• Check the conditions for normal approximation of a sampling distribution of a sample proportion.
• Use the normal distribution to calculate probabilities and percentiles from a sampling distribution.
• Find the sample size needed for a sampling distribution to have a desired standard deviation.

Obesity Rates

In 2017–2018, the National Health and Nutrition Examination Survey estimated that [latex]42.4\%[/latex] of American adults fit the medical definition of obese.^[1] Overweight and obesity are defined as abnormal or excessive fat accumulation that presents a health risk.^[2] Body mass index (BMI) was calculated as weight in kilograms divided by height in meters squared, rounded to one decimal place. A body mass index (BMI) over [latex]25[/latex] is considered overweight, over [latex]30[/latex] is obese, and severe obesity has a BMI of greater than or equal to 40.

National Center for Health Statistics^[3] stated that obesity is associated with serious health risks. Severe obesity further increases the risk of obesity-related complications, such as coronary heart disease and end-stage renal disease.

A large medical clinic instituted a wellness and nutrition program for its patients, where patients could opt in to receive text messages with nutrition and exercise tips or use an app to monitor their diets and activity levels. This clinic would like to determine if, after a year of the program, the proportion of its patients who are obese is less than the national average.

Let’s explore the sampling distribution of sample proportions for varying values of the population proportion and sample size.

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