HYPOTHESIS TESTING WITH ONE SAMPLE

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HYPOTHESIS TESTING WITH ONE SAMPLE
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Overview
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Applications of Inferential Statistics

• Estimate the value of a population parameter.
• Test some claim (or hypothesis) about a
  population.

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Definitions

• In statistics, a hypothesis is a claim or
  statement about a property of a population.
• A hypothesis test (or test of significance) is a
  standard procedure for testing a claim about a
  property of a population.

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Rare Event Rule for Inferential Statistics

• If, under a given assumption, the probability of
  a particular event observed is exceptionally
  small, we conclude that the assumption is
  probably not correct.

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Hypothesis Testing
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Basics of Hypothesis Testing
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Hypothesis Testing

• State the claim (in words).
• State the null and alternative hypotheses.
• Obtain the test statistic and p-value.
• Determine One-tail/Two-tail test, obtain critical
  value(s).
• Reject/Fail to reject H0.
• State conclusion (in words).

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Definitions

• The null hypothesis (denoted by H0) is a
  statement that the value of a population
  parameter (such as proportion, mean, or standard
  deviation) is EQUAL to some claimed value.
• The alternative hypothesis (denoted by H1 or Ha)
  is the statement that the parameter has a value
  that somehow DIFFERS from the null hypothesis.

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Stating the Hypotheses

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Definition

• The test statistic is a value computed from the
  sample data, and it is used in making the
  decision about the rejection of the null
  hypothesis.

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Test Statistics

• Test statistic for proportion
• Test statistic for mean or
  
• Test statistic for standard deviation

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Definitions

• The critical region (or rejection region) is the
  set of all values of the test statistic that
  cause us to reject the null hypothesis.
• The significance level (denoted by ) is the
  probability that the test statistic will fall in
  the critical region when the null hypothesis is
  actually true. Common choices for are 0.05,
  0.01, and 0.10.
• A critical value is any value that separates the
  critical region (where we reject the null
  hypothesis) from the values of the test statistic
  that do not lead to rejection of the null
  hypothesis.

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Critical Values

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Definitions

• The tails in a distribution are the extreme
  regions bounded by the critical values.
• Two-tailed test The critical region is in the
  two extreme regions (tails) under the curve.
• Left-tailed test The critical region is in the
  extreme left region (tail) under the curve.
• Right-tailed test The critical region is in the
  extreme right region (tail) under the curve.

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One-tail vs. Two-tail Test

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Definition

• The P-value (or p-value or probability value) is
  the probability of getting a value of the test
  statistic that is at least as extreme as the one
  representing the sample data, assuming the null
  hypothesis is true.

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P-values

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Decision Criterion

• Traditional Method Reject H0 if the test
  statistic falls within the critical region. Fail
  to reject H0 if the test statistic does not fall
  within the critical region.
• P-value Method Reject H0 if P-value
  (where is the significance level). Fail to
  reject H0 if P-value .

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Wording Conclusions

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Hypothesis Testing

• State the claim (in words).
• State the null and alternative hypotheses.
• Obtain the test statistic and p-value.
• Determine One-tail/Two-tail test, obtain critical
  value(s).
• Reject/Fail to reject H0.
• State conclusion (in words).

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Type I Type II Error

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Definitions

• Type I Error The mistake of rejecting the null
  hypothesis when it is actually true. The symbol
  (alpha) is used to represent the probability of
  a Type I error.
• Type I Error The mistake of failing to reject
  the null hypothesis when it is actually false.
  The symbol (beta) is used to represent the
  probability of a type II error.

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Notation

• (alpha) probability of a type I error (the
  probability of rejecting the null hypothesis when
  it is true)
• (beta) probability of a type II error
  (failing to reject a false null hypothesis)

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Definition

• The power of a hypothesis test is the
  probability of rejecting a false null
  hypothesis, which is computed by using a
  particular significance level , a particular
  sample size n, a particular assumed value of the
  population parameter (used in the null
  hypothesis), and a particular value of the
  population parameter that is an alternative to
  the value assumed in the null hypothesis.

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Testing a Claim about a Proportion
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Hypothesis Testing
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Testing Claims About a Population Proportion P

• Requirements
• The sample observations are a simple random
  sample.
• The conditions for a binomial distribution are
  satisfied.
• The conditions and are both
  satisfied, so the binomial distribution of sample
  proportions can be approximated by a normal
  distribution with and
  .

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Testing Claims About a Population Proportion P
(continued)

• Notation
• n sample size or number of trials
• (sample proportion)
• P population proportion (used in the null
  hypothesis)
• q 1 p
• Test Statistic for Testing a Claim About a
  Proportion

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Testing Claims About a Population Proportion P
(continued)

• P-values Use the standard normal distribution.
• Critical values Use the standard normal
  distribution.

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Testing Claims About a Population Proportion P
Using the Calculator

• 1-PropZTestp0 is the assumed population
  proportion (from the null hypothesis)x is the
  number of successesn is the number of trials
  (sample size)

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Hypothesis Testing

• State the claim (in words).
• State the null and alternative hypotheses.
• Obtain the test statistic and p-value.
• Determine One-tail/Two-tail test, obtain critical
  value(s).
• Reject/Fail to reject H0.
• State conclusion (in words).

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Example

• In Mendels work on the transmission of visible
  traits in pea plants, one experiment he performed
  yielded 8023 seeds, 6022 of which were yellow and
  2001 of which were green. Using a 0.05 level of
  significance, test the claim that the proportion
  of seeds that are yellow is 0.75.

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Testing a Claim About a MeanKnown
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Hypothesis Testing
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Testing Claims About a Population Mean (with
Known)

• Requirements
• The sample is a simple random sample.
• The value of the population standard deviation
  is known.
• Either or both of these conditions is satisfied
• The population is normally distributed, or
• Test Statistic for Testing a Claim About a Mean
  (with Known)

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Testing Claims About a Population Mean (with
Known)

• P-values Use the standard normal distribution.
• Critical values Use the standard normal
  distribution.

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Testing Claims About a Population a Population
Mean (with Known) Using the Calculator

• Z-Test (using summary statistics) is the
  assumed population mean (from the null
  hypothesis) is the given population standard
  deviation is the sample meann is the sample
  size

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Testing Claims About a Population a Population
Mean (with Known) Using the Calculator

• Z-Test (using the data) is the assumed
  population mean (from the null hypothesis) is
  the given population standard deviationList is
  the list name containing the dataFreq is the
  frequency of the data elements

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Hypothesis Testing

• State the claim (in words).
• State the null and alternative hypotheses.
• Obtain the test statistic and p-value.
• Determine One-tail/Two-tail test, obtain critical
  value(s).
• Reject/Fail to reject H0.
• State conclusion (in words).

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Example

• The health of the bear population in Yellowstone
  National Park is monitored by periodic
  measurements taken from anesthetized bears. A
  sample of 54 bears has a mean weight of 182.9 lb.
  Assuming that the population standard deviation
  is known to be 121.8 lb, use a 0.10 significance
  level to test the claim that the population mean
  of all such bear weights is less than 200 lb.

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Testing a Claim About a Mean Not Known
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Hypothesis Testing
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Testing Claims About a Population Mean (with
Not Known)

• Requirements
• The sample is a simple random sample.
• The value of the population standard deviation
  is not known.
• Either or both of these conditions is satisfied
• The population is normally distributed, or
• Test Statistic for Testing a Claim About a Mean
  (with Not Known)

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Testing Claims About a Population Mean (with
Known) (continued)

• P-values and critical values Use Table A-3 and
  df n 1 for the number of degrees of freedom.

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Testing Claims About a Population a Population
Mean (with Unknown) Using the Calculator

• T-Test (using summary statistics) is the
  assumed population mean (from the null
  hypothesis) is the sample meanSx is the
  sample standard deviation n is the sample size

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Testing Claims About a Population a Population
Mean (with Unknown) Using the Calculator

• T-Test (using the data) is the assumed
  population mean (from the null hypothesis)List
  is the list name containing the dataFreq is the
  frequency of the data elements

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Reminder

• Use the Student t distribution when is not
  know and either or both of these conditions is
  satisfied
• The population is normally distributed, or

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Hypothesis Testing

• State the claim (in words).
• State the null and alternative hypotheses.
• Obtain the test statistic and p-value.
• Determine One-tail/Two-tail test, obtain critical
  value(s).
• Reject/Fail to reject H0.
• State conclusion (in words).

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Example

• Researchers are investigating the possibility of
  using drug therapy to treat hypertension. The
  data below represent the systolic blood pressure
  (in mmHg) of 14 patients. Assuming the normality
  of systolic blood pressures, test the claim that
  the mean systolic blood pressure of patients
  undergoing drug therapy for hypertension is less
  that 165 mmHG.

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Testing a Claim about a Standard Deviation or
Variance
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Hypothesis Testing
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Testing Claims About or

• Requirements
• The sample is a simple random sample.
• The population has a normal distribution.
• Test Statistic for Testing a Claim About
  or
• P-values and Critical values Use Table A-4 with
  df n 1 for the number of degrees of freedom.

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Testing Claims About or

• P-values Estimate using Table A-4 with df n
  1 for the number of degrees of freedom or use
  calculator with .
• Right-Tail Testp-Value (test
  statistic, 1000, df)
• Left-Tail Testp-Value (0, test
  statistic, df)
• Two-Tail Testp-Value 2 (test
  statistic, 1000, df) ifp-Value 2
  (0, test statistic, df) if

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Hypothesis Testing

• State the claim (in words).
• State the null and alternative hypotheses.
• Obtain the test statistic and p-value.
• Determine One-tail/Two-tail test, obtain critical
  value(s).
• Reject/Fail to reject H0.
• State conclusion (in words).

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Example

• Researchers wanted to measure the effectiveness
  of recombinant human growth hormone (rhGH) on
  children with total body surface area burns gt 40
  percent. In the study, 16 subjects received daily
  injections at home of rhGH. At baseline, the
  researchers wanted to know the current levels of
  insulin-like growth factor (IGF-I) prior to
  administrations of rhGH. The sample variance of
  IGF-I levels (in ng/ml) was 670.81. Assuming that
  the sample constitutes a simple random sample,
  and that the IGF-I levels are normally
  distributed, test the claim the the population
  variance is not 600.