In a project for a statistics class, a group of students decided to compare a name brand (Brand 1) and a store brand (Brand 2) of microwave popcorn to see which one left fewer unpopped kernels, on average, after the recommended popping time. They knew that microwave ovens can differ, so they asked each of the 50 students in the class to take one bag of each brand home and conduct the test, then report the number of unpopped kernels from each brand.
a. Define the parameter of interest in the context of this situation. Use appropriate notation.
b. Describe the sample statistic of interest in the context of this situation. Use appropriate notation.
c. Would your answers in parts (a) and (b) change if the students had randomly assigned half of the class to test one brand and the other half to test the other brand? Explain.
d. Suppose that, in truth, the store brand leaves an average of ten fewer unpopped kernels and that the standard deviation for the population of differences is five kernels. Describe the sampling distribution of the statistic you defined in part (b), giving numerical values to the extent possible.
e. Draw a picture of the sampling distribution you found in part (d).
f. Using the information provided in part (d) would you be surprised if a sample of 50 differences collected in this way resulted in a higher mean number of unpopped kernels for the store brand than for the name brand? Explain. You can explain in words or use your figure from part (e) to illustrate the situation.