Question

The following is from a web page on April 20 that discussed a Stanford study of percentage of population contracting Covid-19. (Disclaimer: do not treat what you read as factual just because a site has “news” in its title). Please read the paragraphs and answer the question at the end.

https://www.mercurynews.com/2020/04/20/feud-over-stanford-coronavirus-study-the-authors-oweus-all-an-apology/

One major problem with the Santa Clara County study relates to test specificity. It used a kit purchased from Premier Biotech, based in Minneapolis with known performance data discrepancies of two “false positives” out of every 371 true negative samples. Although it was the best test at the time of the study, that’s a high “false positive” rate that can skew results, critics say — especially with such a small sample size.

With that ratio of false positives, a large number of the positive cases reported in the study — 50 out of 3330 tests — could be false positives, critics note. To ensure a test is sensitive enough to pick up only true SARS-CoV-2 infections, it needs to evaluate hundreds of positive cases of COVID-19 among thousands of negative ones.

We assume the rate of false positives is 2 out of 371. To understand the statement that “a large number of 50 out of 3330 tests could be false positives”, we answer the following questions:

(1) What is the probability that there are at least 25 false positives out of 3330 tests? (Hint: use normal distribution approximation to binomial distribution).

(2) What is the probability that there are at least 30 false positives out of 3330 tests?

Answer 1