Two independent
samples
Difference of Means
Differences of Means (Using Independent Samples)
CONDITIONS:
1) The two samples are chosen independently of each
other. OR
The treatments are randomly assigned to individuals
or objects or (vice versa)
2) The sample distributions for both samples should be
approximately normal
- the populations are known to be normal, or
- the sample sizes are large (n  30), or
- graph data to show approximately normal
3) 10% rule – Both samples should be less than 10% of
their respective populations
Differences of Means (Using Independent Samples)
Confidence
Called
intervals:
standard
error
CI  statistic criticalvalueSD of statistic
s
s
x  x   t *

n n
1
2
2
1
2
1
2
2
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 does this
automatically!
Option 2: approximation used by
technology
s s 
2

2
2


n n 

df 
1 s 
1 s
  

n  1 n  n  1 n
1
1
1
2
2
2
2
1
2
1
2
2



Differences of Means (Using Independent Samples)
Hypothesis Statements:
H0: m1 - m2 = hypothesized value
H0 : m 1 = m 2
Ha: m1 - m2 < hypothesized value
H a: m 1 < m 2
Ha: m1 - m2 > hypothesized value
Ha: m1> m2
Ha: m1 - m2 ≠ hypothesized value
Ha: m1 ≠ m2
Differences of Means (Using Independent Samples)
Hypothesis Test:
statistic - parameter
Test statistic 
SD of statistic
 x  x   m  m 
t
State the degrees of freedom
1
2
1
2
2
1
2
1
2
s s

n n
2
Example 1
Two competing headache remedies claim to give fast-acting
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:
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?
Parameters and Hypotheses
μA = the true mean absorption time in minutes for brand A
μB = the true mean absorption time in minutes for brand B
μA - μB = the true difference in means in absorption times in minutes for brands A
and B
H0: μA - μB = 0
Ha: μA - μB  0
Assumptions (Conditions)
1) The samples must be random and independent which is stated in the problem.
2) The sample distributions should be approximately normal. Since it is
stated in the problem that the population is normal then the sample
distributions are aprroximately normal.
3) The samples should be less than 10% of their populations. The
population should be at least 240 people, which I will assume.
4) A and B are both unknown
Since the conditions are met, a t-test for the two-sample means is appropriate.
Calculations
m A  mB  0
xA  20.1
xB  18.9
sA  8.7
sB  7.5
nA  12
nB  12
 = 0.05
x A  xB    m A  m B 

t
s A2 sB2

n A nB
df  21
20.1  18.9   0


8.72 7.52

12
12
 .3619
p  value  2P(t  .3619)  .721
.721  .05
Decision:
Since p-value > , I fail to reject the null hypothesis at the .05 level.
Conclusion:
There is not sufficient evidence to suggest that there is a difference in the true mean
absorption time in minutes for Brand A and Brand B.
#5 Commuting.
A man who moves to a new city sees that there are two routes he
could take to work. A neighbor who has lived there a long time
tells him Route A will average 5 minutes faster than Route B. The
man decides to experiment. Each day he flips a coin to determine
which way to go, driving each route 20 days. He finds that Route
A takes an average of 40 minutes, with a standard deviation of 3
minutes, and Route B takes an average of 43 minutes, with a
standard deviation of 2 minutes. Histograms of travel times for
the routes are roughly symmetric and show no outliers.
a) Find a 95% confidence interval for the difference in average
commuting time for the two routes.
b) Should the man believe the old-timer’s claim that he can save
an average of 5 minutes a day by always driving Route A?
Explain.
Page 567: #5
State the parameters
μA = the true mean time it takes to commute taking Route A
μB = the true mean time it takes to commute taking Route B
μB - μA = the true difference in means in time it takes to commute taking Route B from
Route A
Justify the confidence interval needed (state assumptions)
1) The samples must be random and independent which is state in the
problem.
2) The sample distributions should be approximately normal. It is stated in
the problem that graphs of the travel times are roughly symmetric and show
no outliers, so we will assume the distributions are approximately normal.
3) The samples should be less than 10% of the population. The population
should be at least 400 days, which I will assume.
4) A and B are both unknown
Since the conditions are satisfied a t – interval for the difference of means is
appropriate.
Calculate the confidence interval.
xA  40
xB  43
sA  3
sB  2
nA  20
nB  20
95% CI
xB  x A  t 
sB2 s A2

nB n A
2
2
2
3
43  40  2.043

20 20
1.3599,4.6401
df  33
Explain the interval in the context of the problem.
We are 95% confident that the true mean difference
between the commute times is between 1.3599 and
4.6401 minutes.