{"id":421,"date":"2016-04-21T22:43:39","date_gmt":"2016-04-21T22:43:39","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstats1xmaster\/?post_type=chapter&p=421"},"modified":"2016-04-21T22:43:39","modified_gmt":"2016-04-21T22:43:39","slug":"facts-about-the-chi-square-distribution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-suffolk-introstats1\/chapter\/facts-about-the-chi-square-distribution\/","title":{"raw":"Facts About the Chi-Square Distribution","rendered":"Facts About the Chi-Square Distribution"},"content":{"raw":"

The notation for the chi-square distribution<\/strong> is\u00a0[latex]\\displaystyle\\chi\\sim\\chi^2_{df}[\/latex],\u00a0where df<\/em> = degrees of freedom which depends on how chi-square is being used. (If you want to practice calculating chi-square probabilities then use [latex]\\displaystyle{df}=n-1[\/latex]. The degrees of freedom for the three major uses are each calculated differently.)\n\nFor the \u03c7<\/em>2<\/sup> distribution, the population mean is \u03bc = df<\/em> and the population standard deviation is [latex]\\displaystyle\\sigma_{\\chi^2}=\\sqrt{2(df)}[\/latex].\n\nThe random variable is shown as \u03c7<\/em>2<\/sup>, but may be any upper case letter.\n\nThe random variable for a chi-square distribution with k degrees of freedom is the sum of k independent, squared standard normal variables.\n\n[latex]\\displaystyle\\chi^2=(Z_1)^2+(Z_2)^2+\\dots+(Z_k)^2[\/latex]\n<\/p>

  1. The curve is nonsymmetrical and skewed to the right.<\/li>\n\t
  2. There is a different chi-square curve for each df.\n\"Part<\/li>\n\t
  3. The test statistic for any test is always greater than or equal to zero.<\/li>\n\t
  4. When df<\/em> > 90, the chi-square curve approximates the normal distribution. For [latex]\\displaystyle{X}\\sim\\chi^2_{1,000}[\/latex] the mean, [latex]\\displaystyle\\mu=df=1,000[\/latex]\u00a0and the standard deviation, [latex]\\displaystyle\\sigma=\\sqrt{2(1,000)}[\/latex]. Therefore, [latex]\\displaystyle{X}\\sim{N}(1,000, 44.7)[\/latex], approximately.<\/li>\n\t
  5. The mean, \u03bc, is located just to the right of the peak.\n\"This<\/li>\n<\/ol>

    References<\/h2>\n

    Data from Parade Magazine<\/em>.<\/p>\n

    \u201cHIV\/AIDS Epidemiology Santa Clara County.\u201dSanta Clara County Public Health Department, May 2011.<\/p>\n\n

    Concept Review<\/h2>\nThe chi-square distribution is a useful tool for assessment in a series of problem categories. These problem categories include primarily (i) whether a data set fits a particular distribution, (ii) whether the distributions of two populations are the same, (iii) whether two events might be independent, and (iv) whether there is a different variability than expected within a population.\n\nAn important parameter in a chi-square distribution is the degrees of freedom df in a given problem. The random variable in the chi-square distribution is the sum of squares of df standard normal variables, which must be independent. The key characteristics of the chi-square distribution also depend directly on the degrees of freedom.\n\nThe chi-square distribution curve is skewed to the right, and its shape depends on the degrees of freedom df. For df > 90, the curve approximates the normal distribution. Test statistics based on the chi-square distribution are always greater than or equal to zero. Such application tests are almost always right-tailed tests.\n

    Formula Review<\/h2>\n[latex]\\displaystyle\\chi^2=(Z_1)^2+(Z_2)^2+\\dots(Z_{df})^2[\/latex] chi-square distribution random variable\n\n[latex]\\displaystyle\\mu_{\\chi^2}=df[\/latex] chi-square distribution population mean\n\n[latex]\\displaystyle\\sigma_{\\chi^2}=\\sqrt{2(df)}[\/latex] Chi-Square distribution population standard deviation","rendered":"

    The notation for the chi-square distribution<\/strong> is\u00a0[latex]\\displaystyle\\chi\\sim\\chi^2_{df}[\/latex],\u00a0where df<\/em> = degrees of freedom which depends on how chi-square is being used. (If you want to practice calculating chi-square probabilities then use [latex]\\displaystyle{df}=n-1[\/latex]. The degrees of freedom for the three major uses are each calculated differently.)<\/p>\n

    For the \u03c7<\/em>2<\/sup> distribution, the population mean is \u03bc = df<\/em> and the population standard deviation is [latex]\\displaystyle\\sigma_{\\chi^2}=\\sqrt{2(df)}[\/latex].<\/p>\n

    The random variable is shown as \u03c7<\/em>2<\/sup>, but may be any upper case letter.<\/p>\n

    The random variable for a chi-square distribution with k degrees of freedom is the sum of k independent, squared standard normal variables.<\/p>\n

    [latex]\\displaystyle\\chi^2=(Z_1)^2+(Z_2)^2+\\dots+(Z_k)^2[\/latex]\n<\/p>\n

      \n
    1. The curve is nonsymmetrical and skewed to the right.<\/li>\n
    2. There is a different chi-square curve for each df.
      \n\"Part<\/li>\n
    3. The test statistic for any test is always greater than or equal to zero.<\/li>\n
    4. When df<\/em> > 90, the chi-square curve approximates the normal distribution. For [latex]\\displaystyle{X}\\sim\\chi^2_{1,000}[\/latex] the mean, [latex]\\displaystyle\\mu=df=1,000[\/latex]\u00a0and the standard deviation, [latex]\\displaystyle\\sigma=\\sqrt{2(1,000)}[\/latex]. Therefore, [latex]\\displaystyle{X}\\sim{N}(1,000, 44.7)[\/latex], approximately.<\/li>\n
    5. The mean, \u03bc, is located just to the right of the peak.
      \n\"This<\/li>\n<\/ol>\n

      References<\/h2>\n

      Data from Parade Magazine<\/em>.<\/p>\n

      \u201cHIV\/AIDS Epidemiology Santa Clara County.\u201dSanta Clara County Public Health Department, May 2011.<\/p>\n

      Concept Review<\/h2>\n

      The chi-square distribution is a useful tool for assessment in a series of problem categories. These problem categories include primarily (i) whether a data set fits a particular distribution, (ii) whether the distributions of two populations are the same, (iii) whether two events might be independent, and (iv) whether there is a different variability than expected within a population.<\/p>\n

      An important parameter in a chi-square distribution is the degrees of freedom df in a given problem. The random variable in the chi-square distribution is the sum of squares of df standard normal variables, which must be independent. The key characteristics of the chi-square distribution also depend directly on the degrees of freedom.<\/p>\n

      The chi-square distribution curve is skewed to the right, and its shape depends on the degrees of freedom df. For df > 90, the curve approximates the normal distribution. Test statistics based on the chi-square distribution are always greater than or equal to zero. Such application tests are almost always right-tailed tests.<\/p>\n

      Formula Review<\/h2>\n

      [latex]\\displaystyle\\chi^2=(Z_1)^2+(Z_2)^2+\\dots(Z_{df})^2[\/latex] chi-square distribution random variable<\/p>\n

      [latex]\\displaystyle\\mu_{\\chi^2}=df[\/latex] chi-square distribution population mean<\/p>\n

      [latex]\\displaystyle\\sigma_{\\chi^2}=\\sqrt{2(df)}[\/latex] Chi-Square distribution population standard deviation<\/p>\n\n\t\t\t

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      Candela Citations<\/h3>\n\t\t\t\t\t
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