Do the poll's result show a statistically significant difference between the peoples preference for burger king, wendy's and mcdonalds? explain

Question

The public policy polling organization surveyed 500 registered american voters from february 21st to 24th, 2013. The margin of error for the overall sample is +/-4.4%. The surveyed conclusion was burgerking is americas favorite fast food hamburger restaurant with 22% support, followed by 19% for wendys, and 15% for mcdonalds.

Answer 1

Christina:

 

One quick and easy way to solve this problem is via simultaneous confidence intervals for multinomial proportions.  I am assuming that the sample is random with replacement so that the underlying distribution is multinomial.

 

First, count the number k of "cells".  Here k =  4, namely  "BK", "W", "M" , and "Other (O)" .

 

Find the sample proportion for "O".  It is 1 - (0.22 + 0.19 + 0.15) = 0.64

 

Your problem does NOT explain how the margin of error for the overall sample is obtained. More precisely what significance level is used. It should be the prescribed level of significance divided by k.  Let us assume that that is done correctly.  Otherwise, the answers below WILL NOT APPLY.  OK?

 

Calculate confidence intervals for each population proportion by the formula p_j plus or minus the margin of error for the overall sample, given to be 0.044.

 

BK : 0,22  minus /plus 0.044, that is, 0.176, 0.264

 

W : 0.19  minus/plus  0.044  that is 0.146, 0.234

 

M :  0.15  minus/plus 0.044  that is 0.106, 0.194

 

O : 0.64  minus/plus 0.044   that is 0.596, 0.684

 

If there is an overlap between one pair (one is enough), then the result is NOT statistically significant. Here there is.

 

In context, this means that the data does not provide sufficient evidence to claim that in at least one pair of the population proportions of "favorite foods", these proportions are significantly different. Which, in turn, means that if you consider preference to other foods (such as "In-and-Out, Arby's", etc) it can NOT be concluded at the given level of significance that BK (or M or W for that matter) is more preferred.

 

Of course, you can do  what is called a conditional test, looking only at BK, W and M, but then the observed proportions in that experimental design must add to 1.  Here they do not.

 

Christina, there are many more advanced methods of answering your question.  Take it from me, I know. My areas of specialization are discrete data analysis and information theory.  But these methods are beyond the scope of an intro course.

 

Dattaprabhakar ("Dr. G.")