10'1: Confidence Intervals

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10.1 Confidence Intervals

• Falls under the topic of Inference.
• Inference means we are attempting to answer the
  question, How good is our answer?
• Mathematically Confidence Interval Estimate
  M.O.E
• Conceptually In repeated sampling, it is the
  expected percentage of intervals that would
  trap the parameter.
• Or, We arrived at this interval using a method
  that yields correct results 95 of the time.
• IT DOES NOT MEAN Our answer is 95 correct.
• Our Estimate is our summary statistic, usually
  the sample mean or sample proportion.

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10.1 Confidence Intervals

• The Margin of Error shows how accurate we believe
  our guess is, based on the variability of the
  estimate.
• Recall from the C.L.T. that the sample mean is
  Normal with a mean µ and standard deviation
• Then our Confidence Interval for µ is
• z is determined by the level of confidence we
  desire.
• Remember, the data MUST be from an SRS.
• Outliers can affect the confidence interval.
  (Why?)
• The margin of error only covers the natural
  variation of the data. It DOES NOT help with
  nonresponse, undercoverage, and other user
  errors...

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10.2 Significance Testing

• Answers the question How likely is this sample
  statistic if we think we know the parameter?
• Ho Null Hypothesis - The given parameter value
• HA Alternative Hypothesis - Our expectation or
  hope
• YOU MUST ALWAYS STATE WHAT YOUR HYPOTHESES ARE.
• Test Statistic - Standardized value that measures
  the deviation from our sample statistic to our
  parameter value.
• In the case of means
• P-value Probability that if our null hypothesis
  is true we would get a sample statistic as
  extreme or more extreme than what we observed.

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10.2 Significance Testing

• Small P-value means that the odds of getting the
  sample statistic we did are unlikely given our
  parameter value.
• The smaller the P-value, the stronger the
  evidence AGAINST the null hypothesis.
• Two possibilities Either the null hypothesis is
  wrong or we got an unrepresentative sample.
• We have to decide in advance how small a
  probability will allow us to reject the null
  hypothesis.
• This is called the significance level.
• We use the Greek letter alpha to represent this.
• So, if P alpha, we reject the null hypothesis

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10.2 Significance Testing

• Remember, statistically significant means we
  have evidence to reject the null hypothesis.
• FOUR STEPS FOR PERFORMING SIGNIFICANCE TEST
• State your null and alternative hypotheses
• Calculate the test statistic. (Be sure to show
  formula)
• Find the P-value and state a conclusion.
• EXAMPLES OF PROPER CONCLUSIONS
• At the alpha level of signficance, we have
  sufficient evidence to reject the null
  hypothesis. P-value alpha
• At the alpha level of signficance, we do not
  have sufficient evidence to reject the null
  hypothesis. P-value gt alpha

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10.2 Significance Testing

• In Chapter 10 we are often testing the null
  hypothesis of
• Ho µ µ0 where µ0 is the given parameter value
• So then there are three possibilities for the
  P-value because there are three place where there
  can be error.
• P-value calculation
• HA µ gt µ0 P-value is P(Z z) One
  sided, right tail
• HA µ lt µ0 P-value is P(Z z) One
  sided, left tail
• HA µ ? µ0 P-value is 2P(Z z) Two sided,
  both tails
• Note, for the last one this is the same thing as
  doing
• 2(1- P(Z z))

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10.2 Significance Testing

• EXAMPLE
• Do middle-aged male executives have different
  blood pressure than the general population?
  Suppose an SRS of 72 executives (aged 35 to 44)
  is done and the mean is 126.07. Is this evidence
  that the executive blood pressures differ from
  the national average (for males, 35-44) of 128?
  Test at the 5 level of significance.
• SOLUTION Hypotheses Ho µ 128 HA µ ? 128
  
• Test Statistic z -1.09
• P-value P 2P(Z -1.09)
• 2(1 - P(Z 1.09)) 0.2758
• There is not sufficient evidence that the blood
  pressure of middle-aged male executives differ
  from the general population.

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10.2 Significance Testing

• Remember, you can NEVER accept the null
  hypothesis. We are not showing the null
  hypothesis to be true. We can only reject or
  fail to reject.
• Also, make sure you always state your conclusion
  in terms of the question asked. Saying P-value
  alpha earns zero points
• Be careful about whether your test is one-sided
  or two-sided. If you see words like different
  or changed, then its probably a TWO-SIDED
  test. If you see words like higher, lower,
  better, or worse, then its probably a
  ONE-SIDED test.