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CH 24
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  critical value SD of statistic
 y1  y2  
2
2
1
2
1
2
s
s
t*

n n

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
s s 
2
2
1
2
1
2
2
  
n n 

df 
1 s 
1 s
  

n  1 n  n  1 n
1
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
t
State the degrees of
freedom
 y1  y2   m1  m 2 
2
1
2
2
s
s

n1 n2
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
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 of means are approximately 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
s A  8.7
sB  7.5
nA  12
nB  12
 = 0.05
xA  xB    m A  mB 

t
s A2 sB2

nA nB
20.1  18.9   0


8.7 2 7.52

12
12
df  21
p  value  2 P(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.
• A man who moves to a new city sees that there are two
routes that 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 standard deviation 3
minutes, and Route B takes an average of 43 minutes, with
standard deviation 2 minutes. Histograms of travel times
for the routes are roughly symmetric and show no outliers.
•Find a 95% confidence interval for the difference
in average commuting time for the two routes.
•Is there any evidence to suggest that the man’s
neighbor is correct in his claim that Route A is 5
minutes faster than Route B?
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 stated 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  xA  t 
sB2 s A2

nB nA
22 32
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 Route B is between 1.36
and 4.64 minutes slower than Route A, based on this
sample.
Locals claim that Route A is 5 minutes faster
than Route B. Is this claim supported by
the constructed confidence interval?
No. The entire interval is below 5 minutes’
difference. We are 95% confident that
Route A is between 1.36 and 4.64 minutes
faster than Route B.