one sample inf-1 in statistical testing, we use deductive reasoning to specify what should happen if...
DESCRIPTION
One Sample Inf-3 CLAIM: A new variety of turf grass has been developed that is claimed to resist drought better than currently used varieties. CONJECTURE: The new variety resists drought no better than currently used varieties. DEDUCTION:If the new variety is no better than other varieties (P), then areas planted with the new variety should display the same number of surviving individuals (Q) after a fixed period without water than areas planted with other varieties. CONCLUSION:If more surviving individuals are observed for the new varieties than the other varieties (NOT Q), then we conclude the new variety is indeed not the same as the other varieties, but in fact is better (NOT P). Examples in Testing LogicTRANSCRIPT
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One Sample Inf-1
• In statistical testing, we use deductive reasoning to specify what should happen if the conjecture or null hypothesis is true.
• A study is designed to collect data to challenge this hypothesis.
• We determine if what we expected to happen is supported by the data.
• If it is not supported, we reject the conjecture. • If it cannot be rejected we (tentatively) fail to
reject the conjecture. • We have not proven the conjecture, we have simply
not been able to disprove it with these data.
Statistical Test for Population Mean
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One Sample Inf-2
Statistical tests are based on the concept of Proof by Contradiction.
If P then Q = If NOT Q then NOT P
Analogy with justice system
Logic Behind Statistical Tests
Court of Law1. Start with premise that person is
innocent. (Opposite is guilty.)
2. If enough evidence is found to show beyond reasonable doubt that person committed crime, reject premise. (Enough evidence that person is guilty.)
3. If not enough evidence found, we don’t reject the premise. (Not enough evidence to conclude person guilty.)
Statistical Hypothesis Test1. Start with null hypothesis, status quo.
(Opposite is alternative hypothesis.)2. If a significant amount of evidence is
found to refute null hypothesis, reject it. (Enough evidence to conclude alternative is true.)
3. If not enough evidence found, we don’t reject the null. (Not enough evidence to disprove null.)
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One Sample Inf-3
CLAIM: A new variety of turf grass has been developed that is claimed to resist drought better than currently used varieties.
CONJECTURE: The new variety resists drought no better than currently used varieties.
DEDUCTION: If the new variety is no better than other varieties (P), then areas planted with the new variety should display the same number of surviving individuals (Q) after a fixed period without water than areas planted with other varieties.
CONCLUSION: If more surviving individuals are observed for the new varieties than the other varieties (NOT Q), then we conclude the new variety is indeed not the same as the other varieties, but in fact is better (NOT P).
Examples in Testing Logic
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One Sample Inf-4
1. Null Hypothesis (H0:)2. Alternative Hypothesis (HA:)3. Test Statistic (T.S.)
Computed from sample data.Sampling distribution is known if
the Null Hypothesis is true.4. Rejection Region (R.R.)
Reject H0 if the test statistic computed with the sample data is unlikely to come from the sampling distribution under the assumption that the Null Hypothesis is true.
5. Conclusion:Reject H0 or Do Not Reject H0
H0: ≤ 530
HA: > 530
Other HA: < 530 530
),(~.. * 10N
n
yzST
R.R. z* > z
Or z* < -z
rz*| > z
Five Parts of a Statistical Test
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One Sample Inf-5
Research Hypotheses: The thing we are primarily interested in “proving”.
03A
02A
01A
HHH
:::
Average heightof the class
m61Hm61Hm61H
3A
2A
1A
.:
.:
.:
Null Hypothesis: Things are what they say they are, status quo.
00H : m61H0 .:
Hypotheses
(It’s common practice to always write H0 in this way, even though what is meant is the opposite of HA in each case.)
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One Sample Inf-6
Some function of the data that uses estimates of the parameters we are interested in and whose sampling distribution is known when we assume the null hypothesis is true.
Most good test statistics are constructed using some form of a sample mean.
Test Statistic
Why?The Central Limit Theorem Of Statistics
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One Sample Inf-7
Test Statistic: Sample Mean
Under H0: the sample mean has a sampling distribution that is normal with mean 0.
n
iiy
ny
1
1
n,N~y
),(N~y
0
0
Under HA: the sample mean has a sampling distribution that is also normal but with a mean that is different from 0.
n,N~y
),(N~y
1
1
003A
02A
01A
orHHH
:::
Hypotheses of interest:
Developing a Test Statistic for the Population Mean
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One Sample Inf-8
H0: True
HA1: True
01
What would you conclude if the sample mean fell in location (1)? How about location (2)? Location (3)? Which is most likely 1, 2, or 3 when H0 is true?
(1) (2)(3)
Graphical View
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One Sample Inf-9
0Reject H0 if the sample mean is in the upper tail ofthe sampling distribution.
Rejection Region
),(N n0
C1
01 HjectCyIf Re
How do we determine C1?
H0: Assumed TrueTesting HA
1: 0
Rejection Region
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One Sample Inf-10
Reject H0 if the sample mean is larger than “expected”.
If H0 were true, we would expect 95% of sample means to be less than the upper limit of a 90% CI for μ.
nC 645.10
1
In this case, if we use this critical value, in 5 out of 100 repetitions of the study we would reject Ho incorrectly. That is, we would make an error.
But, suppose HA1 is the true situation, then
most sample means will be greater than C1 and we will be making the correct decision to reject more often.
From the standard normal table.
05.0645.10
n
yP
Determining the Critical Value for the Rejection Region
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One Sample Inf-11
H0 = 0 TrueTesting HA
1: 0Rejection Region
),(N n0
C1
0
Rejection Region
C2
H0: = 0 TrueTesting HA
1: 0
Rejection Region
C3L
H0: = 0 TrueTesting HA
1: 0
C3U
2.5%
5%
5%
2.5%
Rejection Regions for Different Alternative Hypotheses
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One Sample Inf-12
H0: True
0
(1) (2)(3)
Rejection Region
C1
If the sample mean is at location (2) and H0 is actually true, we make the wrong decision.
This is called making a TYPE I error, and the probability of making this error is usually denoted by the Greek letter .
If
= P(Reject H0 when H0 is is the true condition)
nC 645.10
1 If then = 1/20=5/100 or .05
nzC
01 then the Type I error is .
If the sample mean is at location (1) or (3) we make the correct decision.
Type I Error
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One Sample Inf-13
HA1: True
1
Rejection Region
C10
(1) (2)(3)
= P(Do Not Reject H0 when HA is the true condition)
We will come back to the Type II error later.
If the sample mean is at location (1) or (3) we make the wrong decision.
If the sample mean is at location (2) we make the correct decision.
Type II Error
This is called making a TYPE II error, and the probability of making this type error is usually denoted by the Greek letter .
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One Sample Inf-14
H0: True
HA1: True
01
Type I and II Errors
C1
Rejection Region
Type I Error Region
Type II Error Region
• We want to minimize the Type I error, just in case H0 is true, and we wish to minimize the Type II error just in case HA is true.
• Problem: A decrease in one causes an increase in the other.
• Also: We can never have a Type I or II error equal to zero!
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One Sample Inf-15
1. Assume the data come from a population with unknown distribution but which has mean () and standard deviation ().
2. For any sample of size n from this population we can compute a sample mean, .
3. Sample means from repeated studies having samples of size n will have the sampling distribution of following a normal distribution with mean and a standard deviation of /n (the standard error of the mean). This is the Central Limit Theorem.
4. If the Null Hypothesis is true and = 0, then we deduce that with
probability we will observe a sample mean greater than
0 + z /n. (For example, for = 0.05, z=1.645.)
The solution to our quandary is to set the Type I Error Rate to be small, and hope for a small Type II error also. The logic is as follows:
y
y
Setting the Type I Error Rate
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One Sample Inf-16
Following this same logic, rejection rules for all three possible alternative hypotheses can be derived.
Reject Ho if:
For (1-)100% of repeated studies, if the true population mean is 0 as conjectured by H0, then the decision concluded from the statistical test will be correct. On the other hand, in 100% of studies, the wrong decision (a Type I error) will be made.
20
03A
00
2A
00
1A
z
n
yH
z
n
yH
z
n
yH
/:
:
:
Setting Rejection Regions for Given Type I Error
Note: It is just easier to work with a test statistic that is standardized. We only need the standard normal table to find critical values.
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One Sample Inf-17
The value 0 < < 1 represents the risk we are willing to take of making the wrong decision.
But, what if I don’t wish to take any risk? Why not set = 0?
What if the true situation is expressed by the alternative hypothesis and not the null hypothesis?
01
Butthis!
)n
,(N~y 1
Suppose HA1 is really the true
situation. Then the samples come from the distribution described by the alternative hypothesis and the sampling distribution of the mean will have mean 1, not 0.
= 0.05 = 0.01
= 0.001
Risk
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One Sample Inf-18
(at an Type I error probability)
zn
y 0Pr
zn
y
00 ifHReject :Rule
Pretty clear cut when 1 much greater than 0
Do not reject Ho Reject HoC
01
The Rejection Rule
10y C z
n
or if
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One Sample Inf-19
C
0 1
1Hzn
y A0 |Pr
When H0 is true
00 Hz
ny |Pr
A0 Hz
ny |Pr
When HA is true
Error Probabilities
If HA is the true situation, then any sample whose mean is larger than will lead to the correct decision (reject H0, accept HA).
y nz 0
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One Sample Inf-20
Then any sample such that its sample mean is less than will lead to the wrong decision (do not reject H0, reject HA).
ynz 0
Do not reject HAReject HAC1
If HA is the true situation
H0 : True HA : True
Reject H0
Type I Error
Correct 1 -
Accept H0
Correct
1 -
Type II Error
True Situation
Decision
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One Sample Inf-21
A0 Hz
ny |Pr
00 Hz
ny |Pr
1Hzn
y A0 |Pr
Type I Error Probability
Type II Error Probability
Power of the test.
(Reject H0 when H0 true)
(Reject HA when HA true)
(Reject H0 when HA true)
Computing Error Probabilities
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One Sample Inf-22
Sample Size: n = 50Sample Mean: Sample Standarddeviation: s = 5.6
1.40yH0: = 0 = 38
HA: > 38
What risk are we willing to take that we reject H0 when in fact H0 is true?
P(Type I Error) = = .05 Critical Value: z.05 = 1.645
zn
y
0
64516512506538140 ..
..
Rejection Region
Conclusion: Reject H0
Example
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One Sample Inf-23
To compute the Type II error for a hypothesis test, we need to have a specific alternative hypothesis stated, rather than a vague alternative
Vague
HA: > 0
HA: < 0
HA: 0
Specific
HA: = 1 > 0
HA: = 1 < 0
Note:
As the difference between 0 and 1 gets larger, the probability of committing a Type II error () decreases.
Significant Difference: = 0 - 1
Type II Error.
HA: = 5 < 10
HA: = 20 > 10HA: 10
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One Sample Inf-24
H0: = 0
HA: = 1 > 0
nzZ
10Pr
H0: = 0
HA: = 1 < 0
nzZ
10Pr
H0: = 0
Ha: 0
= 1 0
nzZ
10
2Pr
0 1 critical difference
1 - = Power of the test
0 1
0 1
Computing the probability of a Type II Error ()
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One Sample Inf-25
6451z
.05 error) I Type
65s140y
50n
.
Pr(
..
40:
38:
1
00
AH
H
n6451Z
506451Z
nzZ 10
.Pr
.PrPr
18940880Z5065
26451Z .).Pr()(.
.Pr
Assuming s is actually equal to (usually what is done):
Example: Power
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One Sample Inf-26
Power versus (fixed n and )0 1
Pr Z z Pr Z z nn
Type II error
1-As gets larger, it is less likely that a Type II error will be made.
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One Sample Inf-27
Power versus n(fixed and )0 1
Pr Z z Pr Z z nn
Type II error
1-As n gets larger, it is less likely that a Type II error will be made.
n What happens for larger ?
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One Sample Inf-28
n power1 50 .755 .2452 50 .286 .7144 50 .001 .999
n power1 25 .857 .1432 25 .565 .4354 25 .054 .946
0 1Pr Z z Pr Z 1.645 n
n
Power vs. Sample Size
Power = 1-
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One Sample Inf-29
Pr(Type II Error)
1-
0
0 large1-
0
0 large
Power n=10
n=50
Power Curve See Table 4, Ott and Longnecker
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One Sample Inf-30
0 1 0 1Pr Z z Pr Z z n
n
Pr Z z n
1) For fixed and n, decreases (power increases) as increases.
2) For fixed and , decreases (power increases) as n increases.
3) For fixed and n, decreases (power increases) as decreases.
Summary
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One Sample Inf-31
1 <
Decreasing decreases the spread in the sampling dist of
Note: z changes.
Same thing happens if you increase n.
Same thing happens if is increased.
y
N(,n)
N(,n)
Increasing Precision for fixed increases Power
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One Sample Inf-32
1) Specify the critical difference, (assume is known).
2) Choose P(Type I error) = and Pr(Type II error) = based on traditional and/or personal levels of risk.
3) One-sided tests:
4) Two-sided tests:
Example: One-sided test, = 5.6, and we wish to show a difference of = .5 as significant (i.e. reject H0: = 0 for HA: = 1= 0 + ) with = .05 and = .2.
778~7.77784.065.15.6.5 2
2
2
n
Sample Size Determination
22/2
2
)( zzn
22
2
)( zzn