7.52 Fats and Sweets According to a study in the American Journal of Public Health, diets high in fat, sugar, and grains were associated with lower diet costs after adjustment for energy intakes, gender, and age. For most levels of energy intake, each additional 100 g of fat and sweets was associated with a 0.05-0.40 per day reduction in diet costs. In contrast, each additional 100 g of fruit and vegetables was associated with 0.18-0.29 per day increase in diet costs. The study was based on freely chosen diets of 837 French adults that were assessed by a dietary history method. Mean national food prices for 57 foods were used to estimate diet costs. Suppose a random sample of 100 French adults were selected and 46% of them eat fat and sweets due to cost factors.

a. Does the distribution of p, the sample proportion of French adults who eat fats and sweets, have an approximately normal distribution? If so, what are its mean and standard deviation?

b. What is the probability that the sample proportion, p. exceeds 0.5?

c. What is the probability that p lies within the interval 0.35 to 0.55?

d. What might you conclude if the sample proportion were as small as 30%?