Chapter 10: Introduction to Inference

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Chapter 10Introduction to Inference

• If you believe in miracles, head for the Keno
  lounge.
• Jimmy the Greek

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10.1 Estimating with Confidence(pp. 506-526)

• Confidence intervals are important in statistics.
• This chapter provides information on
• How such intervals can be constructed from
  samples.
• How to interpret such intervals.

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C confidence interval for a parameter

• An interval computed from sample data by a method
  that has probability C of producing an interval
  containing the true value of the parameter.
• A confidence interval for an unknown population,
  µ, calculated from a sample size n with mean
  , has the form
• where z is obtained from the normal
  distribution table and s is the standard
  deviation of the population.

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Example Suppose that the following scores
represent a random sample from a population with
a known standard deviation s 3.88

• Find a 95 confidence interval for the mean of
  the population.
• 85, 83, 91, 88, 88, 92, 81, 83, 85, 83, 86, 84
• Calculate the mean.
• Mean 85.75
• Remember that the Central Limit Theorem states
  that the means of sample means of a specific size
  are normally distributed.
• The 95 C.I. for the mean of the population from
  which the sample was chosen is (83.555, 87.945).

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INTERPRETATION OF 95 CONFIDENCE INTERVAL

• If we took 100 random samples from the population
  and computed 95 confidence intervals for each
  sample, we would expect 95 of the 100 samples to
  contain the mean of the population.

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Results of polls produced by polling
organizations (Gallup) are often provided with a
margin of error.

• Example
• 64 of those polled favored Proportion A with a
  margin of error of 3.
• Interpretation
• This determines a 95 confidence interval (0.61,
  0.67).

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Margin of Error Formula

• Margin of error determines the length of the
  confidence level.
• Therefore, the interval for the previous example
  would be (0.61, 0.67).

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Find sample size using margin of error

• Suppose we want a 95 C.I. that is 3 units long
  and the M.E. to be no more than 1.5 units.
• We can do this by taking a larger sample, but how
  large?
• Therefore, we need a sample size of at least 26
  to obtain an M.E. of 1.5 units.

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Using the TI-84

• z values for
• 99 confidence interval 2.576
• 95 confidence interval 1.96
• 90 confidence interval 1.645
• 80 confidence interval 1.28

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Some Cautions

• The data must be an SRS from the population.
• The formula is NOT CORRECT for probability
  sampling designs more complex than an SRS.
• There is no correct method for inference from
  data haphazardly collected with bias of unknown
  size.
• Because is strongly influenced by a few
  extreme observations, outliers can have a LARGE
  EFFECT on the confidence interval.

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Some Cautions (continued)

• If the sample size is small and the population is
  not normal, the true confidence interval will be
  different from the C value used in computing the
  interval.
• You must know the standard deviation s of the
  population.
• If the sample size is large, the sample standard
  deviation s will be close to the unknown s. Then,

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Margin of Error

• The M.E. of a confidence interval gets smaller
  as
• the confidence level C decreases.
• the population standard deviation s decreases.
• The sample size n increases.

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10.2 Test of Significance(pp. 531-556)

• THIS IS ONE OF THE MOST IMPORTANT SECTIONS IN THE
  BOOK!!!!
• IT CONTAINS A GREAT DEAL OF IMPORTANT MATERIAL.
• READ CAREFULLY!!

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Example 1 Tire Manufacturing

• A tire manufacturer advertised that a new brand
  of tire has a mean life of 40,000 miles with a
  standard deviation of 1,500 miles. A research
  team tested a random sample of 100 of these tires
  and obtained a mean life of 39,500 miles.
• Is the manufacturers claim reasonable?
• How likely is it that one would obtain a random
  sample of 100 tires with a mean life of 39,500
  miles from a population with a mean life of
  40,000 miles and a standard deviation of 1,500
  miles?

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Example 1 Tire Manufacturing

• Consider the set of means of ALL samples of size
  100
• The Central Limit Theorem says that the set has a
  mean of 40,000 and a standard deviation of 150.
• The normal distribution table shows that it is
  HIGHLY UNLIKELY that one would obtain such a
  sample if the population is as give.
• 0.043 is the P-VALUE of the test.
• Since the probability is SO SMALL, we would
  likely conclude that the manufacturers claim is
  incorrect and that the mean life of the tires is
  something less than 40,000 miles.

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Example 1 Tire Manufacturing

• Formalized as
• Null hypothesis
• Alternate Hypothesis
• 1-tail test
• Level of significance 1
• Usually determined before the test
• Critical region from invNorm 2.33
• Calculated sample mean 39,500
• Mean of sample means 40,000
• Standard deviation of sample means 150
• Calculated z for sample -3.33
• Pvalue for test .04

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Example 1 Tire Manufacturing

• Conclusion
• Since the calculated z is in the critical region,
  we reject the null hypothesis.
• Since the P-value is less than 1 (the level of
  significance), this supports the rejection of the
  null hypothesis.

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Example 1 Tire Manufacturing

• A 1-tail test was involved because we were
  interested in a deviation in one direction only.
• It wasnt a concern to consumers if the mean life
  of the tires is more than the advertised value of
  40,000 miles.

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Example 2 Parachutes

• An automatic opening device for parachutes has a
  stated mean release time of 10 seconds with a
  standard deviation of 3 seconds. To test the
  claim, a parachute club tested a random sample of
  36 of these devices and found the mean release
  time to be 10.6 seconds. Is the result
  significant at the 5 level of significance?

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Example 2 Parachutes

• Null Hypothesis
• Alternate Hypothesis
• Type of test 2-tail
• Level of significance
• Critical Region
• Calculated sample mean 10.6

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Example 2 Parachutes

• Mean of the sample means 10
• Standard deviation of sample means 0.5
• Calculated z for the sample 1.2
• Since calculated z is NOT in the critical region
  we DO NOT reject the null hypothesis.
• P-Value for the test .8414

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Example 2 Parachutes (TI-84)

• Since the P-value is GREATER than 5, this
  supports the decision to not reject the null
  hypothesis.
• There IS NOT strong evidence to suggest that this
  sample did not come from a population with mean
  10.

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Note well

• A null hypothesis is basically a hypothesis of no
  change.
• If one does not reject a null hypothesis, then
  the test results are not statistically
  significant.
• Accepting a null hypothesis at a LOW LEVEL of
  significance is NOT STRONG evidence that it is
  true.
• Acceptance of a null hypothesis simply means that
  it is not unreasonable to assume that the
  population mean µ is the stated value.
• For all you know, it might be some other number
  even closer to the stated value.

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Note well

• A P-value is the probability that one would
  obtain a statistic as extreme as that which was
  calculated from the sample.
• A small P-value, such as .01, means that the
  statistic is NOT LIKELY the result of pure
  chance.
• A P-value, such as .35, means that the statistics
  is not an unusual or unexpected result.
• The level of significance for a test is usually
  set beforehand.
• If a calculated P-value is smaller than the level
  of significance, then the test statistic is
  statistically significant.

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Note well

• A statistical test could be 2-tailed or 1-tailed.
• The one used is dependent on the purpose and
  nature of the test.
• If both positive and negative deviations from a
  parameter are important, use a 2-tailed test.
• If only positive (or negative) deviations are
  important, use a 1-tailed test.

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Finding P-values
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Note well

• Rejecting a null hypothesis is equivalent to
  saying that the test statistic is statistically
  significant.
• i.e. the calculated test statistic is not a
  likely result of pure chance.
• The null and alternate hypothesis are both stated
  in terms of population parameters, not sample
  statistics.
• You are attempting to use sample statistics to
  come to reasonable conclusions about population
  parameters.

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10.3 Using Significance Tests(pp. 560-567)

• This short section points out things that need
  to be considered when attempting to determine if
  test results are significant.

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10.4 Inference as Decision(pp. 567-577)

• This section introduces three concepts that were
  added to the AP Statistics syllabus for the
  1998-1999 academic year.
• Type I error
• Type II error
• Power of a test

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Type I and Type II Errors
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Lets try an exampleQuality control tests at
the 5 level of significance

• Smiths produces a machine-produced product that
  weighs 1500 lbs. The population of produced
  items has an allowable standard deviation of 40
  lbs. Samples of size 100 are periodically
  examined to see if production standards are being
  maintained.
• Consider the set, M, that consists of mean
  weights of all samples of size 100. The CLT
  states that M will have a normal distribution
  with mean 1500 lbs. and s 4 lbs.

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Quality control tests at the 5 level of
significance

• Null Hypothesis
• Alternate Hypothesis
• Type of test 2-tailed
• Level of significance 5
• 2.5 in each tail
• Critical values of z

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Quality control tests at the 5 level of
significance

• Smiths will reject the null hypothesis if a
  sample of 100 yields a mean that is not within
  1.96 standard deviations of 1500 lbs.
• The null will be rejected if a mean weight is
  less than 1500 1.96(4) which equals 1492.16 lbs
  or if a mean weight is more than 1500 1.96(4)
  which equals 1507.84 lbs.
• In other words, Smiths will accept the null if a
  mean weight is in the range 1492.16 lt x lt 1507.84.

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Quality control tests at the 5 level of
significance

• Related calculator computations

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Suppose a random sample produces a mean of 1509
lbs.

• The null would be rejected.
• There is a suggestion that production standards
  are not being met.
• The probability of obtaining a mean as large as
  1509 is normalcdf(1509,1E99,1500,4) which equals
  .012224 or about 1.2.
• Therefore, if the null is true, there is a 1.2
  chance the Smiths will incorrectly reject it.
• This is a Type I error.
• By setting the level of confidence at 5 PRIOR to
  doing any testing, Smiths is allowing for a 5
  chance of making a Type I error.
• In real life, a Type I error might result in
  stopping production to try to find a problem that
  doesnt exist.

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NOW assume that it is extremely undesirable to
have an item produced that weighs 1515 or more
lbs.

• Smith realizes that things can go wrong in a mass
  production process. Some products may be too
  heavy.
• She is interested in knowing the probability that
  her quality control test will incorrectly accept
  the null if the mean weight somehow shifts to
  1515 lbs.
• If the null is incorrectly accepted, this is a
  Type II error.

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Type II Error (continued)

• The null will be accepted if a sample mean weight
  is less than 1507.84 lbs. and greater than
  1492.16 lbs.
• If the population mean has shifted to 1515 lbs.,
  the z-value for 1507.84 is -1.79.
• Using the normal distribution table, the
  probability that
• z lt -1.79 is .0367.
• THIS IS THE PROBABILITY OF MAKING A TYPE II
  ERROR.
• In other words, if the mean has shifted to 1515
  lbs., Smiths will incorrectly accept the null
  about 3.67 of the time.

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The Power of a Test

• THE POWER OF A TEST IS THE PROBABILITY THAT THE
  NULL WILL BE REJECTED FOR A PARTICULAR VALUE OF A
  POPULATION PARAMETER.
• In this case, the population parameter is µ
  1500 lbs., and the particular alternative value
  is µ 1515 lbs.
• The power of the test for µ 1515 lbs. is
  0.9633.
• (1 0.0367)
• That is, if µ 1515 lbs., Smiths can expect to
  correctly reject the null about 96.33 of the
  time.

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Real-life situations frequently involve Type I
and Type II errors.

• Consider the legal world and a null hypothesis
  The accused man is innocent.
• Type I error occurs when the man is found guilty
  when, in fact, he is innocent.
• Type II error occurs when the man is found not
  guilty, but he is guilty.
• Decreasing the chance of one error type
  frequently increases the chance of the other
  error type.
• In real-world situations, one must often decide
  which error type is more important to minimize.

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Things to Remember

• A Type I error can occur only when a null
  hypothesis is true. You incorrectly reject a TRUE
  null hypothesis.
• A Type II error can only occur when a null
  hypothesis is false. You incorrectly accept a
  FALSE null hypothesis.
• The Power of a Test is 1 - probability (Type II
  error). This is the probability that you
  correctly reject a false null hypothesis.
• One needs an alternative to the null hypothesis
  in order to calculate a Type II error. Without an
  alternative hypothesis, the question What is the
  probability of a Type II error? is meaningless.