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Course: Praxis Core Math > Unit 1

Lesson 3: Statistics and probability

Data representations | Lesson
Data representations | Worked example
Center and spread | Lesson
Center and spread | Worked example
Random sampling | Lesson
Random sampling | Worked example
Scatterplots | Lesson
Scatterplots | Worked example
Interpreting linear models | Lesson
Interpreting linear models | Worked example
Correlation and Causation | Lesson
Correlation and causation | Worked example
Probability | Lesson
Probability | Worked example

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Statistics and probability

Random sampling | Lesson

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What are random samples?

provides information about a

without having to survey the entire group.

To make valid conclusions about a population, we need a sample that recreates the characteristics of the entire population on a smaller scale.

A good sample is representative and random.

Representative means that the sample includes only members of the population being studied.
Random means that every member of the population being studied has an equal chance to be selected for the sample.

To gather information on voter preferences in the next presidential election, we might poll a random sample of voters to draw conclusions about the preferences of all voters. It's impractical to contact every voter!

What skills are tested?

Identifying a valid sampling method
Determining the population for which the conclusions of a sample are valid
Using a random sample to make an estimate

What are some bad ways to sample?

Sampling method refers to how participants are selected for a sample.

Bad sampling methods include those that:

Gather data from outside the population being studied
Gather data that overrepresent or underrepresent a subgroup of the population (not random)

Suppose we are interested in gathering information on how all students at a certain high school feel about proposed cuts to the athletic budget.

The sampling methods below lead to bad samples, ones that are not representative of the feelings of all students at the high school.

Method 1: Surveying all athletes at the high school

This sample overrepresents the feelings of student athletes. Student athletes may feel differently about the proposed cuts than other students.

Method 2: Emailing a survey to all students and using the responses

This sample includes only students who respond to the email. Respondents are likely to have stronger feelings about the topic than students who do not respond. This sample overrepresents students with stronger feelings.

Method 3: Calling a random sample of families with students enrolled in the school district

This sample includes data from outside the population being studied, all students at a certain high school.

What population can we make inferences about using data from a sample?

We can make inferences about only the population from which the random sample was selected.

Suppose a survey randomly samples opinions of many college students from Vermont.

We can make inferences about the opinions of all college students from Vermont.
We cannot make inferences about the opinions of all college students or the opinions of all people from Vermont.

How can we use sample data to make an estimate?

We can use data from a random sample to estimate the number in the population having a particular attribute.

To make an estimate, we need to know the

and the population size.

estimate = population size \times sample proportion

‍

60

‍ seniors at a particular high school are randomly selected for a survey.

6

‍ report driving themselves to school at least once a week. If there are

400

‍ seniors at this high school, what is a reasonable estimate of the total number of seniors that drive themselves to school at least once a week?

The population is all seniors at the high school. The sample proportion is the fraction of seniors who reported driving themselves to school at least once a week.

\begin{aligned} population size & = 400 \\ sample proportion & = \frac{6}{60} \\ estimate & = 400 \times \frac{6}{60} = 40 \end{aligned}

‍

A reasonable estimate is that

40

‍ seniors drive themselves to school at least once a week.

Your turn!

TRY: IDENTIFYING A VALID SAMPLING METHOD

A city manager wants to ask a sample of residents how they feel about different ways to mitigate traffic congestion. Which of the following methods of selecting the sample will yield the most valid information about the feelings of all residents of the city?

Choose 1 answer:

(Choice A)
Survey all attendees at an upcoming sustainability event
(Choice B)
Survey residents picked at random from a list of all residential addresses within the city limits
(Choice C)
Survey all people visiting city hall during a particular month
(Choice D)
Survey everyone attending city council meetings during the next three months
(Choice E)
Survey every tenth visitor to the city library

TRY: IDENTIFYING A VALID CONCLUSION

A veterinarian randomly tested

150

‍ of her canine patients for Lyme disease.

2 %

‍ of the canines tested positive for Lyme disease. Based on this data, which of the following conclusions are valid?

Choose all answers that apply:

(Choice A)
About $2 %$ ‍ of canines in the state have Lyme disease
(Choice B)
About $2 %$ ‍ of the veterinarian's patients have Lyme disease
(Choice C)
About $2 %$ ‍ of the veterinarian's canine patients have Lyme disease

TRY: MAKING AN ESTIMATE

Jian asked a random sample of

700

‍ students at Golden University whether they have a microwave in their dorm room. Of the students surveyed,

210

‍ reported having a microwave in their dorm room. If there are

15,000

‍ students at Golden University, what is a reasonable estimate for the number of students who have a microwave in their dorm room?

Round your answer to the nearest integer.

Your answer should be
an integer, like $6$ ‍
a simplified proper fraction, like $3 / 5$ ‍
a simplified improper fraction, like $7 / 4$ ‍
a mixed number, like $1 3 / 4$ ‍
an exact decimal, like $0.75$ ‍
a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍

TRY: MAKING AN ESTIMATE

50

‍ randomly selected seniors at Lincoln High School were asked to state their intended college major. The bar graph above shows the results of the survey. If there are

600

‍ seniors at Lincoln High School, what is a reasonable estimate of the number of seniors who intend to major in business?

Choose 1 answer:

(Choice A)
$240$ ‍
(Choice B)
$192$ ‍
(Choice C)
$96$ ‍
(Choice D)
$72$ ‍
(Choice E)
$6$ ‍

Things to remember

is a way to gather information about a

without having to survey the entire group.

A good sample must be both
and
!

We can use the results of a good sample to make estimates about the population it was drawn from.

$estimate =$ ‍ $population size \times sample proportion$ ‍

Estimates are valid only for the specific population the sample is drawn from!

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chloelenz123
Posted 2 months ago. Direct link to chloelenz123's post “Is anyone else doing it b...”
Is anyone else doing it backwards?
Instead of 600 multiplied by 6/50 = 72, I simplify the fraction and write it as 600 divided by 25/3 = 72. Same answer, different formula.
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Anwar Mohammad
Posted 6 months ago. Direct link to Anwar Mohammad's post “This is so confusing”
This is so confusing
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Avah Hampton
Posted a year ago. Direct link to Avah Hampton's post “It makes sense”
It makes sense
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Shanta Joshi
Posted 5 years ago. Direct link to Shanta Joshi 's post “Why different scoring for...”
Why different scoring for praxis VA math sub test and core praxis math test?
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