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The Student Success Commons at York County Community College: Quantitative Reasoning

Quantitative Reasoning Resources

Please keep in mind that these videos, calculators, and notes are no substitute for attending class. All the resources on this page are here to supplement the information from your instructor and textbook.

First and foremost please be sure to follow any directions your teacher provides before using these notes or calculators to ensure you meet course requirements.

CHAPTERS 2 & 3

Chapter 3A


UNITS

• The units of a quantity describe what the quantity measures or counts. Units provide crucial context. For example, if you ask a car mechanic how long it will take to have your radiator fixed and they say "5" you need the units to fully understand their meaning. Is it 5 minutes, 5 hours, 5 days, 5 weeks, 5 months? 
Unit analysis (also called dimensional analysis) is the process of working with units to help solve problems.
• You generally cannot add or subtract numbers with different units, but you can combine different units through multiplication, division, or raising to powers. Always perform all operations on both numbers and their associated units.
• You can cancel units that appear in the top and bottom of one fraction, or in the top of one fraction and in the bottom of another as long as the two fractions are being multiplied (see the Unit analysis (cancelling units) PDF)

 

KEY WORD MATHEMATICAL MEANING EXAMPLE
per division "miles per hour" can be written mathematically as "miles ÷ hours" or "mi/hr"
of multiplication "50% of 10" can be written mathematically as "0.50 × 10" (Note: in almost all cases, percentages must be written as their decimal equivalents. To convert a percent to a decimal you must divide the percent by 100 and drop the % symbol.)
is equals "The price is $30" can be written mathematically as "price = $30"
what an unknown value (a variable), usually represented by a letter like x "What is 50% of 10" can be written mathematically as "x = 0.50 × 10"

 

CHAPTERS 5, 6, & 7

SAMPLE VS. POPULATION

POPULATION: The population is the complete set of people or things being studied.

SAMPLE: The sample is a subset of the population. The population is usually too large to look at in its entirety, so a sample that represents the population must be chosen and data can be gathered from the sample.

SAMPLE STATISTICS: Sample statistics are numbers describing characteristics of the sample. This data is gathered directly from the sample.

POPULATION PARAMETERS: Population parameters are specific numbers of interest that describe certain characteristics of the population. The population parameter is usually based on the sample statistic.

For example:

A YCCC student wants to know how many U.S. community college students have pets. She gathers information from 698 community college students and found that 422 of them have pets.

In this example:

• the population is all U.S. community college students.
• the sample is the 698 U.S. community college students that were surveyed.
• the sample statistic is the 422 out of 698 U.S. community college students who have pets, which is about 60.5%.
• the population parameter is the number of all U.S. community college students who have pets. Note that it is not expressly stated, but if the statistical study was performed properly, we can estimate that about 60.5% of all U.S. community college students have pets.


BASIC STEPS IN A STATISTICAL STUDY:

1. State the goal(s) of the study. Determine the population you want to study and what kind of information you want.
2. Make sure a sample is chosen that represents the population.
3. Collect data from the sample and find the sample statistic of interest.
4. Use the sample statistic(s) to infer the population parameter(s).
5. Draw conclusions. Determine what you learned and how it addresses your goals.

MAJOR TYPES OF BIAS

Selection bias occurs whenever the researches select their sample in a way that would tent to make it unrepresentative of the population. For example, if a researchers wants to gather information on all U.S. college students but only chooses males, the sample will not be representative of the whole population.
Participation bias occurs whenever people choose whether to participate. For example, if people must take action to participate in a study rather than being chosen at random, those who feel more strongly about the survey issue are more likely to participate.

SAMPLING METHODS

TYPE
MEANING
NOTES AND EXAMPLES
Simple random This is when the sample is chosen in such a way that every sample of the same size has an equal chance of being selected.

Examples:

  • Names are put in a hat and randomly picked.
  • Every item in the population is assigned a number and numbers are randomly chosen by a computer program
Systematic This is when the sample is chosen according to some kind of mathematical pattern.

Example:

  • Every 10th person on the list is surveyed.
Convenience This is when a sample is chosen because it is convenient.

Examples:

  • Every person who comes out of the post office is surveyed.
  • People watching a certain television show are asked to call in to cast their vote.
  • People are asked to mail in their response to a survey in a magazine.

Note: This type of sampling can often lead to bias.

Stratified

For this sampling method we must first define different subgroups, called strata, within the population, then we must select a sample from each of the subgroups.

Researchers use stratified sampling to ensure specific subgroups are present in their sample

Examples:

  • The population of a town is broken down into different age groups, and 30 people are chosen from each of the different age groups
  • The plants in a nursery are separated into groups based on the amount of sun they require. 

CHAPTER 4

CHAPTERS 8, 9, & 10

 

 

 

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