Question

A tree researcher wishes to test the claim made by a colleague that trees of a certain species in a forest to the north are the same height as the same species in a forest to the south. The researcher collected samples from both forests. In the northern forest, the sample size was 9 trees, the sample average was 32.7 ft, and the sample standard deviation was 6.9 ft. In the southern forest, the sample size was 12 trees, the sample average was 35.4 ft, and the sample standard deviation was 5.6 ft. With alpha = 0.1, test the claim.

Answer 1

The provided sample means are shown below:

Also, the provided sample standard deviations are:

and the sample sizes are

n1​=9 and

n2​=12.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:

Ha: ​

This corresponds to a two-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=0.10, 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 two-tailed test is tc​=1.729, for α=0.10 and df=19.

The rejection region for this two-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∣=0.991≤tc​=1.729, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.3343, and since p=0.3343≥0.10, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1​ is different than μ2​, at the 0.10 significance level.

Graphically

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