Respuesta :
The test statistic value lies to the right of the critical value. So we have sufficient evidence to reject the null hypothesis.
What are null hypotheses and alternative hypotheses?
In null hypotheses, there is no relationship between the two phenomenons under the assumption or it is not associated with the group. And in alternative hypotheses, there is a relationship between the two chosen unknowns.
Students who take a statistics course are given a pre-test on the concepts and skills for the first chapter of a statistics course.
Then they are given a post-test once the professor has concluded lecturing on the material.
Pre-test and post-test scores for 4 students in an elementary statistics class are given below.
Then we have
[tex]\mu _d = \mu _{post} - \mu _{pre}[/tex]
Then the null hypotheses and alternative hypotheses will be
H₀: [tex]\mu _d = 0[/tex]
Hₐ: [tex]\mu _d > 0[/tex]
Then the test statistic will be
[tex]\rm \overline{x} _d = \dfrac{\Sigma x_d}{n} = \dfrac{15+12+10+1}{4}\\\\\overline{x} _d = 9.5[/tex]
Then
[tex]\rm S_d = 6.02[/tex]
The test statistic value is given by
[tex]\rm t = \dfrac{\overline{x} _d }{\dfrac{S_d}{\sqrtn}} \\\\t = \dfrac{9.5}{\dfrac{6.02}{\sqrt4}}\\\\t = 3.16[/tex]
Since this is a right-tailed test, so the critical value is given by
[tex]\rm t_{n-1}(\alpha ) = t_3 (0.05) = 2.353[/tex]
Since the test statistic value lies to the right of the critical value. So we have sufficient evidence to reject the null hypothesis.
Hence, we can conclude that [tex]\mu _d > 0[/tex] that is test scores have improved.
More about the null hypotheses and alternative hypotheses link is given below.
https://brainly.com/question/9504281
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