Elementary Statistics

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Elementary Statistics

• Statistical Decision Making

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Review

• Scientific Method
• Formulate a theory
• Collect data to test the theory
• Analyze the results
• Interpret the results and make a decision
• A theory is rejected if it can be shown
  statistically that the data we observed would be
  very unlikely to occur if the theory were in fact
  true. A theory is accepted if it is not rejected
  by the data.

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Null Hypotheses, Alternative Hypotheses

• Definitions
• The null hypothesis, denoted by H0, is a status
  quo or prevailing viewpoint about a population.
• The alternative hypothesis, denoted by H1, is an
  alternative to the null hypothesis -- the change
  in the population that the researcher hopes is
  true.
• Tip
• The null and alternative hypotheses should both
  be statements about the same population.

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Lets do it 1.2

• Excerpts from the article Stress can cause
  sneezes (The New York Times, January 21, 1997)
  are shown at the right. Studies suggest that
  stress doubles a persons risk of getting a cold.
  Acute stress, lasting maybe only a few minutes,
  can lead to colds. One mystery that is still
  prevalent in cold research is that while many
  individuals are infected with the cold virus,
  very few actually get the cold. On average, up
  to 90 percent of people exposed to a cold virus
  become infected, meaning the virus multiplies in
  the body, but only 40 percent actually become
  sick. One researcher thinks that the
  accumulation of stress tips the infected person
  over into illness.
• The percentage of people exposed to a cold virus
  who actually get a cold is 40. The researcher
  would like to assess if stress increases this
  percentage. So, the population of interest is
  people who are under (acute) stress. State the
  appropriate hypotheses for assessing the
  researchers theory.

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How to make decision?

• Assume the null hypothesis is true
• Assess whether or not the observed result is
  extreme or very unlikely.
• unlikely, rare --- reject the null hypothesis
• not unusual favor or accept null hypothesis

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Example 1.2

• Suppose that you are shown a closed package
  containing five balls. The package is on sale
  because the label, which describes the colors of
  the balls in the package, is missing. The
  salesperson states that most of the packages of
  balls sold in this store contain one yellow ball
  and four blue balls, for a portion of yellow
  balls of 1/5.
• You wish to test the following hypotheses about
  the contents of the package that is missing its
  label.
• H0 The proportion of yellow balls in the
  package is 1/5.
• H1 The proportion of yellow balls in the
  package is more than 1/5.

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Very unlikely or not

• Scenario 1 Suppose that the data are as follows
  The first ball is yellow, the second ball is
  yellow, the third ball is yellow, the fourth ball
  is yellow, and the fifth ball is yellow.
• Q Do you accept or reject the null hypothesis
  that the contents of the package are one yellow
  ball and four blue balls? Why?
• Scenario 2 Suppose that the data are as follows
  The first ball is blue, the second ball is blue,
  the third ball is blue, the fourth ball is blue,
  and the fifth ball is blue.
• Q Do you accept or reject the null hypothesis
  that the contents of the package are one yellow
  ball and four blue balls? Why?

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Statistically significant

• Definition The data collected are said to be
  statistically significant if they are very
  unlikely to be observed under the assumption that
  is true. If the data are statistically
  significant, then our decision would be to reject
  .

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Lets do it 1.3

• Last month, a large supermarket chain received
  many customer complaints about the quantity of
  chips in 16-ounce bags of a particular brand of
  potato chips. Wanting to assure its customers
  they were getting their money's worth, the chain
  decided to test the following hypotheses
  concerning the true average weight (in ounces) of
  a bag of such potato chips in the next shipment
  received from their supplier
• H0 Average weight is at least 16 ounces
• H1 Average weight is less than 16 ounces
• If there is evidence in favor of the alternative
  hypothesis, the shipment would be refused and a
  complaint registered with the supplier.
• Some bags of chips were selected from the next
  shipment and the weight of each selected bag was
  measured. The researcher for the supermarket
  chain stated that the data were statistically
  significant.
• What hypothesis was rejected?
• Was a complaint registered with the supplier?
• Could there have been a mistake? If so, describe
  it.

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Type of Errors

• Definition
• Rejecting the null hypothesis H0 when in fact it
  is true, is called Type I error.
• Accepting the null hypothesis H0 when in fact it
  is not true, is called a Type II error.
• Note
• a Type I error can only be made if the null
  hypothesis is true.
• a Type II error can only be made if the
  alternative hypothesis is true.

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Example 1.4

• Problem
• You plan to walk to a party this evening.
  Are you going to carry an umbrella with you? You
  dont want to get wet if it should rain. So you
  wish to test the following hypotheses
• H0 Tonight it is going to rain.
• H1 Tonight it is not going to rain.
• (a) Describe the two types of error that you
  could make when deciding between these two
  hypotheses.
• (b) What are the consequences of making each type
  of error?
• (c) You learn from the noon weather report that
  there is a 70 chance of rain tonight, what
  should you understand from this information?

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Understand Errors
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Lets do it! 1.5 Which Error is Worse?

• (a)
• H0 The water is contaminated.
• H1 The water is not contaminated.
• A ____________ error would be more serious
  because
• (c)
• H0 The ship is unsinkable.
• H1 The ship is sinkable.
• A ____________ error would be more serious
  because

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Lets Do It! 1.6 -- Testing a New Drug

• H0 The new drug is as effective as the standard
  drug.
• H1 The new drug is more effective than the
  standard drug.
• What are the two types of errors that you could
  make when deciding between these two hypotheses?
• Type I error
• Type II error
• What are the consequences of a Type I error?
• What are the consequences of a Type II error?
• Which error might be considered more severe from
  an ethical
• point of view?

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A little review before we go on

• Population Sample Statistical inference
• null hypothesis H0
• alternative hypothesis H1
• statistically significant

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?? Chance of Error

• Think about it
• If the Type I error is considered very serious,
  why not set the chance of making a Type I error
  to zero?

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Significance level

• The significance level number ? is the chance of
  committing a Type I error, that is, the chance of
  rejecting the null hypothesis when it is in fact
  true.
• In general, the level of ? is fixed in advance
  because it depends on the consequences of
  committing the type I error.

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Power of Test

• The power of the test is defined as 1 - ?. The
  power of the test is the chance of rejecting the
  null hypothesis when the alternative hypothesis
  is true.
• How ? is related to ? ?
• What is a better test?

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Level of significance
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How to make decision?

• Assume the null hypothesis is true
• Assess whether or not the observed result is
  extreme or very unlikely.
• ------ What is extreme?
• unlikely, rare --- reject the null hypothesis
• not unusual favor or accept null hypothesis
• Note we want chance of making an error to be
  small !

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How to set up our decision rule?
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Frequency Plot

• Bag A has a total value - 560, while Bag B
  has a total value 1,890.

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Hypotheses

• Collect data --- Observation
• Definition The number n of observations in a
  sample is called the sample size.
• N1

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How will you decide?

• Think about it
• What if the voucher you select is 60?
• Would this observation lead you to think the
  shown bag is Bag A or Bag B?
• Why?
• How would you answer these questions if the
  voucher you select is 10?

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Decision Rule
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Calculate the chance

• Chance if Chance if
• Face Value Bag A Bag B
• -1,000 1/20 0
• 10 7/20 1/20
• 20 6/20 1/20
• 30 2/20 2/20
• 40 2/20 2/20
• 50 1/20 6/20
• 60 1/20 7/20
• 1,000 0 1/20

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What is rare (unlikely) scenario?
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Decision Rule 1
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Rejection Region
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Chance of errors?
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Chance of error ?
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What can we do? --- Change rule
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Chance of error with rule 2
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Need better result
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Summary from this example
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How to make decision? Classic Approach

• State Null and Alternate Hypothesis
• Write your decision rule
• Calculate ? , ?-- check your decision rule
• Look at your observed value
• Apply your decision rule to your observed value
  and make a decision on which hypothesis you want
  to support

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More on the Direction of Extreme
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More Example--- One-sided Rejection Region to the
Left
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Decision and chance of error
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Summary

• Type of Error
• Significance Level
• How to make decision?
• Direction of extreme
• Rejection Region
• Relationship between the decision rule and
  significance level
• Please work on practice questions of section 1.3