Tests of significance: the basics

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Chapter 14

• Tests of significance the basics

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Reasoning of Tests of Significance

• What would happen if we repeated the sample or
  experiment many times?
• How likely would it be to see the observed
  results if the claim was untrue?
• Do the data give evidence against a claim?

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Stating HypothesesNull Hypothesis, H0

• The statement being tested is called the null
  hypothesis
• The null hypothesis is a statement of no effect
  or no difference
• The test is designed to assess the strength of
  evidence against the null hypothesis.

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Stating HypothesesAlternative Hypothesis, Ha

• The statement we are trying to find evidence for
  is called the alternative hypothesis.
• The alternative hypothesis usually indicates an
  effect or difference
• The alternative hypothesis expresses the hopes or
  suspicions we bring to the data.

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Case Study I Weight Gain
Suppose we know that weight gain after the age
of 30 varies from individual to individual
according to a Normal distribution with standard
deviation s 1 lbs The symbol m represents
the mean or expected weight gain for all
individuals (the parameter) Ten individuals
between the age of 30 and 40 yield an average
gains of 1.02 lbs. Do these data provide
sufficient evidence that people in this age range
tend to gain weight each year?
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Case Study I Weight Gain

• If the claim that m 0 is true (no average
  weight gain), the sampling distribution of
  from 10 individuals is Normal with m 0 and
  standard deviation
• The data yielded 1.02, which is more than
  three standard deviations from m 0. This
  provides strong evidence that people gain weight.
• If the data yielded 0.3, which is less than
  one standard deviations from m 0, there would
  be less convincing evidence that individuals
  gains weight.

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Case Study I
Weight gain
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Statistical hypotheses

• Null H0 m m0
• One sided alternatives
• Ha m gt m0
• Ha m lt m0
• Two sided alternative
• Ha m ¹ m0

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Case Study I
Weight gain
The null hypothesis is no average weight
gain The alternative hypothesis yes, average
weight gain H0 m 0 Ha m gt 0 We use a
one-sided test because we are interested only in
determining weight gain (and not weight loss)
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Test StatisticTesting the Mean of a Normal
Population

• Take an SRS of size n from a Normal population
  with unknown mean m and known standard deviation
  s.
• The test statistic for null hypothesis H0 m m0
  is

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Case Study I
Weight Gain
The test statistic for no average weight gain
is This shows that the sample mean is more
than 3 standard deviations above the hypothesized
mean of 0. This provides strong evidence against
H0.
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P-value

• The P-value provides the probability that the
  test statistic would take a value as extreme or
  more extreme than the value observed if the null
  hypothesis were true.
• The smaller the P-value, the stronger the
  evidence the data provide against the null
  hypothesis

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P-value for Testing Means

• Ha m gt m0
• P-value is the probability of getting a value as
  large or larger than the observed test statistic
  (z) value.
• Ha m lt m0
• P-value is the probability of getting a value as
  small or smaller than the observed test statistic
  (z) value.
• Ha m ¹ m0
• P-value is two times the probability of getting a
  value as large or larger than the absolute value
  of the observed test statistic (z) value.

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Case Study I
Weight Gain
For test statistic z 3.23 and alternative
hypothesisHa m gt 0, the P-value is P-value
P(Z gt 3.23) 1 0.9994 0.0006 Interpretation
s If H0 is true, there is only a 0.0006 (0.06)
chance that we would see results at least as
extreme as those in the sample ? we therefore
have evidence against H0 and in favor of Ha.
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Case Study I
Weight gain
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Statistical Significance

• If the P-value is as small or smaller than the
  significance level a (i.e., P-value a), then we
  say that data are statistically significant at
  level a.
• If we choose a 0.05, we are requiring that the
  data give evidence against H0 so strong that it
  would occur no more than 5 of the time when H0
  is true.
• If we choose a 0.01, we are insisting on
  stronger evidence against H0, evidence so strong
  that it would occur only 1 of the time when H0
  is true.

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Tests for a Population Mean

• The four steps in carrying out a significance
  test
• State the null and alternative hypotheses.
• Calculate the test statistic.
• Find the P-value.
• State your conclusion.
• The procedure for Steps 2 and 3 is on the next
  page.

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Case Study I
Weight gain problem

• Hypotheses H0 m 0 Ha m gt 0
• Test Statistic
• P-value P-value P(Z gt 3.23) 1 0.9994
  0.0006
• Conclusion
• Since the P-value is smaller than a 0.01,
  there is strong evidence that people gain weight
  in this age range.

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Case Study II
Studying Job Satisfaction
Does the job satisfaction of assembly workers
differ when their work is machine-paced rather
than self-paced? A matched pairs study was
performed on a sample of workers, and each
workers satisfaction was assessed after working
in each setting. The response variable is the
difference in satisfaction scores, self-paced
minus machine-paced.
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Case Study II
Studying Job Satisfaction
The null hypothesis no average difference in
the population of assembly workers. The
alternative hypothesis (that which we want to
show is likely to be true) is there is an
average difference in scores in the population
of assembly workers. H0 m 0 Ha m ? 0 This
is considered a two-sided test because we are
interested determining a difference in either
direction.
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Case Study II
Studying Job Satisfaction
Suppose job satisfaction scores follow a Normal
distribution with standard deviation s 60.
Data from 18 workers gave a sample mean score of
17. If the null hypothesis is µ0 0, the test
statistic is
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Case Study II
Studying Job Satisfaction
For test statistic z 1.20 and alternative
hypothesisHa m ? 0, the P-value would
be P-value P(Z lt -1.20 or Z gt 1.20) 2 P(Z
lt -1.20) 2 P(Z gt 1.20) (2)(0.1151) 0.2302
If H0 is true, there is a 0.2302 (23.02)
chance that we would see results at least as
extreme as those in the sample. Therefore do not
have good evidence against H0 and in favor of Ha.
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Case Study II
Studying Job Satisfaction
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Case Study II
Studying Job Satisfaction

• Hypotheses H0 m 0 Ha m ? 0
• Test Statistic
• P-value P-value 2P(Z gt 1.20) (2)(1 0.8849)
  0.2302
• Conclusion
• Since the P-value is larger than a 0.10, there
  is not sufficient evidence that mean job
  satisfaction of assembly workers differs when
  their work is machine-paced rather than
  self-paced.

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Confidence Intervals Two-Sided Tests
A level a two-sided significance test rejects
the null hypothesis H0 m m0 exactly when the
value m0 falls outside a level 1 a confidence
interval for m.
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Case Study II
Studying Job Satisfaction
A 90 confidence interval for m is
Since m0 0 is in this confidence
interval, it is plausible that the true value of
m is 0. Thus, there is insufficient evidence(at
? 0.10) that the mean job satisfaction of
assembly workers differs when their work is
machine-paced rather than self-paced.