Introduction to Inference

1
Introduction to Inference
2
Statistical Hypotheses

• Statistical Hypotheses are statements about
  population parameters.
• Hypotheses are not necessarily true.

3
Examples of Hypotheses

• mean weight of adult males is greater than 160
  pounds
• proportion of UK graduate students who have used
  illegal drugs in the last month is 0.20
• mean lifetime of GE light bulbs is greater than
  mean lifetime of Sylvania light bulbs
• milk drinkers are better lovers

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In statistics, we test one hypothesis against
another

• The hypothesis that we want to prove is called
  the alternative hypothesis, Ha.
• Another hypothesis is formed which contradicts
  Ha.
• This hypothesis is called the null hypothesis,
  Ho. Ho contains an equality statement.

5
Decision

• After taking the sample, we must either Reject
  Ho and believe Ha or Fail to Reject Ho because
  there was not sufficient evidence to reject it.

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Errors
7
Probability of Errors

• P(Type I Error) P(Reject Ho Ho is true) is
  called the level of significance, .
• The smaller is, the harder it is to reject
  Ho.
• We usually choose to be .05.
• P(Type II Error)
• P(Fail to Reject Ho Ho is false).
• It is denoted by . 

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Important Terms

• Test Statistic The statistic we compute to make
  the decision. Sampling distribution of test
  statistic given that Ho is true must be known or
  well approximated.
• Critical Region The values of the test statistic
  such that we Reject Ho and conclude that Ha is
  true.
• This depends on choice of Ha.
• The endpoint of the critical region is called the
  critical point.
• Example If the critical region is Z gt 1.645.
• 1.645 is the critical point.

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Steps of a Hypothesis Test

• State Ha and Ho
• Specify , n , the test statistic and the
  critical region
• Take sample and compute test statistic
• Make decision and interpret the results

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P-value

• The choice of is subjective.
•  
• The smaller is, the smaller the critical
  region. Thus, the harder it is to Reject Ho.
•  
• The p-value of a hypothesis test is the smallest
  value of such that Ho would have been
  rejected.

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Rules

• If P-value , reject Ho.
• If P-value , do not reject Ho.
•  
• Example p-value 0.03
• Reject Ho _at_ .05
• Fail to Reject Ho _at_ .01

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Estimation

• Population mean, , is an unknown parameter.
• Wish to estimate based on a sample.
• is a statistic which estimates .
•  
• Recall
• We call a point estimate because its value is
  a point on the real line. Unfortunately, if we
  sample from a continuous distribution,
• Thus, we are sure that our estimate is wrong.

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Interval Estimates

• Statisticians prefer interval estimates.
• Something depends on amount of variability in
  data and how certain we want to be that we are
  correct.
•  
• The degree of certainty that we are correct is
  known as the level of confidence.
• Common levels are 90, 95, and 99.

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Facts

• Increasing the level of confidence,
• decreases , the probability of error
• increases the critical point
• widens the interval
• Increasing n, decreases the width of the interval