Statistical decisionmaking with means: z and t

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Statistical decision-making with means z and t
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More statistical decision-making.

• Review of sign test.
• Sampling distribution of the mean
• What is it?
• Why is it normally distributed?.
• Hypothesis testing with means and the z
  distribution when m and s are known.
• Hypothesis testing with means and t when m is
  known and s must be estimated.

Reminders Exam 2 is April 2!
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Course notes.

• Chapter 12
• Logic of using the normal curve for hypothesis
  testing.
• Hypothesis testing with one sample when m and s
  are known z.
• Skip pages 279-285.
• Chapter 13
• Hypothesis testing with one sample when m is
  known but s is unknown t.
• Skip pages 305-308.
• Exam 2
• will cover Chapters 8, 10, 12, and 13.
• handed out on 4/02 and due back at the beginning
  of class on 4/09.

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Sign test.

• The sign test can be used when you have two sets
  of scores that are paired because they are scores
  collected from the same person while he/she is
  participating in two different conditions.
• Known as repeated measures, within-subjects
  or correlated measures designs.
• Goal of the sign test is to determine if scores
  collected under one condition differ from those
  collected under the other condition
• Significant as opposed to random
  (non-significant) difference.
• Do any differences between conditions reflect
  chance variation (H0) or do the differences
  suggest that, under one of the conditions, people
  were acting like they were drawn from another
  population (HA).

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Example of sign test nicotine and heart rate.

• Nicotine is a psychomotor stimulant should
  increase heart rate.
• Nicotine is present in tobacco and tobacco smoke
  that smokers inhale.
• If tobacco smokers are taking in nicotine, their
  heart rate should increase during smoking as
  compared to before smoking.
• Heart rate fluctuates naturally over time (random
  variability).
• Sometimes the heart beats faster
• Sometimes the heart beats slower
• Tobacco smoking is expected to increase heart
  rate beyond this natural random variability.

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One way to examine the effect of tobacco smoke on
HR

• Recruit 32 smokers.
• Ask them to abstain from smoking for 8 hours.
• Measure HR before and during smoking.
• Condition 1 Measure HR for 5 minutes before
  smoking (HRbefore)
• Condition 2 Measure HR for 5 minutes during
  smoking (HRduring)

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HA Some questions to think about.

• If tobacco smoking does increase HR (systematic
  change)
• Which should be greater HRbefore or HRduring?
• If you subtracted HRbefore from HRduring
  (HRduring HRbefore) would you expect the
  resulting difference score to be positive () or
  negative (-)?
• Across all subjects would you expect more plusses
  or more minuses?
• Is p(Plus) greater than, less than, or equal to
  0.5?

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H0 Some questions to think about.

• If tobacco smoking does not increase HR (random
  variation)
• Which should be greater HRbefore or HRduring?
• If you subtracted HRbefore from HRduring
  (HRduring HRbefore) would you expect the
  resulting difference score to be positive () or
  negative (-)?
• Across all subjects would you expect more plusses
  or more minuses?
• Is p(Plus) greater than, less than, or equal to
  0.5?

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Decision-making steps

• 1. Define problem Does tobacco smoking increase
  HR?
• 2. Define hypotheses with respect to sign of
  (HRduring HRbefore)
• H0 Tobacco smoking does not increase HR
  p(Plus) 0.50
• HA Tobacco smoking increases HR p(Plus)
  0.50
• 3. Define experiment 32 smokers, measure HR
  before/during smoking.
• 4. Define statistic P, the number of Plusses
  observed with 32 subjects.
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which your
  decision hinges P ??

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Rejection region these outcomes make H0 very
difficult to believe p(P gt 22) 0.0249
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Decision-making steps

• 1. Define problem Does tobacco smoking increase
  HR?
• 2. Define hypotheses with respect to sign of
  (HRduring HRbefore)
• H0 Tobacco smoking does not increase HR
  p(Plus) lt 0.50
• HA Tobacco smoking increases HR p(Plus) gt
  0.50
• 3. Define experiment 32 smokers, measure HR
  before/during smoking.
• 4. Define statistic P, the number of Plusses
  observed with 32 subjects.
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges Reject H0 if P gt 22
• 7. Perform experiment/collect data
• 8. Compare observed statistic to critical value.
• 9. Decide
• 10. Draw conclusion using at least one complete
  sentence

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Decision-making steps

• 1. Define problem Does tobacco smoking increase
  HR?
• 2. Define hypotheses with respect to sign of
  (HRduring HRbefore)
• H0 Tobacco smoking does not increase HR
  p(Plus) lt 0.50
• HA Tobacco smoking increases HR p(Plus) gt
  0.50
• 3. Define experiment 32 smokers, measure HR
  before/during smoking.
• 4. Define statistic P, the number of Plusses
  observed with 32 subjects.
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges Reject H0 if P gt 22
• 7. Perform experiment/collect data P 30
• 8. Compare observed statistic to critical value.
  Is 30 in rejection region?
• 9. Decide Reject H0
• 10. Draw conclusion using at least one complete
  sentence Based on these results, tobacco smoking
  does increase heart rate.

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Example of sign test denicotinized cigs and HR.

• Normal tobacco smoke increases HR in abstinent
  smokers.
• Is it the nicotine or some other smoke
  constituent that causes this HR increase?
• What would happen if smokers smoke denicotinized
  tobacco cigarettes instead of normal tobacco
  cigarettes? Would heart rate increase, decrease,
  or stay the same?

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One way to examine the effect of denic cigs on HR

• Recruit 32 smokers.
• Ask them to abstain from smoking for 8 hours.
• Measure HR before and during smoking of
  denicotinized tobacco cigarettes.
• Condition 1 Measure HR for 5 minutes before
  smoking (HRbefore)
• Condition 2 Measure HR for 5 minutes during
  smoking (HRduring)
• Hypothesize some change in HR, but unknown
  direction (hint a non-directional hypothesis!).

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Decision-making steps

• 1. Define problem Does smoking denicotinized
  tobacco influence HR?
• 2. Define hypotheses with respect to sign of
  (HRduring HRbefore)
• H0 Denicotinized tobacco does not influence
  HR p(Plus) 0.50
• HA Denicotinized tobacco influences HR
  p(Plus) 0.50
• 3. Define experiment 32 smokers, measure HR
  before/during smoking.
• 4. Define statistic P, the number of Plusses
  observed with 32 subjects.
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which your
  decision hinges P ??

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Rejection region these outcomes make H0 very
difficult to believe p(P lt 10) 0.0249
Rejection region these outcomes make H0 very
difficult to believe p(P gt 22) 0.0249
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Decision-making steps

• 1. Define problem Does smoking denicotinized
  tobacco influence HR?
• 2. Define hypotheses with respect to sign of
  (HRduring HRbefore)
• H0 Denicotinized tobacco does not influence
  HR p(Plus) 0.50
• HA Denicotinized tobacco does influence HR
  p(Plus) 0.50
• 3. Define experiment 32 smokers, measure HR
  before/during smoking.
• 4. Define statistic P, the number of Plusses
  observed with 32 subjects.
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges
• Reject H0 if 10 gt P gt 22
• 7. Perform experiment/collect data
• 8. Compare observed statistic to critical value.
• 9. Decide
• 10. Draw conclusion using at least one complete
  sentence

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Decision-making steps

• 1. Define problem Does smoking denicotinized
  tobacco influence HR?
• 2. Define hypotheses with respect to sign of
  (HRduring HRbefore)
• H0 Denicotinized tobacco does not influence
  HR p(Plus) 0.50
• HA Denicotinized tobacco does influence HR
  p(Plus) 0.50
• 3. Define experiment 32 smokers, measure HR
  before/during smoking.
• 4. Define statistic P, the number of Plusses
  observed with 32 subjects.
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges
• Reject H0 if 10 gt P gt 22
• 7. Perform experiment/collect data P 19
• 8. Compare observed statistic to critical value.
  Is 19 in rejection region?
• 9. Decide Fail to reject H0
• 10. Draw conclusion using at least one complete
  sentence Based on these results, there is no
  evidence to support the idea that smoking
  denicotinized tobacco influences heart rate.

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Hypothesis testing with means.

• So far we have covered hypotheses that test the
  probability of the occurrence of a particular
  event
• For example, with the sign test, we calculated
  difference scores and examined H0 p(Plus) lt
  0.50 vs. HA p(Plus) gt 0.50.
• Sign test loses information the actual value of
  the difference score.
• A difference of 15 is the same as a difference
  of 1
• Using the actual values of scores is better
• Uses more information
• Yields a more powerful (sensitive) statistical
  test.
• Also allows us to test sample statistics (i.e.,
  X) that we collect by comparing the statistic to
  a known population parameter (i.e., m)

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Consider this question

• Does cigarette abstinence decrease heart rate in
  experienced smokers to a level below that of a
  non-smoking adult?
• Cant use sign test that test requires that you
  have two scores from the same person, and a
  smoker cannot also be a non-smoker!
• However, if you know the population mean (m 75
  bpm) and standard deviation (s 9.0) of heart
  rate for a non-smoking adult, you could compare
  the heart rate of a sample of non-smokers who
  have abstained from smoking to those population
  values.
• The question becomes Does the sample heart rate
  taken from abstaining smokers look like it was
  drawn from the population of non-smokers with m
  75 and s 9.0, or does it look like it was drawn
  from a different population with a lower m?

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Decision-making steps

• 1. Define problem Do smokers who have abstained
  from smoking have a lower heart rate than
  non-smokers (m 75, s 9.0)?
• 2. Define hypotheses with respect to known
  population mean, m
• H0 Abstaining does not lower HR
• HA Abstaining does lower HR
• 3. Define experiment 32 smokers, measure HR
  after 8 hours non-smoking.
• 4. Define statistic Mean of sample, X.
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges What distribution do we use?
• 7. Perform experiment/collect data
• 8. Compare observed statistic to critical value.
• 9. Decide
• 10. Draw conclusion using at least one complete
  sentence

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Big problem need a probability distribution

• Need a probability distribution that we can use
  to determine our rejection region (given that H0
  is true) and limit p(Type I error) lt .05
• Generally, need to use this distribution for all
  variables on all measurement scales, independent
  of the unit of measurement.
• Should be one that we can use to determine the
  probability of all possible events (like we did
  with the number of heads in the coin problems or
  number of plusses in the sign test.
• The unit normal distribution (z) may help. We
  know, that
• z scores are independent of the unit of
  measurement
• The z distribution can be used to determine
  probability IF we assume that the original
  variable has a normal distribution.
• Are all variables normally distributed? NO! NO!
  NO! NO! NO!

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Why the normal distribution is an appropriate
distribution for hypothesis testing.

• We CANNOT assume that all variables that we
  measure are normally distributed!
• However, there is one distribution that is
  normally distributed, no matter what the shape of
  the parent distribution, provided your sample
  size (N) is large the sampling distribution of
  the mean.

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What is the sampling distribution of the mean?

• The sampling distribution of the mean gives all
  the values the mean of a sample size N can take,
  along with with probability of getting each value
  if sampling is random from the null-hypothesis
  population.
• Imagine a population of scores 2,3,4,5,6.
• Imagine that you sample from that population
  twice (with replacement)
• Sample 1 4
• Sample 2 3.
• You take the two scores that you sampled and
  calculate their average (mean 3.5).
• Do this for all possible samples of size N 2
  Ranges from 2 to 6

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What is the sampling distribution of the mean?

• Sampling distribution of the mean gives all the
  values the mean of a sample size N can take,
  along with with probability of getting each value
  if sampling is random from the null-hypothesis
  population.
• Regardless of the shape of the population of
  scores, the sampling distribution of the mean
  approaches a normal distribution as sample size
  (N) increases.
• The mean of the sampling distribution of the mean
  (mX) is the same as the population mean (m)
• standard deviation of the sampling distribution
  of the mean (sX) is equal to the population
  standard deviation divided by the square root of
  the sample (sX s/ N). For this example, 1.0
  1.41/ 2 )

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So

• If you want to use the z distribution to
  calculate probability, you need to be able to
  assume that the original population is normally
  distributed.
• With large Ns, the sampling distribution of the
  mean is normally distributed!
• In order to convert any score to a z score you
  need the score (X), the population mean (m) and
  the population standard deviation (s)
• X score
• mX m
• sX s/ N
• Now back to our problem . . .

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Decision-making steps

• 1. Define problem Do smokers who have abstained
  from smoking have a lower heart rate than
  non-smokers (m 75, s 9.0)?
• 2. Define hypotheses with respect to known
  population mean, m
• H0 Abstaining does not lower HR mabstaining
  smokers gt 75
• HA Abstaining does lower HR mabstaining
  smokers lt 75
• 3. Define experiment 32 smokers, measure HR
  after 8 hours non-smoking.
• 4. Define statistic z
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges

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Decision-making steps

• 1. Define problem Do smokers who have abstained
  from smoking have a lower heart rate than
  non-smokers (m 75, s 9.0)?
• 2. Define hypotheses with respect to known
  population mean, m
• H0 Abstaining does not lower HR mabstaining
  smokers gt 75
• HA Abstaining does lower HR mabstaining
  smokers lt 75
• 3. Define experiment 32 smokers, measure HR
  after 8 hours non-smoking.
• 4. Define statistic z
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges
• 7. Perform experiment/collect data Xobt 71.04
• 8. Compare observed statistic to critical value.
  Need to convert Xobt to zobt

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Does smoking during pregnancy alter birth weight?
Most folks know what fetal alcohol syndrome is
a constellation of adverse events due to alcohol
abuse during pregnancy. In 1985 Nieburg and
colleagues coined the phrase Fetal Tobacco
Syndrome to describe the adverse events that can
occur when mothers smoke during pregnancy. One
potential adverse event caused by tobacco smoking
during pregnancy is an abnormal birth weight.
Is smoking during pregnancy associated with
abnormal birth weight? The average (m) birth
weight of a baby born to a non-smoking mother is
3,300 grams (s 650). You sample the birth
weight of 438 children born to mothers who
smoked, on average, 14 cigarettes per day during
their pregnancy. The sample mean weight (X) was
3,186 grams for the children born to these
mothers who smoked.
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Decision-making steps

• 1. Define problem Is smoking during pregnancy
  associated with birth weight that differs from
  the population average (m 3,300, s 650)?
• 2. Define hypotheses with respect to known
  population mean, m
• H0
• HA
• 3. Define experiment Measure weight of 438
  children born to smokers.
• 4. Define statistic z
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges zcrit

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Decision-making steps

• 1. Define problem Is smoking during pregnancy
  associated with birth weight that differs from
  the population average (m 3,300, s 650)?
• 2. Define hypotheses with respect to known
  population mean, m
• H0 Smoking is not associated w/ abnormal birth
  weight msmoke 3,300
• HA Smoking is associated w/ abnormal birth
  weight msmoke 3,300
• 3. Define experiment Measure weight of 438
  children born to smokers.
• 4. Define statistic z
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges zcrit gt 1.96
• 7. Perform experiment/collect data Xobt 3186
  grams
• X m
• Zobt
• s/ N

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Decision-making steps

• 1. Define problem Is smoking during pregnancy
  associated with birth weight that differs from
  the population average (m 3,300, s 650)?
• 2. Define hypotheses with respect to known
  population mean, m
• H0 Smoking is not associated w/ abnormal birth
  weight msmoke 3,300
• HA Smoking is associated w/ abnormal birth
  weight msmoke 3,300
• 3. Define experiment Measure weight of 438
  children born to smokers.
• 4. Define statistic z
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges zcrit gt 1.96
• 7. Perform experiment/collect data Xobt 3186
  grams
• 8. Compare observed statistic to critical value.
• 9. Decide
• 10. Draw conclusion using at least one complete
  sentence Based on these results . . .

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What to do if s is unknown (a common occurrence)?

• Absolutely no problem.
• Exact same logic, but estimate s with s.
• Use t distribution table instead of z
  distribution to account for estimation.
• t is actually a family of normal distributions.
• As N increases, t becomes more like z.
• Need degrees of freedom (in this case, N-1) so
  that you use the correct t distribution
• Use degrees of freedom to find tcrit in table.

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Do humans follow an internal, 24-hour clock?
Biological theories often emphasize that humans
have adapted to their physical environment. One
such theory hypothesizes that people should
spontaneously follow a 24-hour cycle of sleeping
and waking even if they are not exposed to the
usual pattern of sunlight. To test this notion,
8 paid volunteers were placed (individually) in a
room in which there was no light from the outside
and no clocks or other indicators of time. They
could turn the lights on and off as they wished.
After a month in the room, each individual tended
to develop a steady light-dark cycle. There
cycles at the end of the study were as follows
(in hours) 25, 27, 25, 23, 24, 25, 26, and 25
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Decision-making steps

• 1. Define problem Do humans follow a 24 hour
  clock?
• 2. Define hypotheses with respect to known
  population mean, m
• H0 m
• HA m
• 3. Define experiment Measure light/dark cycle of
  8 people.
• 4. Define statistic t
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges tcrit

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Decision-making steps

• 1. Define problem Do humans follow a 24 hour
  clock?
• 2. Define hypotheses with respect to known
  population mean, m
• H0 m 24
• HA m 24
• 3. Define experiment Measure light/dark cycle of
  8 people.
• 4. Define statistic t
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges tcrit(7)
• 7. Perform experiment/collect data Xobt 25
  hours, sobt 1.20
• X m
• tobt
• s/ N

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Decision-making steps

• 1. Define problem Do humans follow a 24 hour
  clock?
• 2. Define hypotheses with respect to known
  population mean, m
• H0 m 24
• HA m 24
• 3. Define experiment Measure light/dark cycle of
  8 people.
• 4. Define statistic t
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges tcrit(7) 2.365
• 7. Perform experiment/collect data Xobt 25
  hours, sobt 1.20 tobt 2.357
• 8. Compare observed statistic to critical value.
  tobt
• 9. Decide
• 10. Draw conclusion using at least one complete
  sentence Based on these results . . .

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What have you learned?

• Hypothesis testing with coins
• Logic of hypothesis testing.
• How to determine rejection regions (directional
  and non-directional tests)
• Probability of Type I and Type II errors
• The sign test for use when you have two scores
  from the same person and want to tell if scores
  collected under one condition were different from
  those collected under another condition no means
  required!
• Hypothesis testing with means
• Logic of using the normal distribution.
• The one-sample z test when m and s are known.
• The one-sample t test when m is known and s must
  be estimated.