Transcript 11.2
Chapter 11
Section 2
Inference about Two Means:
Independent Samples
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 1 of 25
Chapter 11 – Section 2
● Learning objectives
1
Test hypotheses regarding the difference of two
independent means
2
Construct and interpret confidence intervals regarding
the difference of two independent means
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 2 of 25
Chapter 11 – Section 2
● Two samples are independent if the values in
one have no relation to the values in the other
● Examples of not independent
Data from male students versus data from business
majors (an overlap in populations)
The mean amount of rain, per day, reported in two
weather stations in neighboring towns (likely to rain in
both places)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 3 of 25
Chapter 11 – Section 2
● A typical example of an independent samples
test is to test whether a new drug, Drug N,
lowers cholesterol levels more than the current
drug, Drug C
● A group of 100 patients could be chosen
The group could be divided into two groups of 50
using a random method
If we use a random method (such as a simple random
sample of 50 out of the 100 patients), then the two
groups would be independent
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 4 of 25
Chapter 11 – Section 2
● The test of two independent samples is very
similar, in process, to the test of a population
mean
● The only major difference is that a different test
statistic is used
● We will discuss the new test statistic through an
analogy with the hypothesis test of one mean
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 5 of 25
Chapter 11 – Section 2
● Learning objectives
1
Test hypotheses regarding the difference of two
independent means
2
Construct and interpret confidence intervals regarding
the difference of two independent means
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 6 of 25
Chapter 11 – Section 2
● For the test of one mean, we have the variables
The hypothesized mean (μ)
The sample size (n)
The sample mean (x)
The sample standard deviation (s)
● We expect that x would be close to μ
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 7 of 25
Chapter 11 – Section 2
● In the test of two means, we have two values for
each variable – one for each of the two samples
The two hypothesized means μ1 and μ2
The two sample sizes n1 and n2
The two sample means x1 and x2
The two sample standard deviations s1 and s2
● We expect that x1 – x2 would be close to μ1 – μ2
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 8 of 25
Chapter 11 – Section 2
● For the test of one mean, to measure the
deviation from the null hypothesis, it is logical to
take
x–μ
which has a standard deviation of approximately
s2
n
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 9 of 25
Chapter 11 – Section 2
● For the test of two means, to measure the
deviation from the null hypothesis, it is logical to
take
(x1 – x2) – (μ1 – μ2)
which has a standard deviation of approximately
s12
s22
n1
n2
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 10 of 25
Chapter 11 – Section 2
● For the test of one mean, under certain
appropriate conditions, the difference
x–μ
is Student’s t with mean 0, and the test statistic
t
x
s2
n
has Student’s t-distribution with n – 1 degrees of
freedom
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 11 of 25
Chapter 11 – Section 2
● Thus for the test of two means, under certain
appropriate conditions, the difference
(x1 – x2) – (μ1 – μ2)
is approximately Student’s t with mean 0, and
the test statistic
t
( x1 x 2) ( 1 2 )
s12 s22
n1 n2
has an approximate Student’s t-distribution
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 12 of 25
Chapter 11 – Section 2
● This is Welch’s approximation, that
t
( x1 x 2) ( 1 2 )
s12 s22
n1 n2
has approximately a Student’s t-distribution
● The degrees of freedom is the smaller of
n1 – 1 and
n2 – 1
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 13 of 25
Chapter 11 – Section 2
● For the particular case where be believe that the
two population means are equal, or μ1 = μ2, and
the two sample sizes are equal, or n1 = n2, then
the test statistic becomes
t
( x1 x 2 )
s2 s2
2
1
n
with n – 1 degrees of freedom, where n = n1 = n2
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 14 of 25
Chapter 11 – Section 2
● Now for the overall structure of the test
Set up the hypotheses
Select the level of significance α
Compute the test statistic
Compare the test statistic with the appropriate critical
values
Reach a do not reject or reject the null hypothesis
conclusion
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 15 of 25
Chapter 11 – Section 2
● In order for this method to be used, the data
must meet certain conditions
Both samples are obtained using simple random
sampling
The samples are independent
The populations are normally distributed, or the
sample sizes are large (both n1 and n2 are at least 30)
● These are the usual conditions we need to make
our Student’s t calculations
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 16 of 25
Chapter 11 – Section 2
● State our two-tailed, left-tailed, or right-tailed
hypotheses
● State our level of significance α, often 0.10,
0.05, or 0.01
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 17 of 25
Chapter 11 – Section 2
● Compute the test statistic
t
( x1 x 2) ( 1 2 )
s12 s22
n1 n2
and the degrees of freedom, the smaller of
n1 – 1 and n2 – 1
● Compute the critical values (for the two-tailed,
left-tailed, or right-tailed test)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 18 of 25
Chapter 11 – Section 2
● Each of the types of tests can be solved using
either the classical or the P-value approach
● Based on either of these methods, do not reject
or reject the null hypothesis
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 19 of 25
Chapter 11 – Section 2
● We have two independent samples
The first sample of n = 40 items has a sample mean
of 7.8 and a sample standard deviation of 3.3
The second sample of n = 50 items has a sample
mean of 11.6 and a sample standard deviation of 2.6
We believe that the mean of the second population is
exactly 4.0 larger than the mean of the first population
We use a level of significance α = .05
● We use an example with μ1 ≠ μ2 to better
illustrate the test statistic
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 20 of 25
Chapter 11 – Section 2
● The test statistic is
t
( x1 x 2) ( 1 2 )
s12 s22
n1 n2
( 7.8 12.9 ) 4.0
3.32 2.62
40
50
1.72
● This has a Student’s t-distribution with 39
degrees of freedom
● The two-tailed critical value is 2.02, so we do not
reject the null hypothesis
● We do not have sufficient evidence to state that
the deviation from 4.0 is significant
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 21 of 25
Chapter 11 – Section 2
● Learning objectives
1
Test hypotheses regarding the difference of two
independent means
2
Construct and interpret confidence intervals regarding
the difference of two independent means
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 22 of 25
Chapter 11 – Section 2
● Confidence intervals are of the form
Point estimate ± margin of error
● We can compare our confidence interval with the
test statistic from our hypothesis test
The point estimate is x1 – x2
We use the denominator of the test statistic (Welch’s
approximation) as the standard error
We use critical values from the Student’s t
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 23 of 25
Chapter 11 – Section 2
● Thus confidence intervals are
Point estimate ± margin of error
( x1 x2 ) t / 2
s12 s22
n1 n2
Point estimate
Standard error
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 24 of 25
Summary: Chapter 11 – Section 2
● Two sets of data are independent when
observations in one have no affect on
observations in the other
● In this case, the differences of the two means
should be used in a Student’s t-test
● The overall process, other than the formula for
the standard error, are the general hypothesis
test and confidence intervals process
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 25 of 25