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Significance Toolbox

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Significance Toolbox Identify the population of interest (What is the topic of discussion?) and parameter (mean, standard deviation, probability) you want to draw ... – PowerPoint PPT presentation

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Title: Significance Toolbox

1
Significance Toolbox
• Identify the population of interest (What is the
topic of discussion?) and parameter (mean,
standard deviation, probability) you want to draw
conclusions about. State the null and
alternative hypotheses.
• Choose the appropriate inference procedure (type
of test) and verify conditions (what kind of
information is given about population/sample. Is
there an SRS? If not, we may be able to perform
the test because of the finite number of
observations central limit theorem but
generalizations may not be necessarily true
especially if the distribution is severely
nonnormal).
• If the conditions are met, carry out the
inference procedure (find the mean, deviation,
and P-value).
• Interpret your results (Is the information
statistically significant?).

2
One-sample z statistic
• H0 µ µ0
• The test uses
• Ha µ gt µ0 is P(Z z)
• Ha µ lt µ0 is P(Z z)
• Ha µ ? µ0 is 2P(Z z)

View graphs on page 573.
3
Fixed Significant Level for Z tests for
Population Mean
• The outcome of a test is significant at level
alpha if P-value .
• Once we have computed the z test statistic,
reject H0 at significant level against a one
sided alternative when Ha µgtµ0 if z z and
Ha µ lt µ0 if z - z
• Reject H0 at significant level alpha against a
two-sided alternative Ha µ?µ0 if z z

4
10.3 Making Sense of Statistical Significance
5
Choosing a level, ?
• Standard the level of significance gives a
clear statement of the degree of evidence
provided by the sample against the null
hypothesis.
• Best practice Decide on a significance level
prior to testing. If the result satisfies the
level, reject the null. If the result fails the
level, find the null acceptable (fail to reject).

6
• If we have a fixed significance level, we
should ask how much evidence is required to
reject H0.
• If H0 represents an assumption people have
believed for years, strong evidence (small )
is needed.
• If rejecting H0 for Ha means making expensive
changeover (products), strong evidence must show
sales will soar.

7
Significant vs Insignificant
• There is no sharp border between significant and
insignificant only increasingly strong evidence
as the P-value decreases.
• When a null hypothesis can be rejected (5 or 1
level), there is good evidence that an effect is
present.
• To keep statistical significance in its place,
pay close attention to the actual data and the
P-value.

8
Statistical inference is not always valid
• Surveys and experiments that are designed badly
will produce invalid results.
• Outliers in the data and testing a hypothesis on
the same data that suggested the hypothesis
invalidates the test.
• Since tests of significance and confidence
intervals are based on the laws of probability,
randomization in sampling or experimentation
ensures these laws apply.

9
Assignment
• Exercises 10.44, 10.57, 10.58, 10.62, 10.64

10
10.4 Inference as Decision
• Reminders
• Tests of significances assess the strength of
evidence ______ (for/against) the null
hypothesis.
• Measurement P-value which is the probability
computed under the assumption that null
hypothesis is ______ (true/false).
• The alternative hypothesis helps us to see what
outcomes count ______ (for/against) the null
hypothesis.

11
Strength ? Decision
• A significance level chosen in advance points to
the outcome of the test as a decision.
• If the result is significant, we reject the null
hypothesis in favor of the alternate.
• If the result is not significant, we fail to
reject the null (null hypothesis is acceptable).
• Making the decision to either fail to reject
(acceptable) or reject results should be left to
the user but at times the final decision is
stated during the interpretation.

12
Acceptance Sampling
• A decision or action must be made as an end
result of inference.
• Failing to reject (Acceptable) or rejecting the
end product.

13
Type I and II Errors
• In tests of significance
• H0 - the null hypothesis
• Ha - the alternative hypothesis
• However, when dealing with Type I and Type II
errors, these hypotheses will represent accepting
one decision and rejecting the other.
• Now
• H0 should be considered the initial hypothesis
• Ha the secondary hypothesis

14
Type I Error
• We have been calculating this type of error all
along.
• If we reject H0 (acceptable Ha) when in fact H0
is true.

15
Type II Error
• If we find that the H0 is acceptable (reject Ha)
when in fact Ha is true.

16
Quick Comparison
H0 True Ha True
Reject H0 Type 1 Error Correct Decision
Fail to reject H0 (acceptable) Correct Decision Type 2 Error
Decision based on sample
17
Cancer Scenario
Ho We suspect that you have cancer Ho is True Ha is True
Reject the null You dont have cancer! Diagnosis Cancer Type I Error Diagnosis No Cancer Correct Decision
Fail to reject the null You have cancer! Diagnosis Cancer Correct Decision Diagnosis No Cancer Type II Error
18
Significance and Type I Error
• The significance level of any fixed level test is
the probability of a Type I error. is the
probability that the test will reject the null
hypothesis H0 when H0 is in fact true.
• Example 10.68, Page 598

19
Example 10.68
• You have an SRS of size n 9 from a normal
distribution with ? 1. You wish to test
H0 µ 0
• Ha µ gt 0.
• You decide to reject H0 if x gt 0 and to accept H0
otherwise.

20
Power
• A significance test measures the ability to
detect an alternative hypothesis.
• The power against a specific alternative is the
probability that the test will reject H0 when the
alternative is true.
• Calculate the power of a specific alternative
subtract the probability of the Type II error for
the alternative from 1.
• Class example 10.68. Accept that the mean, H0,
will be less than or equal to 0 18.4 of the
time however, the mean should be greater than 0
81.6 of the time (100 - 18.4).

21
Power continued
• Power works best for fixed significance levels.
• Larger sample sizes will increase the power for a
fixed significant level.

22
Increase the Power
• If the strength of evidence required for
rejection is too low, increase the significance
level.
• Consider an alternative farther away from µ0.
• Increase the sample size.
• Decrease the standard deviation, ?.

23
Assignment (Work due on Monday, 3/28)
• Exercises 10.67, 10.69, 10.71 and 10.81

24
Scenarios
• OJ Simpson Guilty man goes free.
• Ho OJ is innocent
• Ha OJ is guilty
• Found not guilty Type 2
• Guilty man goes free.
• Movie A Time to Kill
• Ho Father is innocent
• Ha Father is guilty
• Found to be innocent Type 2
• To Kill a Mockingbird Send an innocent man to
jail
• Ho Tom Robinson is innocent
• Ha Tom Robinson is guilty
• Found guilty Type 1
• The Green Mile