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A fast food company uses two management-training methods. Method 1 is a traditional method of training and Method 2 is a supposedly new and innovative method. The company has just hired thirty-one (31) new management trainees. Fifteen (15) of the trainees are randomly selected and assigned to Method 1 and the remaining 16 trainees are assigned to Method 2. After three months of training, the management trainees took a standardized test. The test was designed to evaluate their performance and learning from training. The sample mean score and sample standard deviation of the two methods are given below. The management wants to determine if the company should implement the new training method. 

Conduct an appropriate hypothesis test at the 5% level of significance to determine if there is a difference in the average score between the two methods. Conduct this test using (i) the p-value method, (ii) the critical value method, and (iii) the confidence interval method. Give the conditions under which this test is valid.