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Unformatted text preview: STAT 2023  Holbrook Testing Hypotheses About Proportions
Chapter 20 20  1 STAT 2023  Holbrook Statistical Methods
Statistical
Methods Descriptive
Statistics Inferential
Statistics Estimation 20  2 Hypothesis
Testing STAT 2023  Holbrook Hypothesis Testing Concepts 20  3 STAT 2023  Holbrook What’s a Hypothesis? 1. A Belief about a
Belief
Population Parameter
Population Parameter Is
Parameter
Population Mean,
Population
Proportion, Variance
Proportion, I believe the population
believe
proportion of SFCC students
who have tattoos is .25!
who Must Be Stated
Before Analysis © 19841994 T/Maker Co. 20  4 STAT 2023  Holbrook Population 20  5 Hypothesis Testing STAT 2023  Holbrook Population 20  6 Hypothesis Testing
I believe the
population
proportion is .7
(hypothesis). STAT 2023  Holbrook Population Hypothesis Testing
I believe the
population
proportion is .7
(hypothesis). Random
Random
sample
sample
Prop. ^ = .3
p 20  7 STAT 2023  Holbrook Population Hypothesis Testing
I believe the
population
proportion is .7
(hypothesis). Random
Random
sample
sample
Prop. ^ = .3
p 20  8 Reject
Reject
hypothesis!
hypothesis!
Not close.
Not close. STAT 2023  Holbrook Basic Idea
Sampling Distribution µ
p = 50 .7
20  9 H0 Sample Mean
Sample proportion Basic Idea STAT 2023  Holbrook Sampling Distribution
It is unlikely
It
that we would
get a sample
proportion of
this value ...
this .3
20  10
10 µ
p = 50 .7 H0 Sample Mean
Sample proportion Basic Idea STAT 2023  Holbrook Sampling Distribution
It is unlikely
It
that we would
get a sample
proportion of
this value ...
this
... if in fact this were
the population prop.
.3
20  11
11 µ
p = 50 .7 H0 Sample Mean
Sample proportion Basic Idea STAT 2023  Holbrook Sampling Distribution
It is unlikely
It
that we would
get a sample
proportion of
this value ...
this ... therefore,
...
we reject the
hypothesis
that p = .7
.7
... if in fact this were
the population prop. .3
20  12
12 µ
p = 50 .7 H0 Sample Mean
Sample proportion STAT 2023  Holbrook Null Hypothesis 1. Has Serious Outcome If Incorrect
Has
Decision Made
Decision
2. Always Has Equality Sign: = , ≤ , or ≥
Always
or
3. Designated H0 (Pronounced Hoh)
4. Specified as H0: p = Some Numeric
Value 20  13
13 Specified with = Sign Even if ≤ , or ≥
Specified
or
Example, H0: p = .3 STAT 2023  Holbrook Alternative Hypothesis 1. Opposite of Null Hypothesis
2. Put what we are trying to prove in the
Put
alternative hypothesis
alternative
3.
4.
5.
5. Also called the research hypothesis
Always Has Inequality Sign: ≠ , < , or
or
>
Designated Ha
Specified Ha: p < Some Value 20  14
14 Example, Ha: p < .3
.3 Identifying Hypotheses
Steps STAT 2023  Holbrook 1. Example Problem: Test That the Population
Example
proportion Is Not .3
proportion
2. Steps State the Question Statistically (p ≠ .3)
State
State the Opposite Statistically (p = .3) Select the Alternative Hypothesis (p ≠ .3)
Select 20  15
15 Must Be Mutually Exclusive & Exhaustive
Has the ≠ , <, or > Sign
Has
or Sign State the Null Hypothesis (p = .3) What Are the Hypotheses?
STAT 2023  Holbrook Is the population proportion of SFCC
students who ever drink alcohol .8 ?
students
State the question statistically: p = .8
State
.8
State the opposite statistically: p ≠ .8
State
Select the alternative hypothesis: Ha: p ≠ .8
State the null hypothesis: H0: p = .8
State
20  16
16 What Are the Hypotheses?
STAT 2023  Holbrook Is the population proportion of SFCC
students who ever drink alcohol different
from .8 ?
from
State the question statistically: p ≠ .8
State
State the opposite statistically: p =.8
State
Select the alternative hypothesis: Ha: p ≠ .8
State the null hypothesis: H0: p = .8
State
20  17
17 What Are the Hypotheses?
STAT 2023  Holbrook Is the proportion of SFCC students who
have ever smoked marijuana less than or
equal to .5 ?
equal
State the question statistically: p ≤ .5
State
State the opposite statistically: p > .5
State
Select the alternative hypothesis: Ha: p > .5
Select
State the null hypothesis: H0: p = .5
State
20  18
18 What Are the Hypotheses?
STAT 2023  Holbrook Is the proportion of SFCC students who
have pierced their ears greater than .7 ?
have
State the question statistically: p > .7
State
State the opposite statistically: p ≤ .7
State
Select the alternative hypothesis: Ha: p > .7
State the null hypothesis: H0: p = .7
State
20  19
19 STAT 2023  Holbrook Hypothesis Testing Steps 20  20
20 STAT 2023  Holbrook Step 1
s State H0 s State Ha s Choose n s Collect data 20  21
21 H0 Testing Steps STAT 2023  Holbrook Step 1 H0 Testing Steps
Steps 2 through 5
s (2) Compute the test
(2)
statistic.
statistic. s State H0 s State Ha s (3) Find/interpret the pvalue. s Choose n s (4) Make statistical decision. s Collect data • s 20  22
22 Reject or Fail to Reject H0 (5) Express conclusion in the
(5)
context of the problem.
context STAT 2023  Holbrook Z Test for Proportion 20  23
23 OneSample Z Test for Proportion STAT 2023  Holbrook 1. Assumptions Two Categorical Outcomes
Normal Approximation can be used 20  24
24 ˆ
ˆ
np > 10 and n (1 − p) > 10 OneSample Z Test for Proportion STAT 2023  Holbrook 1. Assumptions
Two Categorical Outcomes
Normal Approximation can be used ˆ
ˆ
np > 10 and n (1 − p) > 10 2. Ztest statistic for proportion
2.
Z≅
20  25
25 p − p0
p0 ⋅ (1 − p0 )
n Hypothesized
Hypothesized
population proportion
population STAT 2023  Holbrook Observed Level of Significance: pvalue 20  26
26 STAT 2023  Holbrook pValue 1. Probability of Obtaining a Test Statistic
Probability
More Extreme (≤ or ≥ ) than Actual
More
Sample Value Given H0 Is True
Sample
2. Called Observed Level of Significance
2. Smallest Value of α H0 Can Be Rejected
Smallest 3. Used to Make Rejection Decision 20  27
27 If pValue < α, Reject H0
If
If pValue ≥ α, Do Not Reject H0
If STAT 2023  Holbrook Examples 20  28
28 STAT 2023  Holbrook One Proportion Z Test Example The present packaging
The
system produces 10%
10%
defective cereal boxes.
Using a new system, a
random sample of 200
200
boxes had 11 defects.
boxes
11
Does the new system
produce fewer defects?
fewer
20  29
29 Checking the Assumption*
STAT 2023  Holbrook ˆ
np > 10 or 200(0.055) = 11 > 10
ˆ
and n (1 − p) > 10 or 200(1 − 0.055) = 189 > 10
So the sample size is large enough.
Note: Our way of checking the assumptions
is slightly different from the book.
is
20  30
30 STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .10
Ha: p < .10
n = 200
Test Statistic: Pvalue:
Pvalue: Decision:
Decision:
Conclusion: 20  31
31 STAT 2023  Holbrook One Proportion Z Test Solution Step 2: Test Statistic Z≅ 20  32
32 p − p0
=
p0 ⋅ (1 − p0 )
n 11
−.10
200
= −2.12
.10 ⋅ (1−.10)
200 STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .10
Ha: p < .10
n = 200 Pvalue:
Pvalue: Test Statistic: Z ≅ −2.12 Decision:
Decision:
Conclusion: 20  33
33 STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .10
Ha: p < .10
n = 200
Test Statistic: Z ≅ −2.12 Pvalue: = 0.0170
Pvalue: 0.0170
1.7% of the time we
1.7%
would have seen data
like this if H0 was true.
like
(rarely)
(rarely)
Decision:
Conclusion: 20  34
34 STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .10
Ha: p < .10
n = 200
Test Statistic: Z ≅ −2.12 Pvalue: = 0.0170
Pvalue: 0.0170
1.7% of the time we
1.7%
would have seen data
like this if H0 was true.
like
(rarely)
(rarely)
Decision:
Reject H0
Conclusion: 20  35
35 STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .10
Ha: p < .10
n = 200
Test Statistic: Z ≅ −2.12 Pvalue: = 0.0170
Pvalue: 0.0170
1.7% of the time we
1.7%
would have seen data
like this if H0 was true.
like
(rarely)
(rarely)
Decision:
Reject H0
Conclusion: 20  36
36 Enough evidence that the
Enough
population proportion of
defects is < .10
defects STAT 2023  Holbrook One Proportion Z Test Thinking Challenge You’re an accounting
You’re
manager. A yearend audit
showed 4% of transactions
4%
had errors. You implement
new procedures. A random
sample of 500 transactions
500
had 25 errors. Has the
25
proportion of incorrect
proportion
transactions changed?
changed
20  37
37 Checking the Assumption*
STAT 2023  Holbrook ˆ
np > 10 or 500(0.05) = 25 > 10
ˆ
and n (1 − p) > 10 or 500(1 − 0.05) = 475 > 10
So the sample size is large enough. 20  38
38 STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .04
Ha: p ≠ .04
n = 500
Test Statistic: Pvalue:
Pvalue: Decision:
Decision:
Conclusion: 20  39
39 STAT 2023  Holbrook One Proportion Z Test Solution Step 2: Test Statistic Z≅ 20  40
40 p − p0
=
p0 ⋅ (1 − p0 )
n 25
−.04
500
= 1.14
.04 ⋅ (1−.04 )
500 STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .04
Ha: p ≠ .04
n = 500 Pvalue:
Pvalue: Test Statistic: Z ≅ 1.14 Decision:
Decision:
Conclusion: 20  41
41 STAT 2023  Holbrook TwoTailed Z Test pValue Solution pvalue is P(Z ≤ 1.14 or Z ≥ 1.14)
pvalue
1/2 pValue 1/2 pValue 1.14
1.50 0 1.14
1.50 20  42
42 Z
Z value of sample
value
statistic (observed)
statistic STAT 2023  Holbrook TwoTailed Z Test pValue Solution pvalue is P(Z ≤ 1.14 or Z ≥ 1.14)
pvalue
.1271
1/2 pValue 1/2 pValue 1.14
1.50 0 1.14
1.50 20  43
43 Z
Z value of sample
value
statistic (observed)
statistic STAT 2023  Holbrook TwoTailed Z Test pValue Solution
Due to symmetry
.1271
1/2 pValue .1271
1/2 pValue 1.14
1.50 0 1.14
1.50 20  44
44 Z
Z value of sample
value
statistic (observed)
statistic STAT 2023  Holbrook TwoTailed Z Test pValue Solution So the pvalue = .1271 + .1271 = .2542
So
.1271
1/2 pValue .1271
1/2 pValue 1.14
1.50 0 1.14
1.50 20  45
45 Z
Z value of sample
statistic (observed)
statistic STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .04
Ha: p ≠ .04
n = 500
Test Statistic: Z ≅ 1.14 Pvalue: = 0.2542
Pvalue: 0.2542
25.42% of the time we
25.42%
would have seen data
like this if H0 was true.
like
(not a rare occurrence)
(not
Decision:
Conclusion: 20  46
46 STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .04
Ha: p ≠ .04
n = 500
Test Statistic: Z ≅ 1.14 Pvalue: = 0.2542
Pvalue: 0.2542
25.42% of the time we
25.42%
would have seen data
like this if H0 was true.
like
(not a rare occurrence)
(not
Decision:
Fail to Reject H0
Conclusion: 20  47
47 STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .04
Ha: p ≠ .04
n = 500
Test Statistic: Z ≅ 1.14 Pvalue: = 0.2542
Pvalue: 0.2542
25.42% of the time we
25.42%
would have seen data
like this if H0 was true.
like
(not a rare occurrence)
(not
Decision:
Fail to Reject H0
Conclusion: 20  48
48 Not enough evidence that the
Not
population proportion of errors
is different from .04
is STAT 2023  Holbrook One Proportion Z Test Example Is drug use running rampant?
Is
200 students at SFCC were
200
asked if they have ever
smoked marijuana and 140
140
responded yes. Is the
population proportion of SFCC
students who have ever
smoked marijuana more than
60% ?
60% 20  49
49 Checking the Assumption*
STAT 2023  Holbrook ˆ
np > 10 or 200(0.7) = 140 > 10
ˆ
and n (1 − p) > 10 or 200(1 − 0.7) = 60 > 10
So the sample size is large enough. 20  50
50 STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .6
Ha: p > .6
n = 200
Test Statistic: Pvalue:
Pvalue: Decision:
Decision:
Conclusion: 20  51
51 STAT 2023  Holbrook One Proportion Z Test Solution Step 2: Test Statistic Z≅ 20  52
52 ˆ
p − p0
=
p 0 ⋅ (1 − p 0 )
n 140
− .60
200
= 2.89
.60 ⋅ (1 − .60)
200 STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .6
Ha: p > .6
n = 200
Test Statistic: Z ≅ 2.89 Pvalue:
Pvalue: Decision:
Decision:
Conclusion: 20  53
53 OneTailed Z Test pValue Solution STAT 2023  Holbrook pValue is P(Z ≥ 2.89)
pValue Use
Use
alternative
hypothesis
to find
direction
direction pValue 0 1.50
2.89 20  54
54 Z
Z value of sample
statistic
statistic STAT 2023  Holbrook OneTailed Z Test pValue Solution pValue is P(Z ≥ 2.89) = 1  .9981 = .0019
pValue Use
Use
alternative
hypothesis
to find
direction
direction .9981 .0019
pValue 0 1.50
2.89 20  55
55 Z
Z value of sample
statistic
statistic STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .6
Ha: p > .6
n = 200
Test Statistic: Z ≅ 2.89 Pvalue: = 0.0019
Pvalue: 0.0019
0.19% of the time we
0.19%
would have seen data
like this if H0 was true.
like
(rarely)
(rarely)
Decision:
Conclusion: 20  56
56 STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .6
Ha: p > .6
n = 200
Test Statistic: Z ≅ 2.89 Pvalue: = 0.0019
Pvalue: 0.0019
0.19% of the time we
0.19%
would have seen data
like this if H0 was true.
like
(rarely)
(rarely)
Decision:
Reject H0
Conclusion: 20  57
57 STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .6
Ha: p > .6
n = 200
Test Statistic: Z ≅ 2.89 Pvalue: = 0.0019
Pvalue: 0.0019
0.19% of the time we
0.19%
would have seen data
like this if H0 was true.
like
(rarely)
(rarely)
Decision:
Reject H0
Conclusion: 20  58
58 Enough evidence that the population
Enough
proportion of students who have
ever smoked marijuana is more than
.6 or 60%
.6 Hypothesis Test Solution
STAT 2023  Holbrook 1. Using the TI83 Hit “STAT” scroll to “TESTS” then “5”, hit “ENTER” “1PropZTest“ Scroll to p0 and enter the hypothesized value For example: p0=0.6 Enter x and n. For example: x=140 and n=200 Scroll and select (by hitting “ENTER”) one of the
”)
following: two tail (≠ ), lower tail (<) or upper tail (>)
following:
),
alternative hypothesis.
alternative 20  59
59 For example: >p0 Select “Calculate” and hit “ENTER” End of Chapter
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