The value of mammography as a screening test for breast cancer has been controversial, particularly among young women. A study was recently performed looking at the rate of false positives for repeated screening mammograms among approximately 10,000 women who were members of Harvard Pilgrim Health Care, a large health-maintenance organization in New England. The study reported that of a total of 1996 tests given to 40-49 year old women, 156 yielded false –positives results.
Some physicians feel a mammogram is not cost-effective unless one can be reasonably certain (e.g., 95% certain) that the false-positive rate is less than 10%. Can you address this issue based on the preceding data? (Hint: Use a CI approach.) PLEASE NEATLY SHOW ALL WORK!
I think it's part a) .98
part b)
.96 to .99
google "single sample confidence interval"
look at the 1st article
To address the issue of the false-positive rate for repeated screening mammograms among 40-49 year old women, we can use a confidence interval approach.
Step 1: Calculate the sample proportion (p-hat) of false positives:
p-hat = (number of false positive results) / (total number of tests)
= 156 / 1996
= 0.078
Step 2: Calculate the standard error (SE) of the proportion:
SE = √((p-hat * (1 - p-hat)) / n)
= √((0.078 * (1 - 0.078)) / 1996)
= √(0.07197 / 1996)
≈ 0.0085
Step 3: Determine the margin of error (ME) using a 95% confidence level:
ME = z * SE
= 1.96 * 0.0085
≈ 0.0166
Step 4: Calculate the lower and upper bounds of the confidence interval:
Lower bound = p-hat - ME
= 0.078 - 0.0166
≈ 0.0614
Upper bound = p-hat + ME
= 0.078 + 0.0166
≈ 0.0946
Part a) The point estimate for the false-positive rate is 0.078, or 7.8%.
Part b) The 95% confidence interval for the false-positive rate is approximately 0.0614 to 0.0946, or 6.14% to 9.46%.
Considering the physician's requirement of being 95% certain that the false-positive rate is less than 10%, the confidence interval (6.14% to 9.46%) does satisfy this criterion.
To address the issue of the false-positive rate in mammography screenings for breast cancer, we can use a confidence interval approach.
Given that 156 out of 1996 tests yielded false positives among 40-49 year old women, we can calculate the proportion of false positives as follows:
Proportion of false positives = 156 / 1996 ≈ 0.078
Now, we can calculate a confidence interval to estimate the true proportion of false positives.
Using the formula for the confidence interval of a proportion, we can calculate the margin of error:
Margin of error = Z * √(p * (1 - p) / n)
Where:
- Z is the z-score corresponding to our desired level of confidence. In this case, we want a 95% confidence level, so Z ≈ 1.96 (obtained from a standard normal distribution table).
- p is the sample proportion of false positives.
- n is the sample size.
Plugging in the values, we have:
Margin of error = 1.96 * √(0.078 * (1 - 0.078) / 1996) ≈ 0.011
To calculate the confidence interval, we can subtract the margin of error from the sample proportion and add it to the sample proportion:
Confidence interval = (0.078 - 0.011, 0.078 + 0.011) = (0.067, 0.089)
Therefore, based on this data, we can say with 95% confidence that the true proportion of false positives for repeated screening mammograms among 40-49 year old women is estimated to be between 0.067 and 0.089.
Now, addressing part b), the 95% confidence interval for the true proportion of false positives is (0.067, 0.089). Since this interval does not include 10%, we can conclude that we cannot be reasonably certain at the 95% confidence level that the false-positive rate is less than 10%.