1. Chapter 9: testing a claim
Ch. 9-1 Significance Tests: The Basics

2. confidence intervals
a significance test
confidence intervals
a significance test
observed data
claim
In the free throw example, I claimed my 𝑝=0.8.
We tried to find evidence “against” this claim.
null hypothesis
𝐻 0
against

3. It is never correct to write a hypothesis about a sample statistic.
alternative hypothesis
𝐻 𝑎
before
It’s “cheating” to look at data first and then frame hypotheses to fit what the data shows.
Free throw ex: 𝑝<.8
greater
less
different from
not equal
Pennies ex: 𝜇≠22
parameters
It is never correct to write a hypothesis about a sample statistic.
Never write 𝑝 <.66 or 𝑥 =19.5. Graders take off points.

4. For each of the following settings, (a) describe the parameter of interest, and (b) state appropriate hypotheses for a significance test.
1. According to the Web site sleepdeprivation.com, 85% of teens are getting less than eight hours of sleep a night. Jannie wonders whether this result holds in her large high school. She asks an SRS of 100 students at the school how much sleep they get on a typical night. In all, 75 of the responders said less than 8 hours.
(a) p = true proportion of students at Jannie’s high school who get less
than 8 hours of sleep at night.
(b) H0: p = 0.85 and Ha: p ≠ 0.85.
2. As part of its 2010 census marketing campaign, the U.S. Census Bureau advertised “10 questions, 10 minutes—that’s all it takes.” On the census form itself, we read, “The U.S. Census Bureau estimates that, for the average household, this form will take about 10 minutes to complete, including the time for reviewing the instructions and answers.” We suspect that the actual time it takes to complete the form may be longer than advertised.
(a) μ = true mean amount of time that it takes to complete the census form.
(b) H0: μ = 10 and Ha: μ > 10.

5. 𝑝-value
stronger
𝛼 is a significance level 𝛼
statistically significant at level 𝛼
alternative hypothesis
Reject 𝐻 0
Fail to reject 𝐻 0
NEVER say “accept 𝐻 0 .” There is usually some evidence that 𝐻 0 is false. If 𝑝-value is large, there is just not enough convincing evidence to rule out random chance.

6. → reject 𝐻 0 → conclude 𝐻 𝑎 (in context)
strong evidence against 𝐻 0
→ reject 𝐻 0 → conclude 𝐻 𝑎 (in context)
→ fail to reject 𝐻 0 → cannot conclude 𝐻 𝑎 (in context)
weak evidence against 𝐻 0
Standard significance level is 𝛼=0.05.
𝑝 =.66
Assuming 𝐻 0 is true 𝑝=.8 , there is a .007 probability of obtaining a
𝑝 value of .66 or lower purely by chance.
This provides strong evidence
against 𝐻 0 and is statistically significant at 𝛼= <.05 .
Therefore, we reject 𝐻 0 and can conclude that the true proportion of
free throws made by Mr. Brinkhus is less than 0.8.
1) Interpret 𝑝-value
2) evidence
3) decision with context

7. 𝑝 =.74
Assuming 𝐻 0 is true 𝑝=.8 , there is a .144 probability of obtaining a
𝑝 value of .74 or lower purely by chance.
This provides weak evidence
against 𝐻 0 and is NOT statistically significant at 𝛼= >.05 .
Therefore, we fail to reject 𝐻 0 and cannot conclude that the true
proportion of free throws made by Mr. Brinkhus is less than 0.8.

8. 𝑝 → true proportion of FT made by Mr. Brinkhus
𝐻 0 : 𝑝=0.80
𝑝 = =0.7
𝐻 𝑎 : 𝑝<0.80
𝜎 𝑝 = 𝑝 1−𝑝 𝑛 =
Sampling Distribution of 𝑝
N 0.8, .0566
=.0566
𝑧= 𝑝 −𝑝 𝜎 𝑝 = .7− =−1.77
0.7
0.8
𝑎𝑟𝑒𝑎=.039
normalcdf(−9999, 0.7, 0.8, .0566)=.039
𝑝-value

9. OUR DECISION Reject 𝐻 0 . Conclude parameter is less than 0.8.
We decided to reject 𝐻 0 , but 𝐻 0 could still be true (it is just unlikely).
TRUTH
For this situation, we can calculate the POWER of the test.
𝐻 0 true
𝐻 𝑎 true
Reject
𝐻 0
Type I
Error
P(Type I) = P(Reject 𝐻 0 | 𝐻 0 true)
Power
Power = P(Reject 𝐻 0 | 𝑝 = alt value)
OUR DECISION
P(Type II) = P(F.t.R. 𝐻 0 | 𝑝 = alt value)
Fail to reject 𝐻 0
Type II
Error
P(Type II) + Power = ___ 1
𝐻 0 is true, but we reject 𝐻 0 .
𝐻 𝑎 is true, but we fail to reject 𝐻 0 .
𝐻 𝑎 is true and we correctly reject 𝐻 0 .
This is the probability that the
test will reject 𝐻 0 when an alternative value of the parameter is true.

10. good shipment
bad shipment
𝐻 0 is true. Reject 𝐻 0 .
The shipment is good (𝑝=0.09) and he sends it
back to manufacturer.
This may upset the manufacturer and he is left
with empty shelves at the store.
𝐻 𝑎 is true. Fail to reject 𝐻 0 .
The shipment is bad (𝑝>0.09) but he accepts
the shipment.
This means the customers have a higher chance of buying
rusty rotors.
???
Type I Error – don’t want empty shelves at store.
Type II Error – don’t want to upset customers that buy rusty rotors.

11. Sampling Distribution of 𝑝 𝐻 𝑎 : 𝑝<0.80 𝑁 .8, .8 .2 50
𝑝= 𝑛=50
Assume 𝐻 0 true
𝐻 0 : 𝑝=0.80
Sampling Distribution of 𝑝
𝐻 𝑎 : 𝑝<0.80
𝑁 .8,
Fail to reject 𝐻 0 if 𝑝 >0.707
Reject 𝐻 0 if 𝑝 <0.707
𝑧= 𝑝 −𝑝 𝜎 𝑝
.05 𝑝
0.8
−1.645= 𝑝 −
0.707
𝑝 =0.707
Recall: Type I Error means you reject 𝐻 0 when 𝐻 0 is true.
invNorm .05,.8,.057 =.707
P(Reject 𝐻 0 | 𝐻 0 true)
=𝛼=0.05
So higher 𝛼 (significance level) means higher chance of getting Type I Error.

12. Assume 𝐻 𝑎 true
Sampling Distribution of 𝑝
𝑁 .7,
Reject 𝐻 0 if 𝑝 <0.707
Fail to reject 𝐻 0 if 𝑝 >0.707
.543
.457
0.7
0.707
normalcdf .707,99999,.7, (.3) 50 =.457
Recall: Power is the probability you reject 𝐻 0 when 𝐻 𝑎 is true.
Recall: Type II Error means you fail to reject 𝐻 0 when 𝐻 𝑎 is true.
So we will make the right decision by rejecting 𝐻 % of the time if 𝐻 𝑎 : 𝑝=0.7 is true, which is Power.
P(Reject 𝐻 0 | 𝐻 𝑎 is true)
= 1− P(Type II Error)
Textbook has P(Type II Error) = 𝛽
= 1−𝛽

13. 𝜇=18.6
𝜇>18.6 8
3.2
0.05
To answer this we will use the Java applet statistical power.

14. Sampling Dist of 𝑥
𝑁 18.6,
Reject 𝐻 0 if 𝑥 >20.46
Fail to reject 𝐻 0 if 𝑥 <20.46
invNorm .95,18.6,
=20.46
.05=𝛼= P(Type I Error)
18.6
20.46
Sampling Dist of 𝑥
𝑁 21,
P(Type II Error) = 𝛽
Power = 0.683=
P(Reject 𝐻 0 | 𝐻 𝑎 is true)
20.46 21

15. So if the alternative 𝐻 𝑎 :𝜇=21 is true, our
0.683
𝐻 𝑎 true. Reject 𝐻 0 .
So if the alternative 𝐻 𝑎 :𝜇=21 is true, our
test will correctly reject 𝐻 % of the time.
Java applet statistical power
higher sample size gives higher power
higher alpha gives higher power
𝐻 𝑎 being farther from 𝐻 0 gives higher power
But higher P(Type I Error)
There’s a tradeoff between Type I & Type II.
This is when researchers weigh which error has a more serious consequence.
Remember:
P(Type I Error) =𝛼
P(Type II Error) =𝛽
Power =1−𝛽