Using the z-distribution, the test statistic is:
c) -2.63.
What are the hypothesis tested?
At the null hypothesis we test if the means are equal, equivalent to a subtraction of 0, hence:
[tex]H_0: \mu_D - \mu_C = 0[/tex]
At the alternative hypothesis, we test if they are different, hence:
[tex]H_1: \mu_D - \mu_C \neq 0[/tex]
What are the mean and the standard error for the distribution of differences?
For each sample, they are given as follows:
- [tex]\mu_D = 12, s_D = \frac{5.2}{\sqrt{73}} = 0.6086[/tex].
- [tex]\mu_C = 14, s_C = \frac{4.1}{\sqrt{81}} = 0.4556[/tex].
For the distribution of differences, it is given by:
- [tex]\overline{x} = \mu_D - \mu_C = 12 - 14 = -2[/tex].
- [tex]s = \sqrt{s_D^2 + s_C^2} = \sqrt{0.6086^2 + 0.4556^2} = 0.76[/tex]
What is the test statistic?
The test statistic is given by:
[tex]z = \frac{\overline{x} - \mu}{s}[/tex]
In which [tex]\mu = 0[/tex] is the value tested at the null hypothesis.
Hence:
[tex]z = \frac{\overline{x} - \mu}{s}[/tex]
[tex]z = \frac{-2 - 0}{0.76}[/tex]
z = -2.63.
More can be learned about the z-distribution at https://brainly.com/question/13873630
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