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