Ch. 21: Error and Power
Read p. 461 – 463
“A trial as a Hypothesis test”
“P-values”
“What to do with an innocent defendant”
Wording on the second sentence of our conclusion
Example:
Ho: p = 0.60 (innocence)
Ha: p > 0.60 (guilt)
p-value = 0.31
Conclusion:
We fail to reject Ho b/c ..... the P-value of .31 >   .05
We have … insufficient evidence that the true percentage of innocence is greater than 60%.
Interpreting the P-Value
Read p. 483 – 484
“How to think about P-values”
Interpreting the P-Value:
There is a P-Value % chance of getting our sample or something more extreme (use Ha) if the Ho is
true.
Examples: Ch. 20 CW
1)
Ho: p = 0.92 (germination rate)
Ha: p < 0.92
p-hat = 171/200 = 0.855
p-value = 0.0004
There is a .04% chance of getting a sample where 85.5% or less seeds germinate if it is actually true
that 92% germinate.
2)
Ho: p = 0.03
Ha: p ≠ 0.03 (% of twin births)
p-hat = 8/469 = 1.706%
p-value = 0.1004
There is a 10.04% chance of getting a sample where 1.706% (or something more extreme) of births are
twins if the true percent of twins births is still really 3%.
Reading:
p.487
p. 491
“Significant vs. important”
“Making Errors”
Types of Error:
Truth of the claim
Reject Ho
Fail to Reject Ho
D
E
C
I
S
I
O
N
Ho True
Ho False
Type I Error
OK
OK
Type II Error
EXAMPLE:
DEFENDANT
Innocent
Guilty
Jury
Decision
Convict (guilty)
Reject Ho
Acquit (not guilty)
Fail to reject Ho
Type I Error
OK
OK
Type II Error
Type I Error =
- We reject the Ho and it is true (we reject Ho and we shouldn’t have)
- We conclude Ha is true, when really Ho is really still true

P(Type I Error) = ALPHA  
Type II Error =
- We fail to reject Ho when it is false (we should have rejected Ho and we didn’t)
- We conclude that Ho is true, when really Ha is true

P(Type II Error) = BETA   
Power =
-
Rejecting the Ho when it is false (we should have rejected Ho and we did)
Concluding Ha is true, and being correct

We want power. It is a good thing!

Formula:
o Power = 1  


Calculating Power – we won’t do it!
Power is considered adequate when: it is above 80%
Increasing Power: 2 things we can do to increase power are….
1. Increase n (sample size)
2. Increase  - gives you a better chance of rejecting Ho
Example 1: In Philadelphia it is believed that 12% of motorists run red lights. The city planners decide to put a
few traffic cameras that will issue tickets to drivers that run red lights. After 2 months they check out a few
corners where cameras had been installed. They found that out of 240 drivers that had the opportunity to run
the red light 15 did so and were issued a ticket.
1. Does this provide evidence of a decline in the percentage of drivers that run the red light? Use a level
of significance of 5%. Assume conditions met already.
2. What is a Type I error in this context? What is its probability?
3. What is a Type II error in this context?
4. Describe what Power means in this context.
5. If they had used a level of significance of 1% what would have happened to the Type I error, the Type II
error, and the Power?
6. How could have they increased the Power without changing the level of significance?
1.
Ho: p = .12
p = .12
q = .88
pˆ 
15
 .0625
240
n = 240
Ha: p < .12
Conditions met Normal Model 1-Prop. Z-Test
Z
.0625  .12
 2.741
(.12)(.88)
240
P( Z  2.741)  .0031
  .05
We reject the Ho because the P-value of .0031 <   .05 .
We have sufficient evidence that the true percentage of motorists who run red lights is less than
12%.
2. Saying that less than 12% of motorists run red lights when really it still is 12%.
P(Type I )    .05
3. Saying that the percent of motorists who run red lights is still 12% when really it is less than 12%.
4. Saying that less than 12% of motorists run red lights and it really is less than 12%.
5.   .05    .01
Type I Error: decrease
Type II Error: increase
Power: decrease
6. Increase the sample size (n)
Example 2: p. 502, #26. Assume conditions have been met already.
a) One tailed: They are looking for an INCREASE in visibility
Ho: Visibility is the same
Ha: Visibility is increased
b) Saying that the visibility of the signs has increased when really it has stayed the same
c) Saying that the visibility of the signs has stayed the same when really the visibility has increased
d) Saying the visibility of the signs has increased and the visibility really did increase
e) The power will decrease. Decreasing alpha decreases the chances of rejecting Ho, thus decreasing the
power.
f) The power will decrease. The smaller the sample size, the lower the power.
VOCAB:
Statistically significant = When we reject the Ho (when P-value < alpha)
Example: If σ = 0.05, which of the following p-values would be significant?
0.023
0.034
0.056
0.089
0.123
Yes
Yes
No
No
No
Example 3: p. 499, #5
There was a significant difference at the   .05 level. At the   .10 level, the device would still have been
significantly better because we would still have reject the Ho. At the   .01 level, the device may not have
been significantly better. It is more difficult to reject the Ho at the   .01 level.
Confidence Levels & Significance Levels:

Confidence levels are… 2 sided

"Match" the … confidence level with the  (significance level) when doing 2 sided test (Ha:  )
  1 – (confidence level)

When doing one sided … “match half the significance level with the confidence level (Ha: < , >)
  1 – (confidence level)/ 2
Ex: 95% confidence interval
Ex: 92% confidence interval
Ha: 
  .05
Ha: 
  .08
Ha: < , >
  .025
Ha: < , >
  .04
Example 4: p. 500, #9. Assume conditions met already.
a) Conditions met  Normal Model  1-Prop Z-Int
.030  (2.326)
(.03)(.97)
 (.01896,.04094)
1302
We are 98% confident that the true percentage of men who believe that the most important measure
of success is their work is between 1.896% and 4.094%
b) Ho: p = .05
Ha: P < .05
Our confidence interval suggests that it has fallen below the 5% mark because 5% is out of the
interval on the upper end.
c) We would use   .01 since it is a one-tailed test with a confidence level of 98%
PRACTICE PROBLEM:
A report on health care in the US states that 28% of Americans have experienced times when they haven’t
been able to afford medical care. A news organization randomly sampled 801 Americans and found that 251
of them reported that there had been times in the last year when they had not been able to afford medical
care.
a. Does this indicate that this problem is more severe than reported (the % has increased)? Use a 5%
level of significance.
b. Are the results statistically significant?
c. Since you rejected the claimed value, use your sample to estimate the true parameter with 92%
confidence.
d. What is a type I error in context?
e. What is a type II error in context?
f. What is the Power in context?
g. Interpret the P-value in context.
a)
Ho: p = .28
p = .28
q = .72
pˆ 
251
 .313
801
n = 801
Ha: p > .28
Conditions met Normal Model 1-Prop. Z-Test
Z
.313  .28
 2.103
(.28)(.72)
801
P( Z  2.103)  .018
  .05
We reject the Ho because the P-value of .018 <   .05 .
We have sufficient evidence that the true percentage of Amercans that have experienced times
where they have been unable to afford healthcare is greater than 28%.
b) Yes, we rejected the Ho
c) Conditions met  Normal Model  1-Prop Z-Int
.313  (1.751)
(.313)(.687)
 (.28467,.34205)
801
We are 92% confident that the true percentage of Americans that have experienced times when they
were unable to afford healthcare is between 28.467% and 34.205%.
d) Saying that the percentage of Americans that have experienced times when they were unable to
afford healthcare is more than 28% when really it still is 28%.
e) Saying that the percentage of Americans that have experienced times when they were unable to
afford healthcare is still 28% when really it was more than 28%.
f)
Saying that the percentage of Americans that have experienced times when they were unable to
afford healthcare is more than 28% and it really is more than 28%.
g) There is a 1.8% chance of getting a sample of Americans that have experienced times when they
were unable to afford healthcare of greater than 28% if it really is 28% of Americans that have
experienced times when they were unable to afford healthcare.
Complete the Ch. 21 practice problems worksheet