Question

2. You are interested in heights of trees on either side of a mountain. You choose a random
sample of ten trees on each side of the mountain and measure their heights in each
sample (Table 1).
Table 1. Heights of trees (m) on the north and south of a mountain
North. South
5.5 3.5
7.0. 3.8
5.0. 4.0
4.0 6.8
6.5 3.3
7.5 6.3
5.8. 6.5
8.0. 7.0
4.0. 6.6
5.4 2.0
Test the data to show whether trees are significantly larger on the northern side of the
mountain at the 5 % significance level. Check that the assumptions are met before choosing
the appropriate test, and give reasons for your choice of test.

Answer 1

We assume that the height of a tree has normal distribution.

We have two small independent samples $(n=10<30, m=10<30)$. Population variances are unknown. First we will prove that population variances are equal.

$H_0: \sigma_x^2=\sigma_y^2, H_1: \sigma_x^2\neq\sigma_y^2\\ \alpha=0.05\\ \text{Using Excel (STDEV.S) we find}\\ s_x^2\approx 1.887\\ s_y^2\approx 3.368\\ \text{We will use the following criterion:}\\ F=\frac{S_b^2}{S_s^2}\text{ where } S_b^2, S_s^2\text{ are sample variances of bigger and smaller}\\ \text{sample respectively}.\\ F\approx \frac{3.368}{1.887}\approx 1.785\\ k_1=n-1=10-1=9\\ k_2=m-1=10-1=9\\ F_{right_{cr}}=F_{cr}(\alpha/2;k_1;k_2)\approx 2.5265.\\ F<F_{right_{cr}}.\text{ So we accept } H_0.\\ \text{Population variances are equal.}\\ \text{Now we can test the hypothesis about population}\\ \text{means}.\\ H_0:\hat{x}_p=\hat{y}_p, H_1:\hat{x}_p>\hat{y_p}\\ \alpha=0.05\\ \text{We will use the following criterion:}\\ T=\frac{\hat{X}-\hat{Y}}{\sqrt{(n-1)S_x^2+(m-1)S_y^2}}\sqrt{\frac{nm(n+m-2)}{n+m}}\\ \text{Using Excel (AVERAGE) we find}\\ \hat{x}=5.87\\ \hat{y}=4.98\\ \text{We have } T\approx 1.228\\ k=n+m-2=18\\ t_{cr}(\alpha;k)\approx 1.7341\\ (1.7341,\infty)\text{ --- critical region}\\ T\approx 1.228 \text{ is not in the critical region. So we accept } H_0.\\ \text{At a significance level 0.05 we can say that population means are}\\ \text{equal.}$

Trees are not significantly larger on the northern side of the mountain at the 5 % significance level.