Question

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)?

Answer 1

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)

$$ \begin{aligned} &\left.\left.D F=\left(\left(S_{1}^{2} / n 1+S_{2}^{2} / n 2\right)\right)^{2} /\left(\left(S_{1}^{2} / n 1\right)^{2} / n 1-1\right)+\left(S_{2}^{2} / n 2\right)^{2} / n 2-1\right)\right) \\ &\left.\left.D F=\left(\left(5.5^{2} / 35+8.6^{2} / 40\right)\right)^{2} /\left(\left(5.5^{2} / 35\right)^{2} / 35-1\right)+\left(8.6^{2} / 40\right)^{2} / 40-1\right)\right) \\ &\mathrm{DF}=67.0 \end{aligned} $$