Tests about a Population Variance

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Tests about a Population Variance
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Overview

• To frame our discussion, consider

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Outline

• Confidence Intervals for s and s2
• Hypothesis Testing One Variance

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• Typically interval estimates for s and s2 are
  based on s and s2.

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• For random samples from normal populations,
• Is a random variable having the chi-square (c2)
  distribution with n-1 degrees of freedom.

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• Like the t-distribution, the actual c2
  distribution is based on the degrees of freedom
  (typically dfn-1).

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Confidence Intervals

• Taking the square root provides the confidence
  interval for s.

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Problem

• You produced an algorithm for determining the
  area of polygonal regions. Your data regarding
  the area of a sample of 50 cm2 regions is
• 51.2, 47.5, 50.8, 51.5, 51.3, 49.5, 51.1, 50.7,
  46.7, 49.2, 52.1, 48.3, 51.6, 49.2, 51.5
• Determine a 90 confidence interval for s2

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Hypothesis Testing

• For many applications, we are concerned the
  standard deviation or variance of the population
  exceeds some specified value.

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• We can base our hypothesis tests on the fact that
  for random samples from a normal population the
  sampling variance is a random variable having
  chi-square distribution with n-1 degrees of
  freedom.

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Critical Regions
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Clarification
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Problem

• A production manager must maintain the standard
  deviation of the diameter of hard disk media to
  less than 2mm. A random sample of 26 disks
  reveal a standard deviation of 1.85mm. If the
  disks diameters are normally distributed, does
  the data indicate that the standard deviation is
  less than 2mm? Use a.05.