Elementary Statistics

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Elementary Statistics

• Sampling Distribution

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Where are we?

• Scientific Method
• Formulate a theory -- hypothesis (Chapter1)
• Collect data to test the theory-sampling method,
  and experimental design (Chapter 2, Chapter3)
• Analyze the results --- graphically, numerically
  (Chapter 4, 5)
• Use models Chapter 6
• Understand the language of probability Chapter 7
• Interpret the results and make a decision
  --p-value approach
• Make a decision on proportion.- Chapter 8,9

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Fact

• is approximated N(p, )
• If the sample size n is sufficiently large.
• np ? 5 and n(1-p) ? 5

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What if we dont know true parameter?

• 68-95-99.7 rule
• We are confident that true parameter is two
  standard deviation away from the sample
  proportion.
• Standard error ----- Estimated Standard deviation

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0.0789,0.1611
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N(30,0.0648)
0.2201
0.1401
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Conclusion

• How tell us about p?
• P is within two standard error from

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Summary
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Make decision Review Hypothesis
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More Practice

• P581 9.8 a)
• P581 9.10 a)
• P607 9.39
• P6099.49 a)

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Notation Most common tests
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Learning about a Population Proportion P568

• Example Preparing for Employment
• Have the skill requirements increased for a
  majority of employers nationwide?
• Appropriate Hypothesis - Restatement
• What proportion of employers say skill
  requirements have been on the increase?
• p population proportion of employers who say
  skill requirements have been on the increase
• Based on a sample of 3000 employers, 57 said
  that skill requirements have been on the
  increase. Calculate the p-value

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• Sampling and Summarize
• Sample of 3000 employers
• 57 said skill requirement increases
• Make decision( Significant level ?0.05)
• Direction of Extreme
• One way to the right
• P-value ??? How? when n is sufficient large

Conclusion p-value lt 0.05, Reject null
hypothesis
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One-Sample Z-test p-value approach

• Example p proportion of Michigan adults who
  smoked in 1993. The department hypothesizes that
  the 1993 proportion is lower than the previous
  years proportion of 0.255.
• Survey based on telephone interviews with 2400
  adults.
• The report from the Michigan Department of
  Public Health showed that 25 percent of Michigan
  adults smoked.
• "The 1993 smoking level (25 percent) compared to
  25.5 percent the previous year (1992)."
• "A spokesman for the Tobacco Institute...said the
  change from 1992 was little to crow about. I
  think they're trying to make a story out of
  something that is really statistically
  indistinguishable from the previous year.
• Who will you support ??

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• Hypothesis
• Sample and Summary
• n 2400
• Make decision one way to the left
• Significant level 0.05
• Mean 0.255
• Standard deviation 0.0089
• Z- score -0.56 -- Z-test statistics
• P-value P(Z? -0.56) 0.288 gtgt 0.05
• Conclusion Accept null hypothesis

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Using Calculator P523

• Using the 1-PropZTest function under the STATS
  TESTS menu.

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Make decision based on sample distribution

• Step1 Hypothesis
• Step2 Collect Sample, Sample size
• Step3 Summary of Sample
• Sample Proportion
• Step4 Make decision
• Direction of Extreme
• Find the sample distribution of sample proportion
• Calculate the p-value of the sample proportion
• Compare the p-value and significant level
• Step5 Conclusion

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Make decision based on confidence interval When
p is unknown

• Step1 Hypothesis
• Step2 Collect Sample, Sample size
• Step3 Summary of Sample
• Sample Proportion
• Step4 Find the confidence interval
• Calculate the standard error
• Find the confidence interval
• Step5 Reject the hypothesis if the value does
  not fall in the confidence interval

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Confidence Interval Estimation

• Point estimate use the sample proportion to
  estimate the population proportion
• Confidence interval estimate - estimate the
  population proportion p is in an interval of
  value
• Confidence level - the probability that the
  interval contains the parameter p.

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68-95-99.7 rule
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Standard Error Revisited

• Standard Error
• approximate 95 confidence interval for is given
  by

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P is unknown Standard Error
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Study Chronic Fatigue Common

• A study was performed to learn about the
  proportion of adults who suffer from chronic
  fatigue syndrome. For this study, 4000 randomly
  selected members of a health maintenance
  organization in Seattle were surveyed for the
  illness. Surveys were mailed asking respondents
  if they felt unusual fatigue that interfered with
  work or responsibilities at home for at least six
  months. Of the 3066 people who responded
  (possible non-response bias), 590 reported
  chronic fatigue.
• We wish to estimate the proportion of adults who
  think they suffer from chronic fatigue syndrome.

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If we repeated this procedure over and over,
yielding many 95 confidence intervals for , we
would expect that approximately 95 of these
intervals would contain and approximately 5
would not.
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Other confidence Level
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Calculator-abbreviated 1-PropZint
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Use calculator for confidence interval

• 1-PropZInt
• X --- number of successes
• N --- sample size
• C-level --- confidence interval
• Click calculate

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Short Examples

• USA today conducted a poll based on telephone
  interviews with 1538 National Adults( From
  October 22-24)
• 51 of those will vote for Bush/Cheney
• Please explain For results based on the total
  sample of National Adults, one can say with 95
  confidence that the margin of sampling error is
  3 percentage points.

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Let's Do It! 9.8 9.8 What Is the Estimate?

• The 99 confidence interval for was calculated
  to be (0.27 , 0.42).
• (a) What is the value for the sample proportion
  ?
• (b) What is the value of the margin of error?
• (c) Give two suggestions for how you might
  reduce the margin of error.