Students in Miss Hampton’s science class are testing the effectiveness of detergent in getting dishes clean. They start by weighing each plate. Then they create a gooey paste of hard-to-clean foods (spaghetti sauce, mustard, mashed potatoes, peanut butter, and grape jelly) and smear $250$ clean dinner plates with a measured amount of the food paste. They weigh each plate and randomly place them into commercial dishwashers. Half the dishwashers have detergent in them, and half have only clean water. After cleaning the dishes, they weigh each plate to determine the portion of food paste that remains. $84$$\%$ of the food is removed from dishes cleaned with detergent, while only $72$$\%$ of the food is removed from dishes cleaned without detergent. Does using detergent really help get dishes cleaner?

1. What is the difference in the proportions (detergent minus plain water)? Express your answer as a decimal.

   Step (a):

   $84$$\%$ $−\ 72$$\%$ $=\:?$

   Answer (a):

   $0.12$

2. Mrs. Hampton’s class runs a computer simulation and determines the sample-to-sample variability of the difference between the proportion of food removed by the detergent compared to plain water. They conclude that the difference in the true proportion of food removed is $0.12\pm0.085$.

   Is a difference of zero a plausible result considering their margin of error? What does a difference of zero mean in the context of this problem?

   Hint (b):

   What is $0.12−0.085$?
   
   A $0$$\%$ difference indicates that there is ...

3. Are you convinced that there is a true difference between cleaning with detergent and cleaning with plain water?

   Hint (c):

   Is a difference of $0$ within the margin of error from part (b)?