You are considering a new delivery system and wish to test whether delivery times are significantly different, on average than your current system. It is well established that the mean delivery time of the current system is 2.38 days. A test of the new system shows that, with 48 observations, the average delivery time is 1.91 days with a standard deviation of 0.43 day. 

a. Identify the null and research hypotheses for a two-sided test, using both words and mathematical symbols. 

b. Perform a two-sided test at the 5% significance level and describe the result. 

c. Perform a two-sided test at the 1% significance level and describe the result. 

d. State the p-value as either p>0.05, p<0.05, p<0.01, or p<0.001. 

e. Summarize the results in a brief memo to management