ST 361: Ch7.5 Estimation --- Interval Estimation for 1   2
Topics:
I. Interval estimation: confidence interval
II. (Two-sided) Confidence interval for estimating population mean  (§7.2, 7.4)
(a) When the population SD  is known: use Z distribution
(b) When the population SD  is NOT known: use t distribution
III. Two-sided confidence interval for estimating population mean difference 1   2 (§7.5)
(a) When the population SD’s  1 ,  2 are known
(b) When the population SD’s  1 ,  2 are NOT unknown
-----------------------------------------------------------------------------------------------------------------------III.
Inference on the difference of two population means:
Motivating example: A public health researcher is interested to learn if the average blood pressure
of blue-collar workers is different from that of white-collar worker.
Scenario I: A random sample of 35 blue-collar workers was collected, and the sample mean systolic
blood pressure and sample SD were 138mmHg and 17, respectively. Suppose that for
the population of white-collar workers, the mean is 145mmHg.
To answer the question of interest, we ________________________________
_______________________________________________________________
_______________________________________________________________
Scenario II: A random sample of 35 blue-collar workers was collected, and the sample mean systolic
blood pressure and sample SD were 138mmHg and 17, respectively. Because the
population mean systolic blood pressure of white-collar workers is not known, another
sample of 40 while-collar workers was collected, and the sample mean and sample SD
were 143mmHg and 20 respectively.
To answer the question of interest, we ________________________________
_______________________________________________________________
_______________________________________________________________
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Set-up: Assume 2 independent samples are obtained from 2 populations:
Population 1 with mean 1 and SD  1 . A sample obtained from Population 1 has mean x1 and SD s1
Population 2 with mean  2 and SD  2 . A sample obtained from Population 2 has mean x 2 and SD s2
 Question of interest: Do the two populations have the same mean, i.e., does
_________________?
(1) A good point estimate for 1   2 is :
_________________________
(2) Sampling distribution of x1  x2  :

 x  x  ___________________
1
(regardless the distribution of x1 and x 2 )
2
So x1  x2  is a biased or unbiased (pick one) estimator of 1   2  .
  x1  x2 =_____________________ (regardless the distribution of x1 and x 2 )

2
x1  x2
  
2
x1
2
x2
  x1  x2    
2
x1

(3)

2
x2
 12
n1


 22
n2
 12
n1

 22
n2
x1  x2  ~ ________________________
Interval Estimation------- assume
if x1 ~Normal and x 2 ~Normal
x1  x2  ~ Normal
Focus on the case of  1 and  2 unknown.
The Confidence Interval for 1   2  is
with degree of freedom (df) =
 SE1 2   SE2 2 


4
4
SE1
SE2

n1  1
n2  1

 
2

where SE1 
s1
s
and SE2  2
n1
n2
 Then ______________________ to the nearest integer.
2
Ex. (Back to the motivating example). What is the 95% confidence interval for the mean difference of the
blood pressure between blue-collar workers and white-collar workers?
Note that df 
 SE1 2   SE2 2 


 SE    SE 
4
1
n1  1
4
2
 72.94
2
n2  1
3
Ex. Gas prices tend to be higher in the West coast. Let 1 be the mean gas price in the East coast, and
 2 be that in the West coast. Data were shown in the table below.
East
West
n (weeks)
25
20
x
2.90
3.05
Sample SD s
0.12
0.15
( Note that df 
 SE1 2   SE2 2 


4
4
SE1
SE2

n1  1
n2  1

 

2
 35.97 )
(a) What assumptions do we need in to have the mean difference follow a normal distribution?
(b) Calculate the 95% confidence interval of the mean difference.
(c) How would you explain your results?
You would suggest that ________________________________ because the 95% CI for 1   2
____________________________________
4