M203J Worksheet
[1] The teenage pregnancy rate in 2006 was 72 teenage girls in every 1000 (http://www.guttmacher.org/pubs/USTPtrends.pdf). Home pregnancy kits (for women who collect and test their own samples) was found to have an overall sensitivity of 75% and a specificity of around 65% (http://www.medicine.ox.ac.uk/bandolier/band64/b64-7.html).
(a) If a pregnant teen uses one of these home pregnancy kits, what's the chance that it will correctly say she is pregnant?
(b) If a pregnant teen uses one of these home pregnancy kits, what's the chance of a false negative (i.e., that it will incorrectly say she is not pregnant)?
(c) If a teen who is not pregnant uses one of these home pregnancy kits, what's the chance that it will correctly say she is not pregnant?
(d) If a teen who is not pregnant uses one of these home pregnancy kits, what's the chance of a false positive (i.e., that it will incorrectly say she is pregnant)?
(e) If a randomly chosen teen uses one of these home pregnancy kits and gets a negative result, what's the chance it is a false negative (i.e., what's the chance she is pregnant in spite of the test saying she's not)?
[2] If you flip a fair coin 100 times and find the fraction p^ of heads, what's the chance that p^ >= 0.6 (i.e., what is the chance that you get at least 60 heads)? Use what we've learned about normal distributions, by assuming that p^ is normally distributed (in this case p = 0.5 since we're assuming the coin is fair). [This is the same problem as choosing a sample of size n = 100 from a normally distributed population with p = 0.50, and asking what percentage of samples have p^ >= 60%.]
[3] Monday's Omaha World-Herald, October 25, 2010, published a poll based on sampling 607 registered voters in Omaha (the polling was done October 17-21). It found that 44% planned to vote for Lee Terry for Congress, 39% planned to vote for Terry's opponent, Tom White, and 17% were undecided or planned not to vote. The claimed margin of error was plus or minus 4%.
(a) What margin of error do you find from the formula assuming p = 0.44?
(b) What margin of error do you find from the formula assuming p = 0.39?
(c) What e would you need in order to have a 99.7% chance that p is in the range p^ plus or minus e% ?
(d) Note that with a margin of error of 4%, it's possible that White is ahead of Terry. If we want to try to tell who's really ahead we might want a smaller margin of error. What sample size would you need to have a margin of error of plus or minus 2% for Terry's p^ = 44% ?