Question

Conservationists have despaired over destruction of tropical rain forest by logging, clearing, and burning. These words begin a report on a statistical study of the effects of logging in Borneo. Here are data on the number of tree species in 12 unlogged forest plots and 9 similar plots logged 8 years earlier:

Unlogged: 22, 18, 22, 20, 15, 21, 13, 13, 19, 13, 19, 15

Logged: 17, 4, 18, 14, 18, 15, 15, 10, 12

Does logging significantly reduce the mean number of species in a plot after 8 years? Do a complete hypothesis test at 5% level of signicance using critical value approach.

Answer 1

The data is:

	Unlogged	Logged
	22	17
	18	4
	22	18
	20	14
	15	18
	21	15
	13	15
	13	10
	19	12
	13
	19
	15
count	12	9
Average	17.5	13.66667
SD	3.5291	4.50

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​ = μ2​

Ha: μ1​ > μ2​

This corresponds to a right-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df=19. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal.

Hence, it is found that the critical value for this right-tailed test is tc​=1.729, for α=0.05 and df = 19.

The rejection region for this right-tailed test is R={t:t>1.729}.

(3) Test Statistics

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:

(4) The decision about the null hypothesis

Since it is observed that t=2.191>tc​=1.729, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0205, and since p=0.0205<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean μ1​ is greater than μ2​, at the 0.05 significance level.

Graphically

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