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Chapter 20 Testing Hypothesis about proportions

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Title: Chapter 20 Testing Hypothesis about proportions


1
Chapter 20 Testing Hypothesis about proportions
  • Example
  • Metal Manufacturer
  • Ingots
  • 20 defective (cracks)
  • After Changes in the casting process
  • 400 ingots and only 17 defective
  • Is this a result of natural sampling variability
    or there is a reduction in the cracking rate?

2
Hypotheses
  • We begin by assuming that a hypothesis is true
    (as a jury trial).
  • Data consistent with the hypothesis
  • Retain Hypothesis
  • Data inconsistent with the hypothesis
  • We ask whether they are unlikely beyond
    reasonable doubt.
  • If the results seem consistent with what we would
    expect from natural sampling variability we will
    retain the hypothesis. But if the probability of
    seeing results like our data is really low, we
    reject the hypothesis.

3
Testing Hypotheses
  • Null Hypothesis H0
  • Specifies a population model parameter of
    interest and proposes a value for this parameter
  • Usually
  • No change from traditional value
  • No effect
  • No difference
  • In our example H0p0.20
  • How likely is it to get 0.17 from sample
    variation?

4
Testing Hypotheses (cont.)
  • Normal Sampling distribution
  • How likely is to observe a value at least 1.5
    standard deviations below the mean of a normal
    model
  • Management must decide whether an event that
    would happen 6.7 of the time by chance is strong
    enough evidence to conclude that the true
    cracking proportion has decreased

5
A Trial as a Hypothesis Test
  • The jurys null hypothesis is
  • H0 innocent
  • If the evidence is too unlikely given this
    assumption, the jury rejects the null hypothesis
    and finds the defendant guilty. But if there is
    insufficient evidence to convict the defendant,
    the jury does not decide that H0 is true and
    declare him innocent. Juries can only fail to
    reject the null hypothesis and declare the
    defendant not guilty

6
The Reasoning of Hypothesis Testing
  • Hypothesis
  • To perform a hypothesis test, we must specify an
    alternative hypotheses. Remember we can never
    prove a null hypothesis, only reject it or retain
    it. If we reject it, we then accept the
    alternative
  • Example Pepsi or Coke
  • p proportion preferring coke
  • H0 p 0.50
  • HA p ? 0.50

7
The Reasoning of Hypothesis Testing (cont.)
  • Plan
  • Specify the model and test you will use
    (proportions, means).
  • We call this test about the value of a proportion
    a one-proportion z-test
  • Mechanics
  • Actual Calculation of a test from the data.
  • P-value the probability that the observed
    statistic value could occur if the null model
    were correct. If the P-value is small enough, we
    reject the null hypothesis

8
The Reasoning of Hypothesis Testing (cont.)
  • Conclusion
  • The conclusion in a hypothesis test is always a
    statement about the null hypothesis. The
    conclusion must state either that we reject or
    that we fail to reject the null hypothesis

9
Alternatives
  • Two-sided Alternative
  • HA p ? 0.50 (Pepsi Coke)
  • The P-value is the probability of deviating in
    either direction from the null hypothesis
  • One-sided Alternative
  • H0 p 0
  • HA p lt 0.20 (Ingots)
  • The P-value is the probability of deviating only
    in the direction of the alternative away from the
    null hypothesis value.

10
Exercises
  • Page 467
  • 1
  • 3
  • 20

11
Chapter 21More About Tests
  • Example Therapeutic Touch (TT)
  • One-proportion z-test
  • 15 TT practitioners 10 trials each
  • H0 p0.50
  • HA pgt0.50
  • Random Sampling
  • Independence
  • 10 condition
  • Success/Failure condition
  • Observed proportion 0.467
  • Find the P-value

12
How to Think About P-values
  • A P-value is a conditional probability. It is the
    probability of the observed statistic given that
    the null hypothesis is true.
  • P-value P(Observed statistic valueH0)

13
Alpha Levels
  • When the P-value is small, it tells us that our
    data are rare given the null hypothesis.
  • We can define a rare event arbitrarily by
    setting a threshold for our P-value.
  • If our P-value falls below that point well
    reject the null hypothesis. We call such results
    statistically significant the threshold is
    called an alpha level or significance level.

14
Alpha Levels (cont.)
  • ? 0.10
  • ? 0.05
  • ? 0.01
  • Rejection Region
  • One Sided Two sided

15
Making Errors
  • Type I error
  • The null hypothesis is true, but we mistakenly
    reject it.
  • Type II error
  • The null hypothesis is false but we fail to
    reject it.

16
Types of errors
  • Examples
  • Medical disease testing
  • I False Positive
  • II False Negative
  • Jury Trial
  • I Convicting an innocent
  • II Absolving someone guilty

17
Probabilities of errors
  • To reject H0, the P-value must fail below ?. When
    H0 is true that happens exactly with probability
    ? so when you choose the level ?, you are setting
    the probability of a Type I error to ?.
  • When H0 is false and we fail to reject it, we
    have made a Type II error. We assign the letter ?
    to the probability of this mistake

18
Power
  • The power of a test is the probability that it
    correctly rejects a false null hypothesis. When
    the power is high, we can be confident that weve
    looked hard enough.
  • We know that ? is the probability that a test
    fails to reject a false null hypothesis, so the
    power of the test is the complement 1 - ?
  • When we calculate power, we have to imagine that
    the null hypothesis is false. The value of the
    power depends on how far the truth lies from the
    null hypothesis value. We call this distance
    between the null hypothesis value p0 and the
    truth p the effect size.

19
Chapter 22Comparing Two Proportions
  • Recall (Ch.16)
  • The variance of the sum or difference of two
    independent random quantities is the sum of their
    individual variances
  • Example of the cereals

20
Comparing Two Proportions (cont.)
  • The Standard Deviation of the Difference Between
    Two Proportions
  • For proportions from the data

21
Assumptions and Conditions
  • Random Sampling
  • 10 condition
  • Independent Samples Condition
  • The two groups we are comparing must also be
    independent of each other (usually evident from
    the way the data is collected).
  • Example
  • Same group of people before and after a treatment
    are not independent
  • Success and failure condition in each sample

22
The Sampling Distribution
  • The sampling distribution for a difference
    between two independent proportions
  • Provided the assumptions and conditions the
    sampling distribution of is modeled
    by a normal model with mean
  • and standard deviation

23
A two-proportion z-interval
  • When the conditions are met, we are ready to find
    the confidence interval for the difference of two
    proportions p1-p2. Using the standard error of
    the difference
  • The interval is
  • The critical value z depends on the particular
    confidence level.

24
Exercises
  • Two-proportion z-interval
  • (page 493, 496)

25
Example
  • Snoring
  • Random sample of 1010 Adults
  • From 995 respondents
  • 37 snored at least few nights a week
  • Splitting in two age categories
  • Under 30 Over 30
  • 26.1 of 184 39.2 of 811
  • Is the difference of 13.1 real or due only to
    sampling variability?

26
Example (cont. snoring)
  • H0 p1 p2 0
  • But p1 and p2 are linked from H0
  • p1 p2
  • Pooling
  • Combining the counts to get an overall proportion

27
Two-Proportion z-test
  • The conditions for the two-proportion z-test are
    the same as for the two-proportion z-interval .
    We are testing the hypothesis
  • H0 p1 p2
  • Because we hypothesize that the proportions are
    equal, we pool them to find
  • And we use the pooled value to estimate the
    standard error

28
Two-Proportion z-test (cont.)
  • Now we find the test statistic using the
    statistic
  • When the conditions are met and the null
    hypothesis is true, this statistic follows the
    standard Normal model, so we can use that model
    to obtain a P-value

29
Example (cont. snoring)
  • Randomization 10 Condition
  • Independent samples condition Success / Failure
  • The P-value is the probability of observing a
    difference greater or equal to 0.131
  • The two sided P-value is 0.0008. This is rare
    enough, so we reject the null hypothesis and
    conclude that there us a difference in the
    snoring rate between this two age groups.

30
Exercise
  • Page 508 16

31
Homework 5
  • Page 423 8, 16
  • Page 443 12, 18
  • Page 467 2, 4, 6, 12
  • Page 491 20
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