Question

A biogeographer is studying the relationship between slope aspect and tree growth. As part of her study, she tests for a statistically significant difference in tree height between a stand of trees growing on a north facing slope, and a stand of trees growing on a south facing slope. A sample of 14 trees from a north-facing slope has a sample mean height of 23.1 meters and a sample variance of 5.3. A sample of 12 trees from a south-facing slope has a sample mean height of 20.6 meters and a sample variance of 3.7. The researcher wants to know if the mean height of tress growing on north facing slopes significantly different than the mean height of trees that grow on south facing slopes.

(Population variance is assumed unequal.

Use the separate variance estimate (SVE) for the standard error. df = smaller of the two values of n1 – 1, n2 – 1) What is the probability from Table C associated with the test statistic? (Report the answer to four decimal places.)

Answer 1

t = 3.016

P-value = 0.0061

Explanation:

Let us assume that the population variance is unequal

Null hypothesis :

`H_0 :` μ
`_1` = μ
`_2`

`H_a :` μ
`_1` ≠ μ
`_2`

The alternative hypothesis :

Here we have

`n_1=14 \\n_2 = 12\\x_1 = 23.1\\s^2_1 = 5.3 \\x_2 = 20.6\\s^2_2 = 3.7`

Since it is not given that variances are equal, degrees of freedom of the test is given as

`df=(((s^2_1)/(n_1)+(s^2_2)/(n_2) )^2)/((((s^2_1)/(n_1))^2)/(n_1-1)+ (((s^2_2)/(n_2))^2)/(n_2-1)) =23`

`t=\frac{x_1-x_2}{\sqrt{(s^2_1)/(n_1)+(s^2_2)/(n_2)}} = 3.016`

The P-value of the test is = 0.0061

Since the p-value is less than 0.05, we will reject the null hypothesis