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SP 225
Lecture 13
Advanced Topics in Hypothesis
Testing
Alternative Hypotheses
 Based on research claims
 Researchers may claim:



A subpopulation mean is not equal to a
population mean
A subpopulation mean is greater than a
population mean
A subpopulation mean is less than a
population mean
Alternative Hypotheses
H1 : 1  2
H1 : 1  2
H1 : 1  2
Results of the Alternative
Hypothesis Choice
 Only some portions of the distribution
support the alternative hypothesis
 ‘Usual’ Values may be:



At the lower end of the distribution
At the upper end of the distribution
At both ends of the distribution
Writing Hypothesis
 Subtract value of the mean from both
sides of the hypothesis statement
 SPSS T-test
 Are the extreme conservatives sampled
in GSS the same age on average as the
general population?
 Meaning of the confidence interval
changes
Testing Option 1
 You know:



Population Mean
Population Standard Deviation
Sample Mean
Option 1 Calculations
 Calculate the standard deviation
of the sampling distribution
x 

n
Testing Option 2
 You know:



Population Mean
Sample Mean
Sample Standard Deviation
Option 2 Calculations
 Calculate the standard deviation
of the sampling distribution
sx 
s
n -1
T-Distribution
 Differences in the sample mean and
population mean cause the standard
score to be t-distributed instead of
normally distributed
 We call the new standard score t instead
of z
T-distribution (cont.)
 K = df = degree of freedom = n-1
 As k gets bigger, the t-distribution gets closer to
the normal distribution
T statistic
 Similar to z-score
t
x
Sx
Reading SPSS Output
One-sample t-test
Reading SPSS Output
One-sample t-test
Hypothesized
value of the mean
Test Value
 Test value is zero
 Hypothesis: the difference between the
related data points has a mean of zero
 Alternative Hypothesis: the difference
between the related data points has a
mean that is not zero
Reading SPSS Output
One-sample t-test
Value of the tstatistic
Reading SPSS Output
One-sample t-test
Degrees of
Freedom n-1
Reading SPSS Output
One-sample t-test
Area under the
curve in a twotailed test
Significance
 The area under the t-distribution
 Distribution of the differences
X-t*s
X
X+t*s
Reading SPSS Output
One-sample t-test
Difference
between the mean
of the data and the
test value
Reading SPSS Output
One-sample t-test
Confidence interval around the
mean difference
Errors in Hypothesis Testing
 Type I Error
 Type II Error
Type I Error
 Rejecting the null hypothesis when it is
true
 Confidence interval percentage is the
chance this doesn’t happen
 100-ci = Type I error rate
 Often called α (alpha)
Type II Error
 Not rejecting null hypothesis when
should have
 Much larger than type I error
 Not easily calculable
 Power of test
Types of Error
The null hypothesis is:
Your Action
True
False
Reject
Type 1 error
Correct
Don’t Reject
Correct
Type 2 error
Decreasing Overall Error
 Reducing type I increases type II
 Reducing type II increases type I
 Increasing the sample size, reduces both