17. Of all the women who test positive, 95/590 or about 16% actually have a malignancy.
18. Of all the women who test positive, 5/9410 or about 0.053% really do have a malignancy.
20. (a) If the disease incidence rate is
1.5%, then (0.015)(4000) = 60 people have the disease,
and 4000-60 = 3940 do not. If the test is 80% accurate
on those who have the disease, in the sense that
80% of those who have it will test positive, then
(0.80)(60) = 48 of these 60 people test positive,
and 60-48=12 do not. Similarly, 80% of those who
do not have the disease will test negative, thus
accounting for (0.80)(3940) = 3152 people. The
remaining 3940-3152=788 people without the disease
test positive. Overall, 48+788=836 people test
positive, and 12+3152=3164 people test negative.
(b) Of the 60 people with the disease, 48 test positive,
giving rise to 48/60 or 80%.
(c) Of the 836 people who test positive, only 48 have
the disease, giving rise to 48/836 or about 5.74%.
This is much lower than the "80% accuracy rate"
verified in part (b): the reason is that here
the 48 true positives are being compared to the 836
positive testees, whereas in the last part they were
only compared to the 60 diseased people.
(d) The chances of the patient who tests positive
having the disease is 5.74%, which is higher than
the overall chance of 1.5% in the absence of a
test result.
26. Consider treatment A: in the first trial its cure rate was 20%, and in the second trial its cure rate was 85%; overall, its cure rate was (40+85)/(200+100), or about 41.67%. Next, consider treatment B: in the first trial its cure rate was 15%, and in the second trial its cure rate was 75%; overall, its cure rate was (30+300)/(200+400) or 55%. Thus treatment A had the higher cure rate in the two trials individually, while treatment B had a higher cure rate overall. This paradox can be resolved by noting that the largest number of people had treatment B in the second trial.
Section 4A:
36. Since Trevor is earning simple interest at an annual rate of 6%, his initial investment of $1000 grows by $60 at the end of each year. After 5 years, his balance is up to $1300, which is a $300 increase in absolute terms. In relative terms, his balance has increased by $300/$1000 = 0.30, i.e., 30%. Kendra, on the other hand, invests the same amount of money at 6% compound interest. After the first year she too gets $60 in interest and has a balance of $1060. However, after the second year, she gets 0.06 × $1060 = $63.60 in interest, resulting in a new balance of $1123.60. After the third year, she gets 0.06 × $1123.60 = $67.42 in interest, resulting in a new balance of $1191.02. Similarly after the fourth and fifth years she gets more "interest on her interest", as tabulated in the spreadsheet http://www.math.wisc.edu/~propp/141/TrevorKendra.xls, ending up with a balance of $1338.23. After 5 years, Kendra's balance is up by $338.23, which in relative terms is $338.23/$1000 = 0.3382, i.e., a 33.82% increase. (Note: to get full precision in the spreadsheet, it was necessary to use Format -> Cells.)
40. If we invest $3000 at an APR of 4% for 12 years, we accumulate a balance of $3000 × (1+0.04)12 = $4803.10.
48. Investing $3000 at an APR of 5% compounded daily for 10 years yields A = $3000 × (1+0.05/365)3650 = $4945.99.
52. Investing $1000 for 1 year at an APR of 4.5% compounded monthly yields A = $1000 × (1 + 0.045/12)12 = $1045.94, which results in an APY of $45.94/$1000 = 0.04594, i.e., 4.59%.
58. If we invest $3000 for 1 year at 4% compounded continuously, we end up with A = $3000 × e0.04 × 1 = $3122.43, and an APY of $122.43/$3000 = 0.04081, i.e., 4.08%. Investing for 5 years we get A = $3000 × e0.04 × 5 = $3664.21, and investing for 20 years we get A = $3000 × e0.04 × 20 = $6676.62.
64. To end up with $10,000 after 10 years at an APR of 4% compounded daily, you must start with P = $10000/[(1+0.04/365)365] = $6703.35.
76. Brian invests $1600 at an APR of 5.5% compounded
annually, so after Y years he has A =
$1600 × (1+0.055)Y. For Y=5 we get $2091.14,
and for Y=20 we get $4668.41.
Celeste invests $1400 at an APR of 5.2% compounded continuously;
after Y years she has A = $1400 × e
0.052 × Y . For Y=5 we get $1815.70,
and for Y=20 we get $3960.90.
Hence, Brian has the higher accumulated balance after 5 years and
after 20 years. Celeste's smaller investment and lower APR leaves
her trailing Brian, despite her more frequent compounding.
80. Investing $1000 at an APR of 7% for Y years yields A = $1000 × (1.07)Y. For Y=5, we get A=$1402.55, which is a bit short of our desired $1500. Trying Y=6 we get A=$1500.73, which is almost exactly what we wanted. Since the interest is being compounded only once a year, we settle for 6 years, and a tiny overshoot of our goal.
_+http://www.math.wisc.edu/~propp/141/S7.htmlYtext/html & D V e { 'U _