Inference

Confidence Interval for p
pˆ 1  pˆ 
pˆ 1  pˆ 
pˆ  z *
to pˆ  z *
n
n
1
Confidence Interval
Plausible values for the
unknown population proportion,
p.
 We have confidence in the
process that produced this
interval.

2
Inference – Using C.I.
The population proportion, p,
could be any of the values in
the interval.
 Values outside the interval are
not plausible values for p.

3
Inference – Hypothesis Test
Propose a value for the
population proportion, p.
 Does the sample data support
this value?

4
Example

A law firm will represent people
in a class action lawsuit against
a car manufacturer only if it is
sure that more than 10% of the
cars have a particular defect.
5
Example
Population: Cars of a particular
make and model.
 Parameter: Proportion of this
make and model of car that
have a particular defect.

6
Example
Null Hypothesis
 H0: p = 0.10
 Alternative Hypothesis
 HA: p > 0.10

7
Example
The law firm contacts 100 car
owners at random and finds out
that 12 of them have cars that
have the defect.
 Is this sufficient evidence for
the law firm to proceed with the
class action law suit?

8
Example

How likely is it to get a sample
proportion as extreme as the
one we observe when taking a
random sample of 100 from a
population with p = 0.10?
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Example

Sampling distribution of p̂
Shape approximately normal
because 10% condition and
success/failure condition are
satisfied.
 Mean: p = 0.10
0.10(0.90)
 0.03
 Standard Deviation:
100

10
Draw a Picture
0.00
0.05
0.10
0.12
0.15
0.20
p-hat
11
Standardize
pˆ  p0
z
p0 1  p0 
n
0.12  0.10
0.02
z

 0.67
0.10(0.90) 0.03
100
12
Use Table Z
z
0.5
0.6
0.7
0.05
0.06
0.07
0.7486
13
1 - 0.7486
= 0.2514
0.7486
0.00
0.05
0.10
0.12
0.15
0.20
p-hat
14
Interpretation

Getting a sample proportion of
0.12 or more will happen about
25% (P-value = 0.25) of the
time when taking a random
sample of 100 from a
population whose population
proportion is p = 0.10.
15
Interpretation


Getting a value of the sample
proportion of 0.12 is consistent with
random sampling from a population
with population proportion p = 0.10.
This sample result does not
contradict the null hypothesis. The
P-value is not small, therefore fail to
reject H0.
16
Interpretation


Even though the sample proportion,
0.12, is larger than the hypothesized
population proportion, 0.10, it is not
large enough for us to believe that
the population proportion is greater
than 0.10.
There is not convincing evidence.
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Conclusion

Based on this sample, the law
firm should not pursue the class
action lawsuit because the
population proportion of
defective cars could be only
10%.
18
Test of Hypothesis

Step 1: State your null and
alternative hypotheses.
 H0: p = po
 HA: p > po
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Test of Hypothesis

Step 2: Check conditions
Independence
 Random sampling condition
 10% condition
 Success/Failure condition

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Test of Hypothesis

Step 3: Calculate the test statistic
value and convert it into a P-value.
z

pˆ  p0
p0 1  p0 
n
Use Table Z.
21
Test of Hypothesis

Step 4: Use the P-value to
reach a decision.
If the P-value is small, then
reject Ho.
 If the P-value is not small, then
fail to reject Ho.

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Test of Hypothesis

Step 5: State your conclusion in
the context of the problem.

What does rejecting, or failing to
reject, Ho mean in the context of
the problem.
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Alternatives

H0: p = po
 HA: p < po, P-value = Pr < z
 HA: p > po, P-value = Pr > z
 HA: p  po, P-value = Pr > |z|
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Another Example


According to the U.S. census, Story
County has 9.7% of its population
classified as non-white.
Of 120 people called for jury duty
in Story County only 3 are nonwhite. Is this convincing evidence
of under-representation of nonwhites?
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Another Example

Step 1: State your null and
alternative hypotheses.
 H0: p = 0.097
 HA: p < 0.097
 p is the proportion of non-whites
among all people in the jury pool for
Story County.
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Test of Hypothesis

Step 2: Check conditions
Independence
 Random sampling condition
 10% condition
 Success/Failure condition

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Test of Hypothesis

Step 3: Calculate the test statistic
value and convert it into a P-value.
pˆ  p0
0.025  0.097
z

p0 1  p0 
0.097 (1  0.097 )
n
120
 0.072
z
 2.67
0.027
P  value  0.0038
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Test of Hypothesis

Step 4: Use the P-value to
reach a decision.

Because the P-value is small, we
should reject Ho
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Test of Hypothesis

Step 5: State your conclusion in
the context of the problem.

This is convincing evidence that
non-whites are underrepresented in the jury pool.
30