Question

The National Restaurant Association reports that 40% of adults living in the U.S. prefer nutritious food over non-nutritious food when eating out at restaurants (even if the non- nutritious food tastes better.)

Joyce owns a restaurant in Iowa City. To plan her restaurant’s menu, she’s curious to know whether or not the national 40% figure holds true for Iowa City residents also. So she randomly surveys Iowa City adults with the question:

Do you prefer nutritious food over non-nutritious food when eating out at restau- rants, even if the non-nutritious food tastes better?

and records an answer of “Yes” or “No” from each survey respondent.

After completing the survey, Joyce also records the survey result x:
x = number of surveyed Iowa City adults who prefer nutritious food at restaurants

Suppose that Joyce surveys 500 Iowa City adults.

What’s the minimum number for x in almost all such surveys, assuming that the national 40% figure applies to Iowa City residents also? (Round to the nearest person.)

Answer 1

Standard error of proportion, SE = = 0.0219089

We know that the sampling distribution of proportion for large sample is Normal dustribution. By Empirical rule, almost all data lies withing 3 standard deviations from the mean.

The proportions of adults living in the U.S. prefer nutritious food will lie between p - 3 *SE and p + 3 * SE

The minimum value of proportion is 0.4 - 3 * 0.0219089 = 0.3342733

Minimum value for X = 500 * 0.3342733 = 167.1367 167

The minimum number for x in almost all such surveys would be 167, assuming that the national 40% figure applies to Iowa City residents.