Consider the following hypothesis test.
H₀: μ₁-μ₂=0
Ha: μ₁-μ₂ ≠ 0
The following results are from independent samples taken from two populations.
Sample 1 | Sample 2 |
n1 = 35 | n2 = 40 |
x̅1 = 13.6 | x̅2 = 10.1 |
s1 = 5.5 | s2 = 8.6 |
a. What is the value of the test statistic (to 2 decimals)?
b. What is the degrees of freedom for the t distribution (to 1 decimal)?
Part a)
Test Statistic:-
$$ \begin{aligned} &t=\left(\bar{X}_{1}-\bar{X}_{2}\right) / \sqrt{\left(S_{1}^{2} / n 1\right)+\left(S_{2}^{2} / n 2\right)} \\ &t=(13.6-10.1) / \sqrt{\left(5.5^{2} / 35\right)+\left(8.6^{2} / 40\right)} \\ &\mathrm{t}=2.1248 \approx 2.12 \end{aligned} $$
Part b)
$$ \begin{aligned} &\left.\left.D F=\left(\left(S_{1}^{2} / n 1+S_{2}^{2} / n 2\right)\right)^{2} /\left(\left(S_{1}^{2} / n 1\right)^{2} / n 1-1\right)+\left(S_{2}^{2} / n 2\right)^{2} / n 2-1\right)\right) \\ &\left.\left.D F=\left(\left(5.5^{2} / 35+8.6^{2} / 40\right)\right)^{2} /\left(\left(5.5^{2} / 35\right)^{2} / 35-1\right)+\left(8.6^{2} / 40\right)^{2} / 40-1\right)\right) \\ &\mathrm{DF}=67.0 \end{aligned} $$
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