of 10 users using a new structured Web
design averaged 44.1 items identified, with a standard
deviation of 10.09. Can we conclude that the mean
number of items identified is greater with the new
structured design?
25
Hypothesis Test with Equal
Variance
Let π1, β¦ , πππand π1, β¦ , πππ
be samples from normal
populations with mean X and Y and standard deviations X
and Y, respectively. Assume the samples are drawn
independently of each other.
Assume that X and Y are known to be equal.
To test a null hypothesis of the form H0: X β Y Ξ0,
H0: X β Y β₯ Ξ0, or H0: X β Y = Ξ0.
Compute π π=ππβ1 π π
2+ ππβ1 π π2
ππ+ππβ2
Compute the test statistic π‘ = πβ π ββ0
π π1
ππ+
1
ππ
26
1/30/2014
14
P-value
Compute the P-value. The P-value is an area under the
Studentβs t curve with v degrees of freedom, which
depends on the alternate hypothesis as follows.
If the alternative hypothesis is H1: X β Y > Ξ0, then the P-value is the area to the right of t.
If the alternative hypothesis is H1: X β Y < Ξ0, then the P-value is the area to the left of t.
If the alternative hypothesis is H1: X β Y Ξ0, then the P-value is the sum of the areas in the tails cut off by t and -t.
27
Section 7.4:
Inference for Paired Data
Set-Up:
Consider paired data. An example is tread wear on tires. A manufacturer wishes to compare the tread wear of tires made of a new material with that of tires made of a conventional material. One tire of each type is placed on each front wheel of 10 front-wheel-drive automobiles. The choice as to which type of tire goes on the right wheel and which goes on the left is made with the flip of a coin. Each car is driven for 40,000, a measurement of tread wear is then made on each tire. The measurements are not independent, since the tires are on the same car.
28
1/30/2014
15
: Paired Data
29
More on Paired Data
Let (π1, π1),β¦, (ππ, ππ) be sample pairs. Let π·π = ππβππ. Let X and Y represent the population means for π and π, respectively. We wish to find a CI for the difference X β Y. Let D represent the population mean of the differences, then D = X β Y. It follows that a CI for D
will also be a CI for X β Y.
Now, the sample π·1, β¦ , π·π is a random sample from a population with mean D, we can use one-sample methods to find CIs for D.
30
1/30/2014
16
Confidence Interval
Let π·1, β¦ , π·π be a small random sample (π < 30) of differences of pairs. If the population of differences is approximately normal, then a level 100(1-)% CI for D is
π· Β± π‘πβ1, πΌ 2
π π·
πIf the sample size is large, a level 100(1-)% CI for D is
π· Β± π§ πΌ 2π π·
In practice, π π· is approximated with π π· π.
31
(7.14)
(7.15)
Hypothesis Tests
with Paired Data We present a method for testing hypotheses involving
the difference between two population means on the basis of such paired data.
If the sample is large, the π·π need not be normally distributed. The test statistic is
π§ = π·βπ0
π π·
π
and a z-test should be performed.
32
1/30/2014
17
Hypothesis Test
Let (π1, π1),β¦, (ππ, ππ) be sample of ordered pairs
whose differences π·1, β¦ , π·π are a sample from a
normal population with mean D.
To test a null hypothesis of the form H0: D 0, H0:
D β₯ 0, or H0: D = 0.
Compute the test statistic
π‘ = π· β π0
π π·
π
33
P-value
Compute the P-value. The P-value is an area under the
Studentβs t curve with n β 1 degrees of freedom, which
depends on the alternate hypothesis as follows.
If the alternative hypothesis is H1: D > 0, then the P-value is the area to the right of t.
If the alternative hypothesis is H1: D < 0, then the P-value is the area to the left of t.
If the alternative hypothesis is H1: D 0, then the P-value is the sum of the areas in the tails cut off by t and -t.
34
1/30/2014
18
Section 7.5: The F Test for
Equality of Variances
Sometimes it is desirable to test a null hypothesis
that two populations have equal variances.
In general, there is no good way to do this.
In the special case where both populations are
normal, there is a method available.
Let π1, β¦ , ππ be a simple random sample from a
N(1, 12) population, and let π1, β¦ , ππ be a simple
random sample from a N(2, 22) population.
Assume that the samples are chosen independently.
The values of the means are irrelevant, we are only
concerned with the variances.
35
Hypotheses
Let π 12 and π 2
2 be the sample variances.
Any of three null hypothesis may be tested.
They are
π»0:π1
2
π22β€ 1, ππ π1
2 β€ π22
π»0:π1
2
π22β₯ 1, ππ π1
2 β₯ π22
π»0:π1
2
π22= 1, ππ π1
2 = π22
36
1/30/2014
19
Test For Equality of Variances
The test statistic is the ratio of the two sample variances:
πΉ =π 1
2
π 22
When H0 is true, we assume that 12/2
2 = 1, or equivalently 1
2 = 22. When π 1
2 and π 22 are, on
average, the same size, F is likely to be near 1.
When H0 is false, we assume that 12 > 2
2. When π 1
2 is likely to be larger than π 22, and πΉ is likely to be
greater than 1.
37
(7.16)
F Distribution
Statistics that have an F distribution are ratios of quantities, such as the ratio of two variances.
The F distribution has two values for the degrees of freedom: one associated with the numerator, and one associated with the denominator.
The degrees of freedom are indicated with subscripts under the letter F.
Note that the numerator degrees of freedom are always listed first.
A table for the F distribution is provided (Table A.7 in Appendix A).
38
1/30/2014
20
Back to the Test
The null distribution of the F statistic is Fπ-1,π-1.
The number of degrees of freedom for the numerator is one less than the sample size used to compute s1
2, and the number of degrees of freedom for the denominator is one less than the sample size used to compute s2
2.
Note that the F test is sensitive to the assumption that the samples come from normal populations.
If the shapes of the populations differ much from the normal curve, the F test may give misleading results.
The F test does not prove that two variances are equal. The basic reason for the failure to reject the null hypothesis does not justify the assumption that the null hypothesis is true.
39
Example 7.9
In a series of experiments to determine the absorption rate of certain pesticides into skin, measured amounts of two pesticides were applied to several skin specimens. After a time, the amounts absorbed (in g) were measured. For pesticide A, the variance of the amounts absorbed in 6 specimens was 2.3, while for pesticide B, the variance of the amounts absorbed in 10 specimens was 0.6. Assume that for each pesticide, the amounts absorbed are a simple random sample from a normal population. Can we conclude that the variance in the amount absorbed is greater for pesticide A than for pesticide B?
40
1/30/2014
21
Summary
Large Sample Inference on the Difference Between Two Population Meansβ Confidence Intervals
β Hypothesis Tests
Inferences on The Difference Between Two Proportionsβ Confidence Intervals
β Hypothesis Tests
Small Sample Inferences on the Difference Between Two Meansβ Confidence Intervals
β Hypothesis Tests
Inferences Using Paired Data
The F Test for Equality of Variance41
Top Related