Problem Solving. Solve the following problems. Show all pertinent solutions. No immediate rounding. Round off final answers to three decimal places.

{optional} Some people claim that they can tell the difference between a diet soda and a regular soda in the first sip. A researcher wanting to test this claim randomly sampled 80 such people. He then filled 4U plain white cups with diet soda, and another 40 cups with regular soda. Each person was randomly assigned a cup. They were asked to take one sip from their cup and identify the soda as diet or regular. 53 participants correctly identified the soda. Does the data provide evidence that these people are able to detect the difference between diet and regular soda? In other words, are the results significantly better than just random guessing? Use a = 0.05