Download Ch23 Notes - Fort Bend ISD

Two-Sample
Inference
Procedures with
Means
Two independent
samples
Difference of Means
Differences of Means (Using Independent Samples)
CONDITIONS:
1) We have 2 SRS from 2 distinct populations
2) Both samples are chosen independently OR the treatments
are randomly assigned to individuals or objects
3) 10% rule – Both samples should be less than 10% of their
respective populations
4) 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
Differences of Means (Using Independent Samples)
Confidence
Called
intervals:
standard
error
CI  statistic  critical value 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
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



Ex1. A man who moves to a new city sees that
there are 2 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 do an 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.
His histogram of travel times are roughly
symmetric and show no outliers. Find a 95%
confidence interval for the difference in the
average commuting time for the 2 routes. Should
the man believe the neighbor’s claim that he can
save an average of 5 minutes by driving Route A?
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 f
rom Route A
Justify the confidence interval needed (state assumptions)
Randomization- Assume two independent random samples of days
10%-The samples should be less than 10% of the populations. The populations
should be at least 200 days for each route, which I will assume.
Nearly normal- 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.
Since the conditions are satisfied a t – interval for the difference of means is
appropriate.
Calculate the confidence interval.
xA  xB  t 
22 32
s A2 sB2
 4.64, 1.36
(40  43)  2.034


20 20
nA nB
in vT (.975,33.1)
df  33.1
(40  43)  2.09
in vT (.975,19)
22 32 (4.687, 1.313)

20 20
df  19
Explain the interval in the context of the problem.
We are 95% confident that the true mean difference between the
commute times is between -4.64 minutes and -1.36 minutes.
The man should not believe the neighbor’s claim that he can save
5 minutes since based on the interval he would only save between
1.36 and 4.64 minutes.
Differences of Means (Using Independent Samples)
Hypothesis Statements:
H0: m1 - m2 = hypothesized value
m1 = m2
Ha: m1 - m2 < hypothesized value
m1 < m2
Ha: m1 - m2 > hypothesized value
m1 > m2
Ha: m1 - m2 ≠ hypothesized value
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) Randomization- Assume two independent random samples
2) 10% - The samples should be less than 10% of their populations. The
populations should be at least 120 people for each drug, which I’ll assume.
3) Nearly normal- The sample distributions should be approximately
normal. Since it is stated in the problem that the population is normal then
the sample distributions are approximately normal.
Since the conditions are met, a t-test for the two-sample means is appropriate.
Calculations
 = 0.05
xA  xB    m A  mB   20.1  18.9   0  .3619


t
s A2 sB2

nA nB
8.7 2 7.52

12
12
p  value  2 P(t  .3619)  .721
df  21.5
p  value  2 P(t  .3619)  .724
df  11
Decision: Since p-value > , I fail to reject the null hypothesis at
the .05 level.
Conclusion:
There is not sufficient evidence to suggest that these drugs differ
in the speed at which they enter the blood stream
MC Answers
1)