Math 2000 Midterm Exam and Solutions
This is a 75-minute open-book, open-notes exam. You must show all of your work clearly to receive full credit. Each problem is worth 20 points.
1. Statistical studies. The table below is taken from a study designed to determine whether people in bicycle accidents have a lower risk of facial injuries if they wear a helmet.
|
|
Facial injuries |
No facial injuries |
Total |
|
Helmet |
30 |
83 |
113 |
|
No helmet |
182 |
236 |
418 |
|
Total |
212 |
319 |
531 |
(Data
from American Journal of Public Health, Vol. 80, No. 12)
a. What is the population of this study?
A statistical study is designed to learn something
about a specific collection of individuals or things. This collection is the population
of the study. In this study, the population is all people who have been in
bicycle accidents. The population cannot be all bicycle riders, because the
table has no data on bicycle riders who have not been in accidents.
b. What is the sample size for this study?
The sample size is the number of people actually observed and included in the table, which is 531.
c. Was the study observational or experimental?
The study is observational because it
involves only data collection on accidents that occurred in the past.
d. If it was experimental, was a placebo used? If it was observational, is it case-control? Explain.
The study is a case-control study because
people in the sample naturally fall into two categories: those who wear helmets
and those who don’t. The researchers did not require participants to be in a
specific group.
e. What percentage of people wearing helmets had facial injuries?
A total of 113 people in the study wore helmets, of which 30 had facial injuries. Thus 30/113 = 26.5% of people wearing helmets had facial injuries.
2. Unit conversions.
a. Suppose your car can travel 33 miles on a gallon
of gasoline on the open road and that gasoline costs $1.65 per gallon. What is
the cost of operating your car in dollars per mile? You must show the use of
units!
The numbers 33 miles per gallon and $1.65 per
gallon must be combined in a way that eliminates gallons and leaves an answer
with units of dollars per mile. This can be done in only one way:
It costs 5 cents per mile to drive the car.
b. Suppose the current exchange rate between marks and dollars is 1.6 marks per dollar. How many dollars are there in one mark?
A conversion factor can be written in three
ways. We can write 1.6 marks = $1 or
If we do the division in the last form (1 ÷
1.6), we see that
There are $0.625 (62.5 cents) in one mark.
c. A German delicatessen sells chocolate for 25 marks per kilogram. Using the exchange rate of 1.6 marks per dollar, what is the cost of the chocolate in dollars per pound? You must show the use of units!
As done many times in class, we can use units
to convert marks to dollars and kilograms to pounds. It looks like this:
The chocolate costs $7.10 per pound.
3. Numbers in perspective. The population of the greater Denver area is about 1,800,000. A football field is 100 yards long and 60 yards wide.
a. Write the population of Denver in scientific notation.
The number 1,800,000, or 1.8 million, is written in
scientific notation as 1.8 × 106.
b. Find the area of a football field in square yards.
The area of the football field is found by
multiplying its length by its width:
c. If all the people of Denver were place shoulder-to-shoulder with one square yards per person, how many football fields would be needed? Use units and scientific notation for your calculations.
With 1 square yard per person, one football field
holds 6,000 people. Done carefully with units, we see that the number of
football fields needed for 1.8 million people is
The people of Denver could be placed (tightly) on 300 football fields.
4. Percentages.
a. The population of New York City is 12.6 million people, 45% more than the population of Chicago. What is the population of Chicago?
The required sentence is
Pop of Chicago + (45% of pop of Chicago) = pop of
NYC = 12.6 million.
Simplifying this sentence a bit we have
145% × (pop of
Chicago) = 12.6 million.
Noting that 145% = 1.45, we divide both sides
by 1.45 to find that
b. Fill in the blank: One meter is _____% longer than one yard.
The question asks for the percent difference
between one meter and one yard, where one yard is the reference
quantity and one meter is the comparison quantity. You can do the problem
in inches if you note that 1 yard = 36 inches and 1 meter = 39.37 inches:
You can do the problem in feet if you note
that 1 yard = 3 feet and 1 meter = 3.28 feet:
You can do the problem in yards if you note
that 1 meter = 1.094 yards:
All of these methods lead to the result that one
meter is 9.4% longer than one yard.
5. Venn diagrams. Consider the Venn diagram below showing the reading preferences of the players on a rugby team. Please answer the following questions. It is important that you explain your reasoning for each answer.
a. How many
players read poetry only?
We can unambiguously say that 12 people read
poetry only (because 12 people are in the poetry circle and in no other
circles).
b. How many
players read fiction or non-fiction?
This one is ambiguous because of the two
interpretations of or. Recall that or can be taken
in the exclusive sense (A or B, but not both A and B), or in the inclusive
sense (A or B or both A and B). Furthermore, the question could mean fiction
or nonfiction, but not poetry, or it could mean fiction or nonfiction,
including poetry. So I see four justifiable answers:
Fiction or nonfiction (inclusive), but no
poetry: 31 people
Fiction or nonfiction (exclusive), but no
poetry: 19 people
Fiction or nonfiction (inclusive), including
poetry: 52 people
Fiction or nonfiction (exclusive), including
poetry: 34 people
Blame the ambiguity in the English language,
not on mathematics!
c. How many
players read fiction and poetry?
Again there is some ambiguity. The number of
people who read fiction and poetry only (excluding nonfiction) is 7. The number
of people who read at least fiction and poetry (including nonfiction) is
13.
d. How many players read poetry?
One could answer that 12 people read poetry
(as in part a), meaning they read only poetry. Perhaps the more usual
interpretation says that 33 people read at least poetry.
