Question

1. How effective is gastric bypass surgery in maintaining weight loss in extremely obese people? A Utah-based study conducted between 2000 and 2011 found that 76% of 418 subjects who had received gastric bypass surgery maintained at least a 20% weight loss six years after surgery.
a. Calculate a 90% confidence interval for the proportion of those receiving gastric bypass surgeries that maintained at least a 20% weight loss six years after surgery. (4 pts)
b. Interpret your interval in the context of the problem – what does the confidence interval mean in regards to this problem?

Answer 1

$a) \hat{p}=0.76\text{ --- point estimate of population proportion}.\\ \alpha=0.1.\\ z_{\alpha/2}=z_{0.05}=1.64.\\ n=418.\\ p=\hat{p}\pm[z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}].\\ p=0.76\pm[1.64\sqrt{\frac{0.76(1-0.76)}{418}}].\\ (0.726;0.794)\text{ --- our confidence interval}.\\ b)\text{We can say that CI } (0.726;0.794) \text{ covers population proportion }\\ \text{with probability } 0.9. \text{ It means that proportion of those receiving}\\ \text{gastric bypass surgeries falls into this interval with probability } 0.9.\\$