Aim: How do we test the
difference between two means?
HW#14: complete slide
Independent Samples
• Samples are independent samples when
they are not related
• A t test is used to test the difference
between means when the two samples are
independent and when the samples are
taken from two normally or approximately
normally distributed populations
Use the t test when…
• Option 1
– Used when the
variances of the
populations are not
equal
• Option 2
– Use when the
variances are equal
Formula for the t test – for testing the
difference between two means of a
small independent sample
• Variances are assumed to be unequal
X

t
1

 X 2   1  2 
s12 s22

n1 n2
Formula for the t test – for testing the
difference between two means of a
small independent sample
• Variances are assumed to be equal:
t
X
1

 X 2   1  2 
 n1  1 s12   n2  1 s22
n1  n2  2
1 1

n1 n2
Where the degree of freedom are equal to
n1  n2  2
Example
• The average size of a farm in Indiana
County, Pennsylvania, is 191 acres.
Assume the data were obtained from two
samples with standard deviations of 38
and 12 acres, respectively, and sample
sizes of 8 and 10, respectively. Can it be
concluded at α = 0.05 that the average
size of the farms in the two countries is
different? Assume the populations are
normally distributed.
Procedure
1.
Need to determine if the variances are equal if they did
not give you any information about variances in the
problem
1.
2.
3.
4.
5.
2.
State the hypothesis
Find the critical value (use f table)
Compute test value with f test
Make a decision
Summarize results
Decide which t test to use based on steps 1-5
1.
2.
3.
4.
5.
State the hypothesis
Find the critical values
Compute test value with t test
Make a decision
Summarize results
Home Work
1. A researcher wishes to determine whether the salaries of professional
nurses employed by private hospitals are higher than those of nurses
employed by government-owned hospitals. She selects a sample of
nurses from each type of hospital and conclude that the private hospitals
pay more than the government hospitals? Assume that the population are
approximately normally distributed. Use the P-value methods
Private
Government
X  $26800
X  $25400
s  $600
s  $450
n  10
n8