Vocabulary:
Population:
Example:
Sample of a population
Example:
Subject:
Example:
Representative
sample:
Biased Sample:
Bias:
Inference:
Valid Inference:
Invalid Inference:
Proportional Relationship:
is the complete set of items being studied.
The members of Mr. Quinones' Drama Club
is a part of the population.
any three members of the club.
any member of a sample.
Reynaldo
is a sample of a population in which The
number of subjects in the sample with the trait
that you are studying is proportional to the
number of members in the population with that
trait.
is an accurate representation of the population
and does not have bias.
The number of subjects in the sample being
studied is not proportional to the number of
members in the population with that trait.
Does not accurately represent the population.
is a tendency towards a particular perspective
that
is different from the overall perspective of the
population.
We have bias towards yes or bias towards no.
is a judgement that is made by interpreting
data.
is true about the population.
is false about the population.
Two quantities x and y have a proportional
relationship if y is always a constant multiple of x.
A relationship is proportional if it can be described
by equivalent ratios.
Convenience sampling
1.
is a sampling method in which you choose
members of the population that are convenient
and available.
You are investigating the lengths of all of the words in
a book. Determine whether each description is a
population or a sample.
To estimate a population we should use our prior
knowledge of Constant of Proportionality. (The ratio of
y/x).
Sample = constant of proportionality * population size
Constant of Proportionality = sample part
sample size
Estimating a Population
1. On a platter of chicken wings there are mild wings and hot wings. In a
representative sample of 15 chicken wings, there are 4 hot wings. If the
platter has 90 chicken wings, estimate the number of hot wings on the platter.
15 90
4
h
2. A bucket of golf balls contains gray golf balls and white golf balls. You
collect the representative sample shown. If the bucket contains 50 golf balls,
about how many golf balls are white?
50 10
w 7
3. You want to find the number of students with October birthdays.
In a school with 1,000 students, the actual number of students with
October birthdays is 120. You do not know this, so you collect three
samples. One sample finds 16 October birthdays in 25 students.
Another sample finds17 October birthdays in 50 students. The third
sample finds 12 October birthdays in 100 students.
a) Which sample best represents the population?
❍ A. 16 students with October birthdays in a sample of 25 students
❍ B. 17 students with October birthdays in a sample of 50 students
❍ C. 12 students with October birthdays in a sample of 100 students
❍ D. All three samples are representative.
b) Use the sample to predict the number of students with October
birthdays in a group of 1,500 students. (use the cross-multiply divide
as above)
4. In a box of 60 pens there are black pens and blue pens. Jenna was
asked to estimate the number of black pens in the box. She incorrectly
said there are about 40 black pens in the box. Use the representative
sample shown.
a) Estimate the number of black pens in the box.
(use the cross-multiply divide as above)
❍ A. She used 10 as the population size instead of 60.
❍ B. She used 2/3 as the constant of proportionality instead of 2/5.
❍ C. She used 2/5 as the constant of proportionality instead of 2/3.
❍ D. She used 60 as the population size instead of 10.
5. In a bowl of fruit salad there are strawberries,
blueberries, grapes, and raspberries. In a representative
sample of 30 pieces of fruit, there are 14 strawberries.
a) If the fruit salad has 270 pieces of fruit, estimate the
number of strawberries in the fruit salad. (use the crossmultiply divide as above)
6.
7. Suppose
you are a news reporter, investigating the town’s opinion of the
new shopping center. What are three ways to choose a convenience sample of
town residents to interview? Are the samples that you chose representative
samples? Explain.
8. Use the convenience sample to estimate how many of the 1,165 cars on
Highway 60 are red. Is your estimate accurate?
9. Suppose you are doing research on the most popular snack foods.
Circle the examples of convenience sampling.
A. You ask your friends.
B. You ask each person shopping at a convenience store.
C. You give a survey to each household in your neighborhood.
D. You ask all the students who ride your bus.
10. You want to find out how many people support the school tax.
Circle the representative sample.
A. You ask every adult in your extended family.
B. You ask the first 25 adults you see at the mall.
C. You ask the parents of your friends.
11. A new industrial plant moves into a city of 300,000 people. You
want to know if the citizens support the industrial development.
Describe a convenience sample and tell whether it would be a
representative sample.
12. A reporter finds that 9 out of the 10 people he interviews at a
concert like the band. Use his data to estimate how many of the
40,000 people in the town like the band. Is it an accurate estimate?
Explain.
13. Which of the following descriptions would be an example of a
convenience sample?
Check all that apply.
❑ A. A person accepts the first 27 people who respond to an email.
❑ B. A person chooses 27 names out of a hat at random.
❑ C. A person accepts the first 27 people who volunteer.
❑ D. None of the descriptions is an example of a convenience
sample.
14. Samuel is doing a report for his math class. He is doing a survey
of what music people listen to. He asks 14 people at the beach. Is the
convenience sample a representative sample of the entire population?
Why?
❍ A. No, it is not representative because the ages of the people are
probably mixed.
❍ B. Yes, it is representative because the ages of the people are
probably mixed.
❍ C. Yes, it is representative because the ages of the people are in a
specific age range.
15. Angela is doing a study to find what is the most popular fruit in her
town. There are 9,000 people living in the town. She uses a
convenience sample and finds that 3 people out of the 20 people
asked say that the pineapple is the best fruit. Estimate the total
number of people in the town who consider the pineapple to be the
best fruit.
A systematic sample is useful when the researcher is able to approach the
population in a systematic, or methodical, way.
Step 1
Assign Numbers to Population
Assign each member of the population a unique number.
Step 2
Find the Interval n
To find the interval n, you need to know the population size and the desired
sample size.
Step 3
Find the Starting Number
To find the starting number, pick a number from 1 to n at random.
Step 4
Collect a Systematic Sample
Collect a systematic sample by choosing the member with the starting
number and then every 4th member until you reach the end of the
population.
A systematic sample takes preparation to collect, but it usually produces a
representative sample. Someone could introduce bias into the sample by
arranging the population to make sure certain members are chosen.
16. A movie theater wants to survey their customers to make sure they are
maintaining a customer-friendly environment. Which sampling description is
an instance of systematic sampling?
I. You ask every movie-goer that walks into the theater.
II. Choose one customer that comes into the theater.
III. Starting with the first movie-goer, choose every 5th movie-goer that
comes into the theater.
IV. Select the first 20 customers that walk into the theater.
17. Suppose you want to determine how many students in your grade
have had a dream about flying. Suppose there are 340 students in
your grade. Describe how you would choose a systematic sample of
20 students. Is the sample a representative sample?
Explain.
18. When might a systematic sample not be a representative sample?
19. Suppose you want to estimate the number of purple lights in a light
display. You decide to take a systematic sample using an interval of 6
lights, starting from the second light. The order of the lights is red,
yellow, green, blue, purple, red, yellow, green, blue, purple, and so on.
Suppose there are a total of 300 light bulbs in the display. Using your
systematic sample, how many purple light bulbs do you estimate are
in the display? How accurate is your estimate? Explain.
20. Caroline is doing research on the heights of players on her soccer
team. Check each description that is an example of systematic
sampling.
❑ A. Caroline chooses every seventh person on the team roster,
starting with the second.
❑ B. Caroline randomly chooses one of her teammates.
❑ C. Caroline chooses the first 5 names on the team roster.
❑ D. Caroline chooses everyone on her team.
❑ E. Caroline has her teammates count off by 6 and chooses the
ones who counted “3.”
21. You are studying the colors of houses on a highly populated
street. Decide whether or not each of the following descriptions is an
example of systematic sampling. Check all that apply.
❑ A. You choose every eighth house on the street, starting with the
second.
❑ B. You choose every house on the street.
❑ C. You choose the first and last 8 houses on the street.
❑ D. You list the houses alphabetically by owner’s last name, and
choose every fifth house on the list, starting at the fourth.
22. The trees in a city park form a line through the center of the park.
The types of trees repeat in the order of elm, spruce, willow, pine, and
birch throughout the line. You decide to take a systematic sample
using an interval of 6 trees, starting from the first tree. In all, there are
150 trees in the park. Using your systematic sample, how many pine
trees do you estimate are in the park?