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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
Truth about the Population
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
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