Todays Goals

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Todays Goals

• Calculate a confidence interval around a point
  estimate
• Perform a hypothesis test
• No more HW this semester
• Article is due Today
• Example in Handout Section of web page
• Mini Project due May 11

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Confidence Interval

• The 99 confidence interval for the mean is
  (79.3, 80.7)
• What is the probability that the true mean is
  between 79.3 and 80.7?
• 1
• 99
• Not enough information
• The true mean is a number, not a random variable

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Confidence Intervals Different cases
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Confidence Intervals Normal, var known

• A 100(1-a) confidence interval for the mean m of
  a normal population when the value of s is known
  is
• Example a 95 confidence interval is

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Confidence interval for normal population when
the variance is unknown

• Let xbar and s be the sample mean and sample
  standard deviation from a random sample of size n
  from a normal population. then the 100(1-a)
  confidence interval is

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Large sample interval, not necessarily normal

• If n is sufficiently large (the CLT applies)
  then
• is a large-sample confidence interval for m with
  confidence level approximately 100(1-a)
• You need at least about 30 observations.

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Large sample interval, not necessarily normal,
unknown variance

• If n is sufficiently large (the CLT applies)
  then
• is a large-sample confidence interval for m with
  confidence level approximately 100(1-a)
• Note that it uses the sample standard deviation.
• You need at least 40 observations.

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Summary
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Hypothesis Testing

• A hypothesis is a claim about a parameter or
  parameters of a probability distribution.
• Examples
• mean 0.75
• p lt .1 (where p is the proportion of defective
  circuit boards)
• mu1 mu2 gt 0, where mu is the breaking strength
  of string (string type one is stronger than type
  two)

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Hypothesis Testing

• We always test two contradictory hypotheses
• H0 mean 0.75
• H1 mean gt 0.75
• H0 p lt .1
• H1 p gt .1
• H0 mu1 mu2 0
• H1 mu1 mu2 gt 0

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Hypothesis Testing

• One of the initial claims is favored over the
  other. This is the null hypothesis.
• The null hypothesis will be rejected only if the
  data is strongly in favor of the alternative
  hypothesis.

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Hypothesis Testing

• One of the initial claims is favored over the
  other. This is the null hypothesis.
• The null hypothesis will be rejected only if the
  data is strongly in favor of the alternative
  hypothesis.
• The result of a hypothesis test is either
• The null hypothesis is rejected
• The null hypothesis fails to be rejected

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Hypothesis Testing

• The result of a hypothesis test is either
• The null hypothesis is rejected
• This is a strong result. It indicates that your
  alternative hypothesis has convincing data behind
  it.
• The null hypothesis fails to be rejected
• This is a weak result.
• It DOES NOT imply that the null hypothesis is
  true.
• Only that there is not a convincing amount of
  data to support the alternative
• If you want to prove something, it should be your
  alternative hypothesis, not your null hypothesis.

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Hypothesis Testing

• Typical claims
• A new method is better than the old method
• The items meet (or do not meet) the
  specifications
• There is a difference in the average quality
  characteristic in the output of two processes
• The counterclaim is stated as the null hypothesis
  H0
• Supposed to be true unless proven otherwise
• The claim is the alternative hypothesis Ha
• The hypothesis test assesses how probable the
  observable differences are assuming H0

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Hypothesis Testing

• A new method is better than the old method
• H0 mn mo
• H1 mn gt mo
• The items meet the specifications
• H0 rangeL gt m or mgt rangeH
• H1 rangeL lt m lt rangeH
• There is a difference in the average quality
  characteristic in the output of two processes
• H0 m1 m2 0
• H1m1 m2 ? 0

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• I claim that the global mean temperature (mu) has
  increased by more than 1 over the last century.
  Which set of hypotheses is best to test this
  claim?

H0 mu gt 1 H1 mu lt 1
A
H0 mu gt 1 H1 mu ? 1
C
H0 mu 1 H1 mu gt 1
H0 mu 1 H1 mu ? 1
D
B
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Test procedure

• A test procedure is specified by
• A test statistic, a function of the sample data
• a rejection region, the set of all test statistic
  values for which the null hypothesis H0 will be
  rejected.

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Test procedure

• A test procedure is specified by
• A test statistic, a function of the sample data
• a rejection region, the set of all test statistic
  values for which H0 will be rejected.
• Example test whether p.1 versus plt.1
• The test statistic x is the number of defective
  boards in a random sample of 200 boards
• The rejection region might be for all xlt15

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Test procedure

• A test procedure is specified by
• A test statistic, a function of the sample data
• a rejection region, the set of all test statistic
  values for which H0 will be rejected.
• Example test whether p.1 versus plt.1
• The test statistic x is the number of defective
  boards in a random sample of 200 boards
• The rejection region might be for all xlt15
• How do we set the rejection region?

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Errors

• Regardless of the rejection region there will
  always be some probability of errors.
• Type I error rejecting the null hypothesis when
  it is in fact true.
• p 0.1, but getting a sample that has only x
  12 defects
• Type II error failing to reject the null
  hypothesis when it is false.
• p .08, but getting a sample that has 16 defects

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Examples of Type I and Type II Errors

• In the prosecution of an accused person,
• H0 the person is innocent
• Ha the person is guilty.
• Type I error is the error of convicting an
  innocent person
• Type II error is the error of not convicting a
  guilty person.
• In diagnostic testing for a rare disease,
• H0 the tested person is disease-free
• Ha the person is diseased.
• Type I error is that the test gives a false
  positive result
• Type II error is that the test gives a false
  negative result.
• In a control chart used to detect deviations of
  the process mean from the target,
• H0 the process mean is on target
• Ha the process mean is off target.
• Type I error is a false alarm
• Type II error is a missed alarm.

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• Null hypothesis anthropomorphic climate change
  is not dangerous
• Alt Climate change is dangerous.
• Result climate change is in fact dangerous, but
  the data available was not strong enough to prove
  it.
• This is a Type I error
• This is a Type II error

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Hypothesis testing

• a pr(Type I error) pr( rejecting the null
  when it is in fact true)
• by setting a value for a, we set the appropriate
  rejection region.
• Usually, we want to minimize the Type I error,
  since we are trying to prove the alternative
  hypothesis.
• So, a is usually chosen to be .05 or .01

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Hypothesis Testing normal distribution with
known variance

• H0 mean m0 (H1 mean gt m0)
• If this were true then the sample mean from a
  sample of size n would satisfy
• We want pr(rejecting the null true) a,
  say .05

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Hypothesis Testing normal distribution with
known variance

• H0 mean m0
• If this were true then the sample mean from a
  sample of size n would satisfy
• We pr(rejecting the null true) a say .05
• Say we set the rejection region to be all Z gt zR
• We want pr(Z gt z.R) .05
• zR z.05 1.645

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Hypothesis Testing normal distribution with
known variance

• H0 mean m0
• Our test statistic is

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Example

• The mean length of a part is expected to be 30mm.
  We are interested in determining whether for the
  month of March, the mean length differs from 30
  mm.
• Null Hypothesis ? Ho m 30mm
• Alternative Hypothesis ? Ha m ? 30mm
• This is a two-tailed test
• Designed to detect departures of a parameter from
  a specified value in both directions
• Assume that the population standard deviation
  s2mm
• A sample of size 36 finds the sample mean length
  to be 29 mm.
• Is this difference statistically significant?

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Example

• m0 30mm
• n 36
• s2mm
• xbar 29
• Calculate
• The rejection region for 1 is -z.005 -2.575
• Since z -3 lt -2.575 the null hypothesis is
  rejected at the 1 level. It is very unlikely
  that the mean is actually 30 mm.