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Chapter 11: The Chi-Square Distribution

11.1 Facts About the Chi-Square Distribution

Learning Objectives

By the end of this section, the student should be able to:

  • define Chi-Square Distribution, and apply the concept to problem solving

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

For the [latex]\chi^2[/latex] distribution, the population mean is [latex]\mu = df[/latex] and the population standard deviation is [latex]\sigma =\sqrt{2(df)}[/latex].

The random variable is shown as [latex]\chi^2[/latex], but may be any uppercase letter.

The random variable for a chi-square distribution with [latex]k[/latex] degrees of freedom is the sum of [latex]k[/latex] independent, squared standard normal variables.

[latex]\chi^2 = (Z_1)^2+(Z_2)^2+...+(Z_k)^2[/latex]

  1. The curve is nonsymmetrical and skewed to the right.
  2. There is a different chi-square curve for each [latex]df[/latex].
    Part (a) shows a chi-square curve with 2 degrees of freedom. It is nonsymmetrical and slopes downward continually. Part (b) shows a chi-square curve with 24 df. This nonsymmetrical curve does have a peak and is skewed to the right. The graphs illustrate that different degrees of freedom produce different chi-square curves.
  3. The test statistic for any test is always greater than or equal to zero.
  4. When [latex]df > 90[/latex], the chi-square curve approximates the normal distribution. For [latex]\chi \sim {\chi }_{1,000}^{2}[/latex] the mean, [latex]\mu = df = 1,000[/latex] and the standard deviation, [latex]\sigma = \sqrt{2\left(1,000\right)} = 44.7[/latex]. Therefore, [latex]X \sim N(1,000, 44.7)[/latex], approximately.
  5. The mean, [latex]\mu[/latex], is located just to the right of the peak.
    This is a nonsymmetrical chi-square curve which is skewed to the right. The mean, m, is labeled on the horizontal axis and is located to the right of the curve's peak.

References

Data from Parade Magazine.

“HIV/AIDS Epidemiology Santa Clara County.” Santa Clara County Public Health Department, May 2011.

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