Sampling Distributions, Confidence Intervals, and Hypothesis Tests

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Sampling Distributions, Confidence
Intervals,andHypothesis Tests
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Sampling Distributions

• Parameter estimation is the process of estimating
  the value of a parameter (e.g., mean or variance)
  associated with a population or process through
  sampling values of that population/process
• The sampling could involve i.i.d. repetitions of
  the process or sampling without replacement from
  the population
• If the population is large with respect to the
  sample size, then the sampling can be regarded as
  nearly i.i.d., otherwise correction factors can
  be used

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Sampling Distributions

• To evaluate the goodness of the parameter
  estimate, something about the sampling
  distribution of the estimation procedure needs to
  be known
• For example, if have i.i.d. sample from a
  Gaussian process, with variance ?2, then the
  sampling distribution for the sample mean is
  normal

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Sampling Distributions

• If have i.i.d. sample from a process, not
  necessarily Gaussian, with variance ?2, then the
  sampling distribution for the sample mean is
  approximately normal for large n

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Sampling Distributions

• If have i.i.d. sample from a Gaussian process,
  with variance ?2, then the sampling distribution
  for the sample variance sample is
• If have i.i.d. sample from a Gaussian process,
  with unknown variance, then

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Confidence Intervals

• A confidence interval for a parameter estimate
  provides a measure of the accuracy of the
  estimate
• An x confidence interval is a random interval
  (derived from the sample) that has a x
  probability of containing the population parameter

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Components of a Confidence Interval Calculation

• Sample statistic. The sample statistic serves as
  a point estimate for the corresponding population
  parameter
• Population variance. Large population variance
  will imply larger confidence interval, small
  variance implies smaller confidence interval
• If population variance is known, it will be used
  in the confidence interval calculation
• If the population variance is not known, then
  sample variance (with appropriate correction
  factor) will be used to estimate population
  variance

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Components of a Confidence Interval Calculation

• Standard error. The standard error of the sample
  statistic is the standard deviation of the
  sampling distribution
• Confidence Level. The confidence level indicates
  the probability that the confidence interval
  contains the population parameter
• This is often just an estimate, since the
  sampling assumptions are usually not met

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CI Sample Mean

• General form of a confidence interval is
• (sample statistic) /- (sampling distribution
  score)(SE)
• For estimating the population mean

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CI Sample Mean

• For , let be that value
  such that
• So

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CI Sample Mean

• The confidence interval
  for the sample mean, assuming simple random
  sampling and large n, is given by
• The confidence
  interval for the sample proportion , assuming
  simple random sampling, is given by

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Examples

• Chapter 7 problems
• 3, 4, 5, 9, 10, 16, 17, 18

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Tests for Significance

• Tests for significance, or hypothesis tests
  address the question of whether an observed
  difference what would be expected under a
  specified model is real or just due to chance
  variation
• For example, is there a statistically significant
  difference between the response rate of one type
  of cancer treatment versus another type of
  treatment or is the difference just due to
  chance variation

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Components of a Significance Test

• Null and alternative hypothesis
• The null hypothesis says that an observed
  difference reflects chance variation. Denoted by
  H0.
• The alternative hypothesis says that the observed
  difference is real. Denoted by HA

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Components of a Significance Test

• Test statistic
• A test statistic is used to measure the
  difference between the data and what is expected
  according to the null hypothesis
• To perform the hypothesis test, assumptions are
  made about the sampling distribution of the test
  statistics IF the null hypothesis was true
• Test statistic often has the general form

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Components of a Significance Test

• Significance level
• The observed significance level is the chance of
  getting a test statistic as extreme or more
  extreme than the observed one
• This chance (p-value) is computed on the basis
  that the null hypothesis is correct
• The smaller this chance is, the stronger the
  evidence against H0
• The p-value is not the chance that H0 is right
• A significance level that must be achieved to
  reject the null hypothesis is generally set
  before performing a significance test

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Type I and Type II Errors

• If the alternative hypothesis is accepted
  whenever the observed p-value is below the
  specified significance level, a, then a
  represents how often this would be done when in
  fact the null hypothesis holds (Type I error)
• Balanced against the probability, b, of not
  rejecting null hypothesis when it is false (Type
  II error)
• Commonly take (or 1)
• is often not straightforward to calculate
• Note that b increases as a decreases
• Ideally, experiments are designed, ahead of time,
  to achieve a given power

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Notes

• Required significance levels are somewhat
  arbitrary
• Even if statistically significant, results can
  still be due to chance
• With large samples, even a small difference can
  be statistically significant. That does not
  necessarily make it important. Conversely, an
  important difference may not be statistically
  significant for a small sample

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Notes

• Every legitimate test of significance involves a
  chance model. The test addresses whether the
  observed difference is real or just a chance
  variation.
• If the entire population has been surveyed, then
  a significance test is irrelevant
• If the sample can not be viewed as a random
  sample of the population, then a significance
  test is inappropriate

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Test on the Mean of a Sample

• A common problem is determining whether the group
  represented by a sample data set is significantly
  different from a specified population
• Or whether data obtained from an experiment
  represent a significant departure from a
  hypothesis
• This type of problem is often phrased in terms of
  the difference of the sample mean and a
  hypothesized mean

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Test on the Mean of a Sample

• For a large sample, the central limit theorem
  gave the sampling distribution for the sample
  mean
• With known population mean and standard
  deviation ,

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Test on the Mean of a Sample

• In practice, as long as the sample is
  approximately normal and the sample size is
  large, then often assume that the z-test
  statistic is
• And a test on the mean is conducted as if
  probability that the observed mean would have
  come from a population with mean can be
  calculated accordingly

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Test on the Mean of a Sample

• Thus, for large n, could test the hypothesis
• by calculating the probability that the
  observed mean would have come from a population
  with mean

H0 Data came from population with mean Ha
Data came from population with mean gt (lt
)
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Test on the Mean of a Sample

• A frequent gambler at the Sands Casino in Las
  Vegas believes that one of the casinos roulette
  wheels is not balanced. The gambler records 2000
  plays of the wheel. 1000 of plays of the wheel
  come up black. Is there statistically
  significant evidence that the wheel is
  unbalanced?

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Test on the Mean of a Sample

• The national average number of years that it now
  takes a college student to graduate is 5.5 years.
  A sample of 300 UNT student records shows an
  average graduation time of 6 years with an
  standard deviation of 1 year. Is the difference
  between the national average of 5.5 years and the
  UNT average of 6 years real?

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Test on the Mean of a Sample

• If the underlying distribution is known to be
  normal, then for known population mean
• where s is the sample standard deviation
• Note that for large n, the sampling distribution
  is approximately normal
• So, again, can test a hypothesis on the mean by
  calculating the probability that the observed
  mean would have come from a population with mean

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Chi-Square Test for Homogeneity

• Consider the contingency table
• Was the treatment really effective? Or, was the
  treatment useless and the results were merely the
  result of chance?
• Chi-square test for homogeniety can be applied to
  provide an answer to these questions

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Chi-square Test for Homogeneity

• The null hypothesis of the chi-square for
  homogeneity is that the row distributions are the
  same and are given by the column marginal
• That is H0 for
  all i,k, and j
• Under the null hypothesis, the maximum likelihood
  estimates for ?ji are obtained from the data by
  

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Chi-square Test for Homogeneity

• The chi-square test statistic is given by

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Chi-square Test for Homogeneity

• The distribution of X2 under the null hypothesis
  is chi-square with (I-1)(J-1) degrees of freedom
• For the polio example the X2 statistic is
  calculated using

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Chi-square Test for Homogeneity

• The chi-square test yields

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Hypothesis Test and Confidence Intervals

• One way to view a hypothesis test is as a check
  whether the value for a parameter specified in
  the null hypothesis lies in the
  confidence region (interval)
• In other words, a
  confidence region for a parameter, , consists
  of those values, , for which a null
  hypothesis that will not be
  rejected at the ? significance level

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Hypothesis Test and Confidence Intervals

• Example
• Suppose X1, X2,,Xn is a sample from a normal
  distribution having unknown mean and known
  variance . Suppose
• H0
• HA
• Let ?? be the specified significance level
• Then a hypothesis test would accept the null
  hypothesis if

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Hypothesis Test and Confidence Intervals
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Test of Equality of Means from Two Independent
Samples

• Suppose that X1, X2,..,Xn is an independent
  sequence of random variables
  and Y1, Y2,..,Ym is an independent sequence of
  random variables. Then the
  statistic
• has a t distribution with mn-2 d.f., for

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Test of Equality of Means from Two Independent
Samples

• Thus, the confidence
  interval on the difference between the means is
• Accordingly, would reject the null hypothesis
  that
• (i.e.,
  ), at the significance level for the
  two-sided test ( level if one-sided
  test of or
  ), if 0 was outside of the confidence interval

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Test of Equality of Means from Two Independent
Samples

• That is, would reject the hypothesis that
  (two-sided) at the level if
• Would reject the hypothesis that
  (one-sided) and accept the alternative
  at the level if

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Test of Equality of Means from Two Independent
Samples

• If the variances of the populations are not
  assumed equal then sampling distribution of the
  difference of the sample means is no longer a t
  distribution
• However, if the SE of the sample means difference
  is estimated by

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Test of Equality of Means from Two Independent
Samples

• Then, the statistic
• is approximately t with degrees of freedom equal
  to