Question

A study published in JAMA in 2004 examined past results of other studies on bariatric surgery. Bariatric surgery is done to replace the size of the stomach in various ways. It is typically used on only obese patients, and one the traditional form of the surgery, gastric bypass has a 1% mortality (death) rate caused by the surgery. However, in the studies reporting on the effects of this surgery on Type II diabetes, 1417 of 1846 diabetic patients recovered completely from diabetes after surgery. These patients no longer need medication like insulin (which is injected into the skin) or pills to help manage blood sugar.

(a) What is the percentage of patients who recovered from diabetes in 2004?

(b) What is the 90%, 95% and 99% Confidence Interval for the percentage of recovery from diabetes (2004) data? (to four decimal places for the proportion - two decimal places if you give your answer as a percentage). In a study released a decade later, the percentage of patients who recover from diabetes has increased to 80%. W will determine whether this is a significant change from the 2004 data. The sample size in 2014 was 2000.

(c) What is the percentage of patients who recovered from diabetes in 2014? What is the number of patients in the sample that have recovered?

(d) What is the 90%, 95% and 99% Confidence Interval for the percentage of recovery from diabetes (2014) data? (to four decimal places for the proportion - two decimal places if you give your answer as a percentage).

(e) Construct the 95% confidence interval for the difference of the two proportions. Does the interval appear to indicate a significant difference in the two intervals. (Remember when that a confidence interval for the difference of two proportions contains a zero - it is permissible that there is no difference between the two.) Do the results change at the 90% confidence interval?

(f)Test the claim that by performing a hypothesis test (at the 5% level of significance) that the proportion of patients is greater in the second study. State your null and alternative hypothesis. Use technology to calculate your p-value. Do these results change at difference levels of significance?

(g) If you had a morbidly obese relative with diabetes what would you tell him or her about bariatric surgery? Explain. Would you recommend it? Why/why not?

Answer 1

a)

b)

90% Confidence interval

The confidence interval for the proportion is obtained using the following formula,

Where,

95% Confidence interval

99% Confidence interval

In 2014, the proportion of patients who recover from diabetes = 0.80

Since the proportion of patients who recovered from diabetes in 2014 is outside the upper confidence interval limit for all of the three confidence interval limits, there is a significant increase in the proportion of patients who recovered from diabetes.

c)

In 2014, the percentage of patients who recover from diabetes = 80%

Sample size = 2000

d)

90% Confidence interval

95% Confidence interval

99% Confidence interval

e)

The 95% Confidence Interval of the difference of two proportions is obtained using the following formula,

Conclusion: Since the confidence interval doesn't contain zero, there is a significant difference between the two proportions.

f)

Hypothesis:

The null and alternative hypothesis are,

This a left-tailed test

The significance level,

Test statistic

The z-test is used to compare two population proportions.

The z-statistic is,

Where,

P-value

The P-value for z = 0.5158 is obtained in excel using the function =NORM.S.DIST(-2.441,TRUE)

Since the p-value is less than 0.05, the null hypothesis is rejected.

The result can be changed if we reduce the significance level such that at significance level = 0.005 we can not reject the null hypothesis.