Students in Miss Hampton’s science class tested the effectiveness of detergent in getting dishes clean. They started by weighing each plate. Then they created a gooey paste of hard-to-clean foods (spaghetti sauce, mustard, mashed potatoes, peanut butter, and grape jelly) and smeared $250$ clean dinner plates with a measured amount of the food paste. They weighed each plate and randomly placed them into commercial dishwashers. Half the dishwashers had detergent in them, and half had only clean water. After cleaning the dishes, they weighed each plate to determine the portion of food paste that remained. $84\%$ of the food was removed from dishes cleaned with detergent, while only $72\%$ of the food was 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):

   $84\%-72\%=?$

2. Mrs. Hampton’s class ran a computer simulation and determined the sample-to-sample variability of the difference between the proportion of food removed by the detergent compared to plain water. They concluded that the difference in the true proportion of food removed was $0.12 ± 0.085$.

   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)?