Question

Plant	Sample size	Sample mean	Sample sd
A	50	50.3	0.2
B	50	50.7	0.3

(a) (7 points) Is there evidence, at 5% significance level, to show that the mean diameter of these two plants are different? Write down the hypotheses, test statistic, P-value and your conclusion.

Answer 1

The sample means are shown below:

Also, the sample standard deviations are:

and the sample sizes are n1​=50 and n2​=50.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​ = μ2​. The mean diameter of these two plants is the same.

Ha: μ1​ ≠ μ2​. The mean diameter of these two plants is different.

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.

Testing for Equality of Variances

A F-test is used to test for the equality of variances. The following F-ratio is obtained:

The critical values are FL​=0.567 and FU​=1.762, and since F = 0.444, then the null hypothesis of equal variances is rejected.

(2) Rejection Region

The significance level is α=0.05, and the degrees of freedom are df = 85.371. In fact, the degrees of freedom are computed as follows, assuming that the population variances are unequal:

Hence, it is found that the critical value for this two-tailed test is tc​=1.988, for α=0.05 and df = 85.371.

The rejection region for this two-tailed test is R = { t : ∣t∣ >1.988}.

(3) Test Statistics

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

(4) Decision about the null hypothesis

Since it is observed that ∣t∣ = 7.845 > tc ​= 1.988, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p =5.42416e-12 ~ 0, and since p = 0 < 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 the population mean μ1​ is different than μ2​, at the 0.05 significance level.

Therefore, there is enough evidence to claim that the mean diameter of these two plants is different at the 0.05 significance level.