Question

A new SAT study course is tested on 12 individuals. Pre-course and post-course scores are recorded. Of interest is the average increase in SAT scores. The following data is collected. Conduct a hypothesis test at the 5% level. Pre-course score NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

Pre-course score	Post-course score
1230	1340
940	930
1090	1140
840	880
1100	1070
1250	1320
860	860
1330	1370
790	770
990	1040
1110	1200
740	850

PART A) State the null hypothesis.  H₀: μ_d ≤ 0

PART B) State the alternative hypothesis. H_a: μ_d = 0

PART C) In words, state what your random variable X_d represents.-->The variable X_d represents the sample mean difference in SAT scores before the course and after the course.  

PART D) State the distribution to use for the test. t={12-1}

PART E) What is the test statistic?

PART F) What is the p-value?  If

H₀ is true, then there is a chance equal to the p-value that the sample average difference between the post-course scores and pre-course scores is at least 41.67.

PART G) Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value

PART H) Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion.

α = 0.05. reject the null hypothesis. Since p-value < α, we reject the null hypothesis.   

There is sufficient evidence to show that the average post-course SAT score is larger than the average pre-course SAT score.

PART I) Explain how you determined which distribution to use.

Answer 1