A study is done to determine if drinking coffee before taking a math exam has an impact on performance. The experiment is designed as follows: 

1 class of thirty students is treated as a control group and are instructed to not drink any coffee before the exam. 

A separate class of thirty-two students is treated as the experimental group: each student is given a cup of coffee and told to drink it before their exam. 

We take as our null hypothesis that there is no difference in performance whether or not the students drink coffee and 5% as our significance level.

1.After the test, it is found that the average score of students from the control group is 57.4 with a standard deviation of 12. It is also found the average score of students from the experimental group is 54.1 with a standard deviation of 7.

(a)Calculate the test statistic corresponding to these values. Show the formula with proper values plugged in.

(b)What value should you use as your degrees of freedom to find the p-value?

2.You may assume the correct p-value is 0.19388. 

(a)What should we conclude about the situation? Use the context of coffee and students; not just "reject" or "not reject."

(b)The test statistic associated with a p-value of 0.05 is2.045 in this scenario. What is the highest the experimental group could average on the test, assuming the same standard deviation, while still having the researchers come to the conclusion to not reject the null hypothesis?