OneSample Hypothesis Tests

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Chapter 9

• One-Sample Hypothesis Tests

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Hypothesis Testing

• A hypothesis is a statement about a population
  parameter
• We use sample data (and probabilities regarding
  that sample data) to test the validity of the
  hypothesis or claim
• Hypothesis Testing uses sample data to test a
  claim

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GENERIC PROCEDURE for Hypothesis Testing

• 3 Basic Steps
• Define the hypothesis
• Calculate the test statistics
• State the conclusions

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GENERIC PROCEDURE for Hypothesis Testing

• 3 Basic Steps
• Define the hypothesis
• What are you trying to prove?
• You need to provide clear hypothesis statements
• Calculate the test statistic
• State the conclusions

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Define the hypothesis

• formulate the hypotheses to be tested
• Null hypothesis
• Alternative hypothesis

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Define the hypothesis

• formulate the hypotheses to be tested
• (null alternative)
• Null hypothesis, H0 states the obvious
• We are testing the null hypothesis
• The null must bear a reference to an exact value
  in order to be tested (the null hypothesis must
  bear an equality sign)
• The null defines the acceptance region
• Alternative hypothesis, H1

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Define the hypothesis

• formulate the hypotheses to be tested
• (null alternative)
• Null hypothesis, H0 states the obvious
• Alternative hypothesis, H1 bears the burden of
  proof
• We are seeking to prove the alternative
  hypothesis
• We have done so ONLY IF the data provides enough
  evidence
• includes the remainder of the population, that
  not covered by the null
• The alternative defines the rejection region

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Define the hypothesis

• Null hypothesis, H0 states the obvious
• Alternative hypothesis, H1 bears the burden of
  proof
• One-tailed vs. two-tailed hypothesis
• The type of test depends on what you want to
  prove
• Hypotheses are stated in terms of population
  values (we are trying to test a population)

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Define the hypothesis

• Null hypothesis, H0 states the obvious
• Alternative hypothesis, H1 bears the burden of
  proof
• One-tailed vs. two-tailed hypothesis
• The type of test depends on what you want to
  prove
• One-tailed rejection region in one direction of
  the null
• Null ? ? a or
• Null ? ? a

Sample hypotheses???
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Define the hypothesis

• Null hypothesis, H0 states the obvious
• Alternative hypothesis, H1 bears the burden of
  proof
• One-tailed vs. two-tailed hypothesis
• The type of test depends on what you want to
  prove
• One-tailed rejection region in one direction of
  the null
• Two-tailed rejection region in both directions
  of the null
• Null ? a

Sample hypotheses???
EXAMPLE 9.16, 9.17, 9.18, 9.19 (state the
hypotheses only, one-tailed vs. two-tailed?)
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GENERIC PROCEDURE for Hypothesis Testing

• 3 Basic Steps
• Define the hypothesis
• Calculate the test statistic
• State the conclusions

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GENERIC PROCEDURE for Hypothesis Testing

• 3 Basic Steps
• Define the hypothesis
• Calculate the test statistic
• test statistics are based on the sample data
  (observed values)
• critical measures are based on a significance
  level
• State the conclusions

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Calculate the test statistic

• Observed and Critical values
• Observed values are based on the data
• observed values are z-scores or t-scores for the
  sample
• Significance of data

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Calculate the test statistic

• Observed and Critical values
• Observed values are based on the data (zOBS,
  tOBS)
• Significance of data
• Set up a p-value to determine the significance of
  the data
• p-value prob. of seeing another sample more
  rare than the observed one,
• (probability of making a mistake in our
  conclusion)
• we calculate the p-values based on the observed
  values

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Calculate the test statistic

• Observed and Critical values
• Significance level
• Critical Measure

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Calculate the test statistic

• Observed and Critical values
• Significance level (maximum allowable error)
• Set up a significance level to control the
  possible frequency errors
• a-level prob. of incorrectly rejecting the null
  hypothesis (typically 0.05)
• Critical Measure

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Calculate the test statistic

• Observed and Critical values
• Significance level (How accurate do you want to
  be?)
• Set up a significance level (a-level) to control
  the possible frequency of incorrectly rejecting
  the null hypothesis
• Significance of data

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Calculate the test statistic

• Observed and Critical values
• Statistical Errors (Type I and Type II Errors)
• Significance level (How accurate do you want to
  be?)
• Set up a significance level (a-level) to control
  the possible frequency of Type I errors
• Use this a-level to determine a critical value
• Significance of data

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Calculate the test statistic

• Observed and Critical values
• Observed values are zOBS or tOBS
• Critical values are zCRIT or tCRIT
• Significance level, a-level
• Significance of data, p-value

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GENERIC PROCEDURE for Hypothesis Testing

• 3 Basic Steps
• Define the hypothesis
• Calculate the test statistic
• State the conclusions
• compare the observed values (test statistics) to
  a critical value (based on an error level)
• compare the probability of the observed values to
  the critical significance level (?-level)

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State the conclusions

• We state our conclusions based on rejecting or
  not rejecting the null

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State the conclusions

• We state our conclusions based on rejecting or
  not rejecting the null
• DECISION RULES
• Compare observed values to critical values
• reject the null if obs gt crit
• Compare p-values to a-level
• reject the null if p-value lt ?-level

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GENERIC PROCEDURE for Hypothesis Testing

• 3 Basic Steps
• Define the hypothesis
• Calculate the test statistic
• State the conclusions

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Hypothesis Tests for the Population Mean (s known
or n gt 120)

• One-sample hypothesis tests compare the mean from
  one population and to some standard

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Hypothesis Tests for the Population Mean (s known
or n gt 120)

• One-sample hypothesis tests compare the mean from
  one population and to some standard
• Define the hypothesis
• formally state the null alternative hypotheses
• Null ? a
• Alternative ? ? a
• Calculate the test statistics
• State the conclusions

(two-tailed )
or one-tailed?
? ? a or ? ? a
? lt a or ? gt a
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Hypothesis Tests for the Population Mean (s known
or n gt 120)
What does this mean for the sampling distribution
of the sample mean? NORMAL

• One-sample hypothesis tests compare the mean from
  one population and to some standard
• Define the null alt hypothesis (one-tailed or
  two-tailed?)
• Calculate the test statistics
• zOBS STANDARDIZE( , ?, ?/?n)
• State the conclusions

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Hypothesis Tests for the Population Mean (s known
or n gt 120)

• One-sample hypothesis tests compare the mean from
  one population and to some standard
• Define the null alt hypothesis (one-tailed or
  two-tailed?)
• Calculate the test statistics
• zOBS STANDARDIZE( , ?, ?/?n)
• zCRIT NORMSINV(1-?/ of tails)
• State the conclusions

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Hypothesis Tests for the Population Mean (s known
or n gt 120)

• One-sample hypothesis tests compare the mean from
  one population and to some standard
• Define the null alt hypothesis (one-tailed or
  two-tailed?)
• Calculate the test statistics
• zOBS STANDARDIZE( , ?, ?/?n)
• zCRIT NORMSINV(1-?/ of tails)
• p-value ( of tails) (1- P(zgtzOBS))
• ( of tails) (1-NORMSDIST(zOBS))
• State the conclusions

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Hypothesis Tests for the Population Mean (s known
or n gt 120)

• One-sample hypothesis tests compare the mean from
  one population and to some standard
• Define the null alt hypothesis (one-tailed or
  two-tailed?)
• Calculate the test statistics (zOBS, zCRIT,
  p-value)
• State the conclusions
• zOBS gt zCRIT?
• p-value lt a-level?
• If (both) true then reject the null (otherwise,
  cannot reject the null)

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Hypothesis Tests for the Population Mean (s known
or n gt 120)

• Define the null alt hypothesis (one-tailed or
  two-tailed?)
• Calculate the test statistics (zOBS, zCRIT,
  p-value)
• State the conclusions
• zOBS gt zCRIT? p-value lt a-level?
• If (both) true then reject the null (otherwise,
  cannot reject the null)
• EXAMPLES 9.29, 9.45, 9.30, 9.47

zOBS STANDARDIZE(x, ?, ?/?n)
zCRIT NORMSINV(1-?/ of tails)
p-value ( of tails) (1-NORMSDIST(zOBS))
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Hypothesis Tests for the Population Mean (s
unknown or n lt 120)

• One-sample hypothesis tests compare the mean from
  one population and to some standard

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Hypothesis Tests for the Population Mean (s
unknown or n lt 120)

• One-sample hypothesis tests compare the mean from
  one population and to some standard
• Define the hypothesis (one-tailed or two-tailed?)
• formally state the null alternative hypotheses
• Null ? a, ? ? a or ? ? a
• Alternative ? ? a, ? lt a or ? gt a
• Calculate the test statistics
• State the conclusions

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Hypothesis Tests for the Population Mean (s known
or n lt 120)
What does this mean for the sampling distribution
of the sample mean? t-distribution

• One-sample hypothesis tests compare the mean from
  one population and to some standard
• Define the null alt hypothesis (one-tailed or
  two-tailed?)
• Calculate the test statistics
• tOBS STANDARDIZE(x, ?, s/?n)
• State the conclusions

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Hypothesis Tests for the Population Mean (s
unknown or n lt 120)

• One-sample hypothesis tests compare the mean from
  one population and to some standard
• Define the null alt hypothesis (one-tailed or
  two-tailed?)
• Calculate the test statistics
• tOBS STANDARDIZE(x, ?, s/?n)
• tCRIT TINV(2a/ of tails,df)
• State the conclusions

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Hypothesis Tests for the Population Mean (s
unknown or n lt 120)

• One-sample hypothesis tests compare the mean from
  one population and to some standard
• Define the null alt hypothesis (one-tailed or
  two-tailed?)
• Calculate the test statistics
• tOBS STANDARDIZE(x, ?, s/?n)
• tCRIT TINV(2a/ of tails,df)
• p-value TDIST(tOBS, df, of tails)
• State the conclusions

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Hypothesis Tests for the Population Mean (s
unknown or n lt 120)

• One-sample hypothesis tests compare the mean from
  one population and to some standard
• Define the null alt hypothesis (one-tailed or
  two-tailed?)
• Calculate the test statistics (tOBS, tCRIT,
  p-value)
• State the conclusions
• tOBS gt tCRIT?
• p-value lt a-level?
• If (both) true then reject the null (otherwise,
  cannot reject the null)

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Hypothesis Tests for the Population Mean (s
unknown or n lt 120)

• Define the null alt hypothesis (one-tailed or
  two-tailed?)
• Calculate the test statistics (tOBS, tCRIT,
  p-value)
• State the conclusions
• tOBS gt tCRIT? p-value lt a-level?
• If (both) true then reject the null (otherwise,
  cannot reject the null)
• EXAMPLES 9.58, 9.60

tOBS STANDARDIZE(x, ?, s/?n)
tCRIT TINV(2?/ of tails)
p-value TDIST(tOBS, df, of tails)
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Hypothesis Tests for the Population Proportion

• One-sample hypothesis tests compare the
  proportion from one population and to some
  standard

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Hypothesis Tests for the Population Proportion

• One-sample hypothesis tests compare the
  proportion from one population and to some
  standard
• Define the hypothesis (one-tailed or two-tailed?)
• formally state the null alternative hypotheses
• Null ? a, ? ? a or ? ? a
• Alternative ? ? a, ? lt a or ? gt a
• Calculate the test statistics
• State the conclusions

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Hypothesis Tests for the Population Proportion

• Test the proportion from one population and
  compare it to some standard
• Define the null alt hypothesis (one-tailed or
  two-tailed?)
• Calculate the test statistics
• zOBS
• STANDARDIZE(p, p, )
• State the conclusions

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Hypothesis Tests for the Population Proportion

• Test the proportion from one population and
  compare it to some standard
• Define the null alt hypothesis (one-tailed or
  two-tailed?)
• Calculate the test statistics
• zOBS STANDARDIZE(p, p,
  )
• zCRIT NORMSINV(1-?/ of tails)
• State the conclusions

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Hypothesis Tests for the Population Proportion

• Test the proportion from one population and
  compare it to some standard
• Define the null alt hypothesis (one-tailed or
  two-tailed?)
• Calculate the test statistics
• zOBS STANDARDIZE(p, p,
  )
• zCRIT NORMSINV(1-?/ of tails)
• p-value of tails(1-NORMSDIST(zOBS))
• State the conclusions

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Hypothesis Tests for the Population Proportion

• Test the proportion from one population and
  compare it to some standard
• Define the null alt hypothesis (one-tailed or
  two-tailed?)
• Calculate the test statistics (zOBS, zCRIT,
  p-value)
• State the conclusions
• zOBS gt zCRIT?
• p-value lt a-level?
• If (both) true then reject the null (otherwise,
  cannot reject the null)

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Hypothesis Tests for the Population Proportion

• Define the null alt hypothesis (one-tailed or
  two-tailed?)
• Calculate the test statistics (zOBS, zCRIT,
  p-value)
• State the conclusions
• zOBS gt zCRIT? p-value lt a-level?
• If (both) true then reject the null (otherwise,
  cannot reject the null)
• EXAMPLES 9.70, 9.71

zOBS STANDARDIZE(p, p, )
zCRIT NORMSINV(1-?/ of tails)
p-value ( of tails) (1-NORMSDIST(zOBS))
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