Apr. 5

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Apr. 5

• Stat 100

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To do

• Read Chapter 21, try problems 1-6
• Skim Chapter 22

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Thought Question

• Suppose we think mean classes missed per week is
  higher for males than for females.
• How would we determine whether this is the case?

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Basic Strategy

• Collect data on missed classes for males and
  females
• Calculate mean for each sex and determine the
  difference.
• Then what? Suppose the mean for 50 males was 0.5
  higher than mean for 50 females.
• Could we generalize that for all students, mean
  is higher for males?

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An Issue

• Did the observed difference occur just by chance
  or does it reflect a true difference?
• Whats the likelihood the observed difference
  would be 0.5 if theres really no difference in
  the populations?

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Significance Test

• Data used to decide between two competing
  statements (hypotheses) about the population
• Statements are called null hypothesis and
  alternative hypothesis

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The two hypotheses

• null hypothesis a nothing happening statement
• no difference between groups, or no relationship,
  or nothing new
• alternative hypothesis "something's
  happening
• there is a difference, or there is a
  relationship, or somethings new

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Notation

• H0 represents null hypothesis
• HA represents alternative hypothesis

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Missed classes by men and women

• H0 No difference in mean classes missed for men
  and women
• HA There is a difference between mean missed
  classes by men and women

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Deciding between the hypotheses an example

• Students asked to randomly a number between 1 and
  10
• In past, a bias toward picking 7 has been noticed
• With random picking, what proportion would pick
  7?
• Answer 1/10 0.1 because there are 10 numbers

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Are students picking randomly?

• let p represent proportion of all students would
  pick 7
• null random picking , p 0.10
• alternative bias toward 7 , pgt 0.10

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The sample data

• Suppose 24 of 100 students in the class pick 7.
• The proportion picking 7 is 24/100 0.24 (much
  higher than .10)
• Could this have happened through random picking?
• Or, does it reflect bias toward 7?

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With true random picking

• Distribution of possible sample proportions would
  be bell curve
• Centered at 0.10 (chance of randomly picking 7)
• SD Sqr root(0.1)(1-0.1)/100 0.03
• About 99.7 chance that sample proportion would
  fall in range 0.10 ? 3(.03), or .01
  to .19

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Where the observed statistic falls

• Sample value of 0.24 is outside interval of what
  might normally occur with random picking
• Reasonable to reject the hypothesis that picking
  is random

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Where the observed statistic falls

• Observed 0.24 picked 7
• Find percentile rank of 0.24 in bell curve with
  mean 0.10 and SD 0.03
• z score (0.24-0.10) / 0.03 0.14 / 0.03 4.67
• Page 137 table, proportion to left of z is around
  0.995

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Statistically Significant

• A result is called statistically significant when
  a null hypothesis is rejected
• In our example, the result was statistically
  significant

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Thought Question

• Imagine all the criminal trials in the United
  States
• In each trial, what is the null hypothesis?
• What is the alternative hypothesis?
• Overall criminal trials what are the two types of
  mistakes that can b made?

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Statistical Errors

• Type 1 rejecting the null hypothesis when you
  should not (like a convicting innocent person)
• Type 2 not rejecting the null hypothesis when
  you should (like failing to convict a guilty
  person)

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Example

• Experiment is done to see if new treatment for
  depression is better than old treatment
• null hyp new treatment not better
• alternative hyp new treatment is better
• Type 1 error deciding new treatment is better
  when it is not
• Type 2 error deciding new treatment is not
  better when it actually is