Test of Significance

INSTRUCTIONS

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In this lab, we consider another way to test claims, namely, what we called "tests of significance". This method is again based on the fact that the Z statistic
                          Z=(XBar-μ)/(σ/sqrt(n))

and the one-sample t statistic
                         t=(XBar-μ)/(s/sqrt(n))

should have values close to 0 if XBar is close to μ.
Also, the two-sample t statistic
                        t= ((X1bar - X2bar) - (μ₁ - μ₂))/sqrt(s₁²/n₁ + s₂ ²/n₂)),

where the degrees of freedom
                        df = (s₁²/n₁ +s₂²/n₂)² / ((s₁²/n₁)² x 1/(n₁-1) + (s₂²/n₂)²x 1/(n₂-1))

and pooled-two-sample-t statistic
                        t = ((X1bar - X2bar) - (μ₁ - μ₂))/sqrt(S_p²/n₁ + S_p²/n₂)),

where
                       S_p² = ((n₁-1)s₁² + (n₂-1) s₂²)/(n₁ + n₂ - 2)
and the degees of freedom
                        df = n₁+ n₂ -2
should have values close to 0 if the difference between the sample means is close to the difference between the population means.
Further, the chi-squared statistic
                        Χ² = (n-1)s²/σ²
should be close to n-1 if the sample variance s² is close to the population variance σ².
Finally, the F statistic
                        F = (s₁²/σ₁²)/(s₂²/σ₂²)
should be close to 1 if the sample variances of two samples are close to the population variances of the two populations they were sampled from.