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Confidence interval for difference in two unknown normal means, variances unknown

  1. Apr 7, 2012 #1
    I have two samples, one with 21 measurements and one with 23 measurements, from a Normal distribution and I am constructing a 95% CI for the difference in the two means. I do not know the variances of the two Normal distributions.

    I know that [itex]\overline{X_1}-\overline{X_2}[/itex]~[itex]N(\mu_1-\mu_2, \sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}})[/itex]

    But then as [itex]\sigma_1[/itex],[itex] \sigma_2[/itex] are unknown I use the sample variances, but then does it then follow that,

    [itex]\frac{(\overline{X_1}-\overline{X_2})-(\mu_1-\mu_2)}{\sqrt{\frac{S_1^2}{n_1}+\frac{S_2^2}{n_2}}}≈N(0,1)[/itex]

    Even though my samples are both less than 30 measurements?
     
    Last edited: Apr 7, 2012
  2. jcsd
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