2uData ⮞Hypothesis Testing ⮞Tests on the variance

Tests on the variance

Consider a population whose variance of random variable `X` is `sigma^2`. Drawing a sample with size `n`, whose variance is `s^2`. We have to decide some statements about the relation of the variance `sigma^2` with a value `a^2`, with confidence level (`1–alpha`) (or significance level `alpha`).

Tests on variance is conducted similarly as that on the mean or on the proportion:

• null hypothesis has the form : Ho : `sigma^2=a^2`
• depend on context, alternative hypothesis can be `sigma^2!=a^2` ; or `sigma^2>a^2` ; or `sigma^2< a^2`.
• test statistic is :
  

  `chi^2=((n-1)s^2)/a^2`(7)

• probability distribution of test statistic is chi-square distribution

Example

The maximum standard deviation of filled volume for a filling machine is 10 ml. The control of 20 bottles taken from company C show that the value of this parameter is 12 ml. Does this machine satisfy the requirement with significance level of 0,05 ?

From the information above:

• the population is all the bottles produced by company C
• the parameter of interest is the variance of filled volumes of bottles `X` in this population `sigma^2`,
• we have to compare `sigma^2` with `10^2=100`
• sample size `n=20`, variance of `X` in the sample is `s^2=12^2=144`

We have to conduct an hypothesis test on variance with the following steps.

• Pair of hypotheses :
•   Ho : `sigma^2=100`
•   Ho : `sigma^2>100`
• test statistic is
  

    `chi^2=((n-1)s^2)/a^2`

• probability distribution of test statistic is chi-square distribution
• `alpha=0,05` ; rejection region is in the right of `chi^2`*
• Using percentage point table of chi-square distribution: `chi^2"*"=chi_(0,05,20-1)^2=30,144`
• From the collected data :   `chi_o^2=(19xx144)/100=27,36`
• Because `chi_o^2< chi^2`*, we cannot reject Ho.
• Therefore, filling machine of company C satisfies the requirement.