Download Math 101 - Review for Quiz 3 - Answers Practice Problems 1. Find

Math 101 - Review for Quiz 3 - Answers
Practice Problems
1. Find the five-number summary and the interquartile range for the given set of numbers. Also draw
the box plot.
20, 25, 31, 38, 42, 47, 51, 54, 56
min = 20, Q1 = 31, M = 42, Q3 = 51, max = 56
20
25
30
35
40
45
50
55
2. The following are the approximate lengths of Beethoven’s nine symphonies and Mahler’s nine symphonies in minutes. Determine the range, the mean, and standard deviation for each data set and
discuss how they are different and what that means.
Beethoven: 28 36 50 33 30 40 38 26 68
Mahler:
52 85 94 50 72 72 80 90 80
Beethoven: range = 40, mean = 38.78, sd = 13.13
Mahler: range = 44, mean = 75, sd = 15.44
Beethoven’s symphonies are shorter on average with slightly less variation in length.
3. After recording the pizza delivery times for two different pizza restaurants, you determine that
Restaurant A has a mean delivery time of 45 minutes with standard deviation of 3 minutes, and
Restaurant B has a mean delivery time of 42 minutes with a standard deviation of 20 minutes. Interpret these numbers and decide which restaurant you would order from assuming you like both equally.
One standard deviation away from the mean for Restaurant A is between 42 and 48 minutes. One
standard deviation away from the mean for restaurant B is between 22 and 62 minutes. Restaurant
B has much more variation in its delivery times. If you want more consistent deliveries, even though
slightly longer on average, choose Restaurant A.
4. An incoming first-year took her college’s placement exams in French and math. In French she scored
82, and in math she scored 86. The overall results on the French exam had a mean of 72 and a
standard deviation of 8. The mean math score was 68 with a standard deviation of 12. On which
exam did she do better as compared to the other incoming first-years?
zf rench =
82−72
8
= 1.25 and zmath =
86−68
12
= 1.5
She did better on the math exam as compared to the other students.
5. A individual records the speed of cars driving past his house, where the speed limit is 20 mph. The
mean of 100 readings is 23.84 mph with a standard deviation of 3.56 mph.
(a) What percentage of the cars were traveling under the speed limit?
z=
20−23.84
3.56
= −1.08 ⇒ 13.57%.
(b) If a car is going faster than 90% of the cars, how fast in the car going?
z = 1.3 ⇒ 1.3 =
x−23.84
3.56
⇒ x = 28.47 mph
(c) Which is more unusual, a car traveling 34 mph or a car traveling 10 mph?
z10 =
10−23.84
3.56
= −3.89 and z34 =
34−23.84
3.56
= 2.85
10 mph is more rare.
6. In a random survey of 226 college students, 20 reported having no siblings. Estimate the proportion
of students nationwide who have no siblings and construct the 95% confidence interval. Interpret
what you find.
.088 ± .038.
We are 95% confident that the percent of college students without any siblings is between 5 and
12.6%.
7. A fair coin is tossed two times and the number of heads is observed.
(a) What is the sample space?
S = {0, 1, 2}.
(b) Determine the probability distribution.
Outcome
0
1
2
Probability
1/4
1/2
1/4
(c) What is the probability of having exactly 2 heads? 1/4
8. An experiment consists of selecting a number at random out of the set of numbers {1, 2, 3, 4, 5, 6,
7, 8, 9}. Find the probability that the number is
(a) less than 4. → 1/3
(b) odd. → 5/9
(c) less than 4 or odd. → 2/3
9. Three horses, A, B, and C, are going to race. The probability that A wins is 31 and the probability
that B wins is 21 . What is the probability that C will win? Draw the probability distribution.
Outcome
A
B
C
Probability
1/3
1/2
1/6
10. A couple decides to have four children. What is the probability that they will have more girls than
boys? What is the probability that they will have more boys?
P(more girls)=5/16
P(more boys)=5/16
11. The probability that a prize appears in a cookie package is 0.01. What is the probability that 2
packages will contain at least 1 prize?
Assume independence due to a large number of packages.
P(at least 1 prize) = 1-P(neither contains a prize) = 1 - (.99)(.99) = 1 - .9801 = .0199.
12. The proportion of individuals in a certain city earning more than $40,000 per year is .25. The
proportion of individuals earning more that $40,000 and having a college degree is .10. Suppose
that a person is randomly chosen and he turns out to be earning more than $40,000, what is the
probability that he is a college graduate?
P (college | > $40K) =
P (college ∩ >$40K)
P (>$40K)
=
.1
.25
= .4
13. In a certain agricultural region, the probability of a drought during growing season in 0.2, the
probability of a cold spell is 0.15, and the probability of both is 0.1. Find the probability of
(a) not having a drought
P (no drought) = 1 − P (drought) = 1 − .2 = .8
(b) either a drought or a cold spell
P (drought ∪ cold) = P (drought) + P (cold) − P (drought ∩ cold) = .2 + .15 − .1 = .25
14. Consider the following data on the percent of adult sparrowhawks in a colony that return from the
previous year and the number of new adults that join the colony. Create the scatter plot. Do the
two variables appear correlated? Describe the relationship.
Percent return
New adults
74
5
66
6
81
8
52
11
73
12
62
15
52
16
45
17
62
18
46
18
60
19
46
20
38
20
60
40
50
New adults
70
80
Scatter plot
5
10
15
20
Percent return
The variables appear somewhat negatively correlated. There are no strong outliers. (The correlation
coefficient actually comes out to: -0.7484673).