Question

researchers at Vanderbilt University were curious about whether consumers are able to use the information on nutritional labels. They surveyed 200 primary care patients at the University Medical Center. In one question, respondents were asked to calculate the number of grams of carbohydrates in a single serving of soda, using the information on the label of a 20-ounce bottle (2.5 servings). Only 32% answered correctly. With 95% confidence, what is the margin of error in this estimate? ______

If the researchers want to reduce the margin of error to .05 or less, with 95% confidence, how many patients should they survey? (Use p* = 0.32 from the survey above)_____

DMPA is an injectable contraceptive drug that lasts for 3 months. In a study of birth control methods, 902 users experienced 2 unintended pregnancies. Give a 90% confidence interval for the probability of an unintended pregnancy for users of DMPA using the plus four method.

lower limit =

Upper limit =

Answer 1

Solution:

Question 1)

Given: Researchers at Vanderbilt University were curious about whether consumers are able to use the information on nutritional labels. Thus they surveyed 200 primary care patients at the University Medical Center.

Respondents were asked to calculate the number of grams of carbohydrates in a single serving of soda, using the information on the label of a 20-ounce bottle (2.5 servings). Only 32% answered correctly.

Sample size = n = 200

Sample proportion =

Part a) We have to find Margin of Error with 95% confidence level

We need to find z_c value for c=95% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.95) /2 = 1.95 / 2 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is : Z_c = 1.96

Thus

Part b) If the researchers want to reduce the margin of error to .05 or less, with 95% confidence, how many patients should they survey? (Use p* = 0.32 from the survey above)

Given: E = Margin of Error = 0.05

c = confidence level = 95% then z value for 95% confidence level = 1.96

p* = 0.32

Thus sample size formula is:

Question 2)

Given: In a study of birth control methods, 902 users experienced 2 unintended pregnancies.

Give a 90% confidence interval for the probability of an unintended pregnancy for users of DMPA using the plus four method.

Plus four method means we add 4 additional observations in sample size n.

Two as success and two as failures.

thus we get: new sample size = n + 4 = 902 + 906

and x = number of an unintended pregnancy for users of DMPA = 2 + 2 = 4

Thus sample proportion of an unintended pregnancy for users of DMPA  

Formula:

where

Z_c is z critical value for c = 90% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.90) / 2 = 1.90 / 2 = 0.9500

Look in z table for Area = 0.9500 or its closest area and find corresponding z value.

Area 0.9500 is in between 0.9495 and 0.9505 and both the area are at same distance from 0.9500

Thus we look for both area and find both z values

Thus Area 0.9495 corresponds to 1.64 and 0.9505 corresponds to 1.65

Thus average of both z values is : ( 1.64+1.65) / 2 = 1.645

Thus Z_c = 1.645

Thus

Thus

Lower Limit=

Lower Limit

Lower Limit

and

Upper Limit=

Upper Limit

Upper Limit