Download Two-Sample Hypothesis Test with Means

Two-Sample
Hypothesis Test
with Means
Suppose we have a population of
adult men with a mean height of
71 inches and standard deviation
of 2.6 inches. We also have a population of
adult women with a mean height of 65 inches
and standard deviation of 2.3 inches. Assume
heights are normally distributed.
Describe the distribution of the difference in
heights between males and females (malefemale).
Normal distribution with
mx-y =6 inches & sx-y =3.471 inches
Female
65
Male
71
Difference = male - female
6
Remember:
m  m m
x y
s
x y
x
y
 s s
2
2
x
y
We will
be
interested
in the
difference
of means,
so we will
use this to
find
standard
error.
Two-Sample Procedures
with means
• The goal of these inference
procedures is to compare the
responses to two treatments or
to compare the characteristics
of two populations.
• We have INDEPENDENT samples
from each treatment or
population
Assumptions:
• Have two SRS’s from the
populations or two randomly
assigned treatment groups
• Samples are independent
• Both populations are normally
distributed
– Have large sample sizes
– Graph BOTH sets of data
• s’s known/unknown
Formulas
Since in real-life, we
will NOT know both s’s,
we will do t-procedures.
Degrees of Freedom
Option 1: use the smaller of the two
values n1 – 1 and n2 – 1
This will produce conservative
results – higher p-values & lower
confidence.
Calculator
Option 2: approximation used bydoes this
automatically!
technology
2
 s12 s22 
  
n1 n2 

df 
2 2
2 2
1  s1 
1  s2 
  
 
n1  1  n1  n2  1  n2 
Two competing headache remedies claim to give fastacting relief. An experiment was performed to
compare the mean lengths of time required for bodily
absorption of brand A and brand B. Assume the
absorption time is normally distributed. Twelve people
were randomly selected and given an oral dosage of
brand A. Another 12 were randomly selected and given
an equal dosage of brand B. The length of time in
minutes for the drugs to reach a specified level in the
blood was recorded. The results follow:
mean
SD
n
Brand A
20.1
8.7
12
Brand B
18.9
7.5
12
Describe the shape & standard error for sampling
distribution of the differences in the mean speed of
absorption. (answer on next screen)
How is Two Sample Test Different From Matched Pair Test?
The Difference is when the data is subtracted
Hypothesis Statements:
H0: m1 =
- m2 = 0
Ha:
Ha:
H
Haa::
m1<- mm22 < 0
m1>- mm22 > 0
mm11 -≠ mm22 ≠ 0
Be sure
to define
BOTH m1
and m2!
Hypothesis Test:
Test statistic 
Since we usually
assume H0 is true,
statistic
parameter
then this equals 0 –
can usually
SDsoofwestatistic
leave it out
 x  x   m  m 
t
1
2
1
2
2
1
2
1
2
s s

n n
2
Pooled procedures:
• Used for two populations with the
same variance and small sample size
• When you pool, you average the
two-sample variances to estimate
the common population variance.
• DO NOT use on AP Exam!!!!!
We do NOT know the variances of the population,
so ALWAYS tell the calculator NO for pooling!
The length of time in minutes for the drugs
to reach a specified level in the blood was
recorded. The results follow:
Brand A
Brand B
mean
20.1
18.9
SD
8.7
7.5
n
12
12
Is there sufficient evidence that these
drugs differ in the speed at which they
enter the blood stream?
Assume.: Have 2 independent SRS from volunteers
State assumptions!
Assume population is larger than 240
Given the absorption rate is normally distributed
Hypotheses & define variables!
s’s unknown
H0: mA= mB
Ha:mA= mB
Where mA is the true mean absorption time
for Brand A & mB is the true mean
absorption time for Brand B
Using Technology, two sample, t-test of means
p  value  .7210 df  21.53   .05
Conclusion in context
Since p-value > a, I fail to reject H0. There is not
sufficient evidence to suggest that these drugs differ in
the speed at which they enter the blood stream.
Suppose that the sample mean of Brand
B is 16.5, then is Brand B faster?
p  value  .2896 df  21.53   .05
No, I would still fail to reject the null
hypothesis.
A modification has been made to the process
for producing a certain type of time-zero film
(film that begins to develop as soon as the
picture is taken). Because the modification
involves extra cost, it will be incorporated only
if sample data indicate that the modification
decreases true average development time by
more than 1 second. Should the company
incorporate the modification?
Original 8.6 5.1 4.5 5.4
Modified 5.5 4.0 3.8 6.0
6.3 6.6
5.8 4.9
5.7 8.5
7.0 5.7