DE and STATS

1. Let
dy 1
dt
= y 0
1
= 3y 1 ? 4y 2
dy 2
dt
= y 0
= 2y 1 ? 6y 2 ,
where y 1 (0) = 1 and y 2 (0) = 3. Find the solution of the system and comment on
its stability. [10 marks]
2. At a shopping mall, 80% of the customers come to shop, 15% come to eat, and
5% come to socialize (assume that each customer comes to do only one of these
activities). Of those who come to shop, 95% are older than age 20 years; of those
who come to eat, 65% are older than 20 years; and of those who come to socialize,
10% are older than 20.
(a). What is the probability that a randomly selected customer comes is younger
than 20 years of age? [4 marks]
(b). If one person is randomly selected, find the conditional probability that the
person is there to shop given that the person is 18 years old. [3 marks]
(c). What is the probability that a randomly picked customer visited the shop-
ping mall to socialize, given that the person is older than age 20 years?
[3 marks]
[10 marks]
3. Medical tests for the presence of drugs are not perfect. They often give false
positives, where the test indicates the presence of the drug in a person who has not
used the drug, and false negatives, where the test does not indicate the presence
of the drug in a person who has actually used the drug.
Suppose a certain marijuana drug test gives 13.5% false positives and 2.5% false
negatives. Suppose the principal of a high school with 1000 students believes there
is a marijuana drug problem in the school and so decides to give all students this
drug test. Further suppose that 50 students actually use marijuana and 950
students have never used marijuana.
(a). Approximately how many students will get a false-positive result? How many
will get a false-negative result? [2 marks]
(b). Approximately how many students will get “caught”, and how many will “get
away with it”? How many will be falsely accused of using marijuana?
[9 marks]
(c). Based on these calculations, would you recommend giving the test to every
student? Explain why or why not. [1 marks]
[12 marks]
4. If X is N(12,1.4 2 ), use R to find:
(a). P(X > 13.4) [1 marks]
(b). P(10.5 < X  14.5) [3 marks]
(c). P(|X ? 12| > 0.9) [5 marks]
(d). P(X 2 + 10 < 110) [2 marks]
(e). P(X > 14|X > 13.4) [3 marks]
(f). the 95 th percentile [2 marks]
[16 marks]
5. A machine is designed to produce bolts with a 3-in diameter. The actual diam-
eter of the bolts has a normal distribution with mean of 3.002 in and standard
deviation of 0.002 in. Each bolt is measured and accepted if the length is within
0.005 in of 3 in; otherwise the bolt is scrapped. Find the percentage of bolts that
are scrapped. [5 marks]
6. Student marks for a Mathematics course follow a Normal distribution with a mean
of 65 marks and a standard deviation of 20 marks.
(a). What is the probability that a randomly selected student who follows the
Maths course will score between 45 to 60 marks? [2 marks]
(b). Calculate the probability that the randomly selected student will fail the
course, if the pass mark is 49. [2 marks]
(c). What should be the cuto↵ mark for a Higher Distinction (HD) if the facili-
tator wants only 15% HDs in the o↵er. [2 marks]
[6 marks]
7. IQ scores have a normal distribution with mean µ = 100 and variance ? 2 = 152.
(a). If one person is randomly chosen, find the probability that her or his IQ is
greater than 110. [1 marks]
(b). If a sample of 25 people is randomly chosen, find the probability that the
sample mean is greater than 110. [3 marks]
[4 marks]
8. A random sample of 10 run times from top-10 finishers in cross-country meets
from the Great Plains Athletic Conference (GPAC) has a mean of 1606 seconds
with a standard deviation of 19.8 seconds. Assuming the population is normally
distributed, construct a 90% confidence interval estimate of the mean run time
of all top-10 finishers in the GPAC. Interpret your confidence interval.
[5 marks]
9. People have claimed that the ratio of a person’s overall height to navel height
equals the golden ratio (a number approximately equal to 1.618). To test this
claim, a statistic professor has measured the overall height and navel height of 39
students and calculated the ratio as (overall height/navel height). The 39 values
of this ratio have a mean of 1.6494 units and a standard deviation of 0.0474 units.
(a). Use these data to calculate a 95% confidence interval estimate for the mean
ratio of all students at this university. Assume the sample is a random sample
of students at this university. [5 marks]
(b). Do the results support the claim? Why or why not? [2 marks]
[7 marks]