Chapter 20 Testing Hypothesis about proportions

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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?

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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.

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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?

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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

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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

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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

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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

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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

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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.

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Exercises

• Page 467
• 1
• 3
• 20

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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

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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)

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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.

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Alpha Levels (cont.)

• ? 0.10
• ? 0.05
• ? 0.01
• Rejection Region
• One Sided Two sided

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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.

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Types of errors

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

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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

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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.

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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

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Comparing Two Proportions (cont.)

• The Standard Deviation of the Difference Between
  Two Proportions
• For proportions from the data

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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

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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

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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.

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Exercises

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

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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?

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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

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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

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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

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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.

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Exercise

• Page 508 16

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Homework 5

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