The following summary statistics are available for two dependent random samples:

s₁² = 100, s₂² = 225, n = 122, r₁₂ = .50.

Test the following hypotheses at the .05 level of significance:

$\begin{array}{l}H_{0}: \sigma_{1}^{2}-\sigma_{2}^{2}=0 \\H_{1}: \sigma_{1}^{2}-\sigma_{2}^{2} \neq 0\end{array}$

Question

The following summary statistics are available for two dependent random samples:

s₁² = 100, s₂² = 225, n = 122, r₁₂ = .50.

Test the following hypotheses at the .05 level of significance:

\begin{array}{l}H_{0}: \sigma_{1}^{2}-\sigma_{2}^{2}=0 \\H_{1}: \sigma_{1}^{2}-\sigma_{2}^{2} \neq 0\end{array}

Quizplus · Accepted Answer

The Answer is The degrees of freedom (v) are equal to n - 2 = 122 - 2 = 120. The standard deviations are, respectively, 10 and 15 (i.e., the square root of the variance terms provided).

t=\frac{s_{1}^{2}-s_{2}^{2}}{2 s_{1} s_{2} \sqrt{\frac{1-r_{12}^{2}}{v}}}=\frac{100-225}{2(10) 15 \sqrt{\frac{1-(50)^{2}}{120}}}=\frac{-115}{13.693}=-8.398

_{From Appendix Table 2, and using an alpha level of .05 and degrees of freedom (}_v_{) of}₁₂_{0, we determine the critical values to be}_+/_-_1.98_{. As the test statistic}₍_-_8.398_{) exceeds the critical value, our decision is to reject the null hypothesis.}

The Following Summary Statistics Are Available for Two Dependent Random H0:σ12−σ22=0H1:σ12−σ22≠0\begin{array}{l}H_{0}: \sigma_{1}^{2}-\sigma_{2}^{2}=0 \\H_{1}: \sigma_{1}^{2}-\sigma_{2}^{2} \neq 0\end{array}H0​:σ12​−σ22​=0H1​:σ12​−σ22​=0​

The Following Summary Statistics Are Available for Two Dependent Random $\begin{array}{l}H_{0}: \sigma_{1}^{2}-\sigma_{2}^{2}=0 \\H_{1}: \sigma_{1}^{2}-\sigma_{2}^{2} \neq 0\end{array}$