Question

In a study on the role of music and the growth of tomato plants, you planted 32 tomato plants. Half of the plants grew up without music, while the other half grew up with music. When the plants reached their full maturity you measured them and you obtained as an average 19.54 cm and 23.39 cm for the group without music and the group with music respectively. In addition, you have calculated the standard deviations of the two groups which are 3.68 cm for the group without music and 4.93 cm for the group with music. From these results, what conclusion can you draw for a 5% threshold?

Answer 1

The provided sample means are shown below:

Also, the provided sample standard deviations are:

and the sample sizes are

n1​=16

n2​=16.

(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=0.05, and the degrees of freedom are df=30. In fact, the degrees of freedom are computed assuming that the population variances are equal.

Hence, it is found that the critical value for this two-tailed test is tc​=2.042, for α=0.05 and df=30.

The rejection region for this two-tailed test is R={t:∣t∣>2.042}.

(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∣=2.503>tc​=2.042, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.018, and since p=0.018<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 different than μ2​, at the 0.05 significance level.

Graphically

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