Chi-Squared confidence intervals

Chi-Squared confidence intervals


To estimate a population variance a Chi-Squared distribution is used,
• Chi-Squared: X2=(n1)s2σ2X^2=\frac{(n-1)s^2}{\sigma ^2}
nn: sample size
ss: sample standard deviation
σ\sigma: population standard deviation
(n1)(n-1): is also called “degrees of freedom”
• Chi-Square table gives critical value area to the right

The Confidence interval for the variance is given by:
(n1)s2XR2\frac{(n-1)s^2}{X_R^2} < σ2\sigma ^2 < (n1)s2XL2\frac{(n-1)s^2}{X_L^2}
  • Introduction
    What are Chi-Squared Confidence Intervals?

  • 1.
    Determining Degrees of Freedom
    How many degrees of freedom does a sample of size,
    7 have?

    20 have?

  • 2.
    Determining the Critical Value for a Chi-Square Distribution (XR2(X_R^2 and XL2)X_L^2)
    If a Chi-Squared distribution has 8 degrees of freedom find XR2X_R^2 and XL2X_L^2, with a
    95% confidence level

    99% confidence level

  • 3.
    Determining the Confidence Interval for Variance
    Road and racing bicycles have an average wheel diameter of 622mm. From a sample of 15 bicycles it was found that the wheel diameters have a variance of 10mm. With a 90% confidence level give a range where the variance of all road and racing bicycle wheels lie.

  • 4.
    Determining the Confidence Interval for Standard Deviation
    A Soda-pop company “Jim’s Old Fashion Soda” is designing their bottling machine. After making 41 bottles they find that their bottles have an average of 335mL of liquid with a standard deviation of 3mL. With a 99% confidence level what is the range of standard deviation that this machine will output per bottle?