Question

You’re running an experiment to determine whether students perform better if they are rewarded with candy or vegetables. You sample 5 students from your class and you first reward the students with vegetables when they get right answers in the class. Then you test them and record their scores. Next, with a different 5 students you reward these students with cookies when they get right answers in the class. Then you test them with another test and record their scores. The following table depicts grades for the two conditions:

Candy	Vegetables
3.5	2.5
3	2.5
3.75	3
4	2.75
4	3

The standard deviation for the candy group is 0.25, and the mean is 3.75. The standard deviation for vegetable group is 0.25, and the mean is 2.75.

a. Identify the name of the test you will be employing.

b. Conduct a formal null hypothesis significance test

Answer 1

The provided sample means are shown below:

Xˉ1​=3.75

Xˉ2​=2.75

Also, the provided sample standard deviations are:

s_1 = 0.25

s2​=0.25

and the sample sizes are n_1 = 5 and n_2 = 5

.

(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 = 8

Hence, it is found that the critical value for this right-tailed test is

t_c = 1.86

The rejection region for this right-tailed test is

R={t:t>1.86}.

(3) Test Statistics

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

​​=

(4) Decision about the null hypothesis

Since it is observed that

t=6.325>tc​=1.86,

it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is p = 0.0001,

and since p = 0.0001<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 greater than μ2​, at the 0.05 significance level.

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