Null Hypothesis for Tests of Differences in Means and Proportions

Discussion 1: Coaching companies claim that their courses can raise the SAT scores of high school students. Of course, students who retake the SAT without paying for coaching generally raise their scores, too. A random sample of students who took the SAT twice found 427 who were coached and 2733 who were uncoached.19 Starting with their Verbal scores on the first and second tries, we have these summary statistics:
The summary statistics for Gain are based on the changes in the scores of the individual students. Let’s first ask if students who are coached increased their scores significantly. You could use the information on the Coached line to carry out either a two-sample t-test comparing Try 1 with Try 2 for coached students, or a matched pairs t-test using Gain. Which is the correct test? Why? Carry out the proper test. What do you conclude?
Discussion 2: What we really want to know is whether coached students improve more than uncoached students and whether any advantage is large enough to be worth paying for. Use the information in the previous exercise to answer these questions. Is there good evidence that coached students gained more on average than uncoached students? How much more do coached students gain on average? Based on your work, what is your opinion: do you think coaching courses are worth paying for?
Discussion 3: The data you used in the previous two problems came from a random sample of students who took the SAT twice. The response rate was 63%, which is pretty good for nongovernment surveys, so let’s accept that the respondents do represent all students who took the exam twice. Nonetheless, we can’t be sure that coaching caused the coached students to gain more than the uncoached students. Explain briefly but clearly why this is so.
Discussion 4: The Abecedarian Project is a randomized controlled study to assess the effects of intensive early childhood education on children who were at high risk based on several sociodemographic indicators. The project randomly assigned some children to a treatment group that was provided with early educational activities before kindergarten and the remainder to a control group. A recent follow-up study interviewed subjects at age 30 and evaluated educational, economic, and socio-emotional outcomes to learn if the positive effects of the program continued into adulthood. The follow-up study included 52 individuals from the treatment group and 49 from the control group. Out of these, 39 from the treatment group and 26 from the control group were considered “consistently” employed (working 30+ hours per week in at least 18 of the 24 months prior to the interview). Does the study provide significant evidence that children who had early childhood education have a higher proportion of consistent employment than those who did not? How large is the difference between the proportions in the two populations that are consistently employed? Do inference to answer both questions. Be sure to explain exactly what inference you choose to do.