Mathematics-Statistics

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Colonoscopy is a medical procedure that is designed to find and remove precancerous lesions in the colon before they become cancerous. In a sample of 51, 460 people without colorectal cancer, 5043 had previously had a colonoscopy, and in a sample of 10,292 people diagnosed with colorectal cancer, 720 had previously had a colonoscopy. 
Construct a 95% confidence interval for the difference in the proportions of people who had colonoscopies between those who were diagnosed with colorectal cancer and those who were not. 
Does the confidence interval contradict the claim that the proportion of people who have had colonoscopies is the same among those with colorectal cancer and those without? 

The concentration of benzene was measured in units of milligrams per liter for a simple random sample of five specimens of untreated wastewater produced at a gas field. The sample mean was 7.8 with a sample standard deviation of 1.4. Seven specimens of treatment wastewater had an average benzene concentration of 3.2 with a standard deviation of 1.7. It is reasonable to assume both samples come from populations that are approximately normal. Can you conclude that the mean benzene concentration is less in treated water than in untreated water? Use the α =0.05 level of significance. 

The National Health and Nutrition Examination Surveys (NHANES) are designed to assess the health and nutritional status of adults and children in the United States. According to a recent NHANES survey, the mean height of adult men in the United States is 69.7 inches, with a standard deviation of 3 inches. A sociologist believes that taller men may be more likely to be promoted to positions of leadership, so the mean height µ of male business executives may be greater than the mean height of the entire male population. A simple random sample of 100 male business executives has a mean height of 69.9 inches. Assume that the standard deviation of male executive heights is σ = 3 inches. Can we conclude that male business executives are taller, on the average, than the general population at the α =0.05 level?