Outline

1
Outline

• Hypothesis Test for ? Small Samples
• t-test Example 1
• t-test Example 2
• Hypothesis Test for the Population Proportion p
  Large Samples
• Population Proportion Example 1
• Population Proportion Example 2

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Hypothesis Testing Small Samples

• Central Limit Theorem (review)
• For large enough samples, sampling distribution
  of the mean will be normal even if the raw data
  are not normally-distributed.
• But what do we do when our sample size is not
  large enough?

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For n lt 30, we face two problems

• Central Limit Theorem does not apply
• Z does not give probability of finding X in some
  range relative to µO, when n lt 30.

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µ0
When sampling distribution of the mean is normal,
the Z table gives us the probability that we will
find a sample mean in some range (for samples
of size n, with µ µ0).
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µ0
But when n is lt 30, we cannot be sure that the
sampling distribution of the mean is normal. So,
how do we obtain the probability that a sample
mean will be in a given range?
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For n lt 30, we face two problems

• s is not a good estimator of ?.
• That is, variability in the sample is not a good
  source of information about variability in the
  population.

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Hypothesis Testing Small Samples

• To measure the probability of finding the mean of
  a small sample in a given range relative to µO,
  we use a different probability distribution the
  t distribution.
• t ?
• s/vn

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Hypothesis test for small samples

• CAUTION
• The t distribution changes shape with sample
  size, becoming more like the SND as n gets
  larger.
• For a hypothesis test, to find the critical
  value of t in the t table, you need to know 2
  things a and degrees of freedom.
• For the one-sample t test, d.f. n 1.

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Hypothesis Testing Small Samples

• Very important point about testing with n lt 30
• If an exam question with n lt 30 gives you the
  population standard deviation, ?, then use Z.
• Large n ( 30) use Z
• Small n, ? known use Z
• Small n, ? unknown use t

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Hypothesis Testing Small Samples

• H0 ? ?0
• HA ? ?0 HA ? ? ?0
• or HA ? ?0
• (One-tailed test) (Two-tailed test)
• Test Statistic t - ?0
• s/vn

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Hypothesis Testing Small Samples

• Rejection Region
• One-tailed test Two-tailed test
• tobt gt ta tobt gt ta/2
• or tobt lt -ta
• Where ta and ta/2 are based on d.f. (n 1)
• Remember to report your decision explicitly!!

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Confidence Interval Small Samples

• C.I. ta/2 (s/vn)
• Notes ta/2 is based on d.f. (n 1). Use this
  C.I. when n lt 30 and ? not known.

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Example 1 t-test

• In a recent pollution report, a team of
  scientists expressed alarm at the dihydrogen
  monoxide levels in fish in Ontario lakes.
  Historically, the average dihydrogen monoxide
  level has been 2.65 parts per thousand (ppt).
  This year, samples of fish from 20 lakes in
  Ontario turned up an average dihydrogen monoxide
  count of 2.98 ppt with a variance of 36.
• Note for more information on DHMO, visit
  http//www.dhmo.org/

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Example 1 t-test

• a. Does it appear that dihydrogen monoxide levels
  are increasing in fish in Ontario lakes? (a
  .01)
• b. The dihydrogen monoxide levels of fish in 5
  lakes in the Timmins area were 2.84, 3.96, 4.40,
  1.60, and 2.63. Construct a 95 confidence level
  for the average dihydrogen monoxide counts in
  fish in the Timmins area.

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Example 1a t-test

• H0 ? 2.65
• HA ? gt 2.65
• Test Statistic t - ?0
• s/vn
• Rejection region tobt gt t.01,19 2.539

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Example 1a t-test

• 2.98 s2 36 n 20
• tobt 2.98 2.65
• v36/20
• tobt 0.246
• Decision Do not reject H0. There is not enough
  evidence to conclude that dihydrogen monoxide
  levels are increasing.

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Example 1b t-test

• SX 15.43 15.43/5 3.086
• SX2 52.584 (SX)2 15.432 238.085
• S2 52.584 238.085 1.242
• 5
• 4
• S v1.242 1.114 S 1.114/v5 .498

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Example 1b t-test

• C.I. ta/2 (s/vn)
• For 95 C.I., a .05. Associated t.025,4 2.776
• C.I. 3.086 2.776 (.498) (1.703 ? 4.469)

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Example 2 t-test

• There have been claims that provincial funding
  cuts for hospitals have led to an increase in the
  average waiting time for elective surgery. A
  search of past records reveals that that average
  waiting time for elective surgery was 38.5 days
  prior to the 2003 Ontario election. A random
  sample of patients scheduled for elective surgery
  is identified, and the waiting time until surgery
  for each patient is measured. The data are shown
  below as of days intervening between when
  surgery is ordered and when it occurred.
• Is there evidence in these data that average
  waiting time has increased under the current
  provincial government? (a .01)

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Example 2 t-test

• Patient Waiting time (days)
• 1 43
• 2 28
• 3 55
• 4 38
• 5 30
• 6 45
• 7 51
• 8 39

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Example 2 t-test
Why one-tailed?

• H0 ? 38.5
• HA ? gt 38.5
• Test Statistic t ?0
• s/vn
• Rejection region tobt gt t.01,7 2.998

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Example 2 t-test

• Sx 329 n 8 41.125
• Sx2 14149
• s2 14149 (329)2
• 8
• 7
• s2 88.411
• s v88.411 9.402

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Example 2 t-test

• tobt 41.125 38.5
• 9.402/v8
• tobt 0.787
• Decision Do not reject H0. There is not
  sufficient evidence to conclude that waiting
  times have increased.
• Note Be sure to give the full decision.

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Example 3 t-test

• It is known that the mean of errors made on a
  particular pursuit rotor task is 60.9. A
  physiologist wishes to know if people who have
  had a spinal cord injury but who are apparently
  recovered perform less well on this task. In
  order to test this, a random sample of 8 people
  who have had spinal cord injuries is chosen and
  they are administered the pursuit rotor task. The
  of errors each made is
• 63, 66, 65, 62, 60, 68, 66, 64

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Example 3 t-test

• a. Is there evidence to support the belief that
  recovered patients are impaired in performing
  this task? (a .01)
• b. Form the 90 C.I. for the mean number of
  errors committed by recovered patients.

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Example 3a t-test

• Note these words
• It is known Population information
• The mean of errors This is ?0.
• is 60.9
• In order to test this Hypothesis Test!!
• a random sample of 8 n lt 30, ? not known
• This calls for a t-test

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Example 3a t-test
Why greater than?

• H0 ? 60.9
• HA ? gt 60.9
• Test Statistic t - ?0
• s/vn
• Rejection region tobt gt t.01,7 2.998

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Example 3a t-test

• 64.25 s 2.55 n 8
• tobt 64.25 60.9
• 2.55/v8
• tobt 3.72
• Decision Reject H0. Recovered patients perform
  worse than normals on this task.

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Example 3b t-test

• C.I. ta/2 (s/vn)
• For 90 C.I., a .10. Corresponding t.05,7
  1.895.
• C.I. 64.25 1.895 (2.55/v8)
• (62.542 ? 65.958)