Apr. 27 Statistic for the day:
Americans eat, on average, 15
quarts of ice cream per year.
Power of a test
The power of a statistical test refers to its ability to
detect when something is happening – in other words,
to reject the null hypothesis when it should be rejected.
Assignment:
Take practice exams
Stop by to pick up your old exams
Power as it relates to type 2 errors
Influences on power
If the null hypothesis is false:
ƒ The closer the observed statistic is to the null value, the
smaller the power.
ƒ The larger the sample, the larger the power.
ƒ Rejecting it is a good thing.
ƒ Failure to reject is a type 2 error.
ƒ The probability of rejecting is the power.
Thus, high power corresponds to low probability of
type 2 error.
Eliminating one type of error:
A couple of really stupid tests
Test A: Always reject the null hypothesis, no matter
what the data are.
This test will never make a type 2 error! It has
perfect power! (But it’s stupid.)
Test B: Never reject the null hypothesis, no matter
what the data are.
This test will never make a type 1 error! But it
has no power, and it’s stupid.
Return to chi-squared statistics
Suppose we are interested in the following research question:
Is there a significant difference between men and women in
STAT 100 with respect to the proportion who have smoked
marijuana?
According to the survey for this class, 58.8% of women (out
of 114) versus 61.4% of men (out of 101) have smoked
marijuana.
1
Rows: sex
How to measure the distance between what the
research advocate observes in the table and what
the skeptic expects:
Columns: marijuana
top lines are observed counts
bottom lines are expected counts
No
47
45.60
Yes
67
68.40
All
114
114.00
Male
39
40.40
62
60.60
101
101.00
All
86
86.00
129
129.00
215
215.00
Female
Add up the following for each cell:
χ2 =
(obs − exp) 2
exp
(47 − 45.6) 2 (67 − 68.4) 2 (39 − 40.4) 2 (62 − 60.6) 2
+
+
+
= 0.152
45.6
68.4
40.4
60.6
Chi-squared distribution with 1 degree of freedom:
2.0
How about a p-value for the marijuana
test?
The key is to take the square root of the chi-squared statistic and
treat that as the standardized score!
1.0
1.5
If chi-squared statistic is
larger than 3.84, it is
declared large and the
research advocate wins.
Null: No difference between men & women
0.5
Cutoff=3.84
Alternative (2-sided): A difference exists
5% on
this side
Test statistic: 0.152 = 0.39
0.0
95% on
this side
0
1
2
3
4
5
6
But our chi-squared is 0.152, so the research advocate does not win.
There is NOT a statistically significant difference between men and women.
2-sided p-value:
2×.35 = .70
Decision: WE HAVE NO EVIDENCE OF ANY DIFFERENCE IN THE
PERCENTAGE WHO HAVE SMOKED MARIJUANA.
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