INTERVAL ESTIMATION - TWO SAMPLE FORMULAS Confidence Interval for the by sja20118

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									INTERVAL ESTIMATION – TWO SAMPLE FORMULAS

Confidence Interval for the Difference between Two Proportions
                                   ) )    ) )                                                  ) )    ) )
           )    )                  p1 q1 p 2 q 2                    )    )                     p1 q1 p 2 q 2
         ( p1 − p 2 ) − zα / 2          +        p ( p1 − p 2 ) p ( p1 − p 2 ) + zα / 2             +
                                    n1     n2                                                   n1     n2

                )               )
(Verify that np          and   nq are greater than 5, for both samples. Use the sample estimators)

Confidence Interval for the Difference between Two Means in case of
Independent Samples.
     General Case
                                                                    2
                                                              s12 s 2                                          2
                                                                                                         s12 s 2
                                      ( x1 − x 2 ) − tα / 2      +    p µ1 − µ 2 p ( x1 − x 2 ) + tα / 2    +
                                                              n1 n2                                      n1 n2

Recommended formula to be used when both populations are normally distributed or for large samples of non-
normal populations. The recommended degrees of freedom d.f. for t is the smaller of n1 − 1 and n 2 − 1
(conservative approach).
         A particular Case: Both Populations have Equal Variances

                                                                    1  1                                          1  1
                                      ( x1 − x 2 ) − tα / 2 s 2 (
                                                              p       + ) p µ1 − µ 2 p ( x1 − x 2 ) + tα / 2 s 2 ( + )
                                                                                                               p
                                                                    n1 n2                                         n1 n 2

               (n1 − 1) s12 + (n2 − 1) s 2
                                         2
where s p =
          2

                     n1 + n2 − 2
Recommended formula that requires the additional assumption that                 σ 12 = σ 2 . In this case de degrees of freedom
                                                                                          2


is d . f . = n1 + n 2 − 2



Confidence Interval for Matching Pairs

                    sd                       sd
         d − tα          p µd p d + tε
                2    n                  2     n

The sample consists of n matching pairs (dependent samples). Recommended formula to be used when the
population of differences is normally distributed or it is a large sample of a non-normal population. The degrees of
freedom here is d . f . = n − 1

								
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