Statistics: Understanding Variables, Measures of Center, and Measures of Spread, Study notes of Business Statistics

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Introductory Material

Descriptive Statistics / Graphs

MATH 2283

Population vs Sample

• A population is an entire group we wish to

study.

 (^) A population can be real: such as all adults residing in North Carolina.  (^) A population can be theoretical: such as all potential ball bearings made by a certain machine or all potential readings of a scale when a ten pound weight is placed upon it.

• A sample is a subset of the population.

Characteristics of an object

• A variable is any characteristic of an object.

 (^) Variables are numerical or categorical descriptions of objects.  (^) For example: height, weight, gender.

• When a specific variable is considered, the

population essentially goes from a being a

collection of objects to being a collection of

numbers or words.

Parameter vs Statistic

• A parameter is any number that summarizes

a population.

 (^) For example: the population mean μ for a specific variable. μ is a Greek letter (pronounced “mu”).  (^) Parameters are often unknown.

• A statistic is any number that summarizes a

sample.

 (^) For example: the sample mean.  (^) Statistics are often used to estimate unknown parameters.

x

Example

• (^) The survival times in days of 72 guinea pigs after

being injected with TB (tubercle bacilli) in a medical

experiment is recorded. The average survival time

of the 72 guinea pigs is 162 days.

 (^) Population?  (^) Variable?  (^) Sample Size? Unknown : an average guinea pig will survive 170 days after injection.  (^) 170 days is a _________, as it describes a _________.  (^) 162 days is a _________, as it describes a _________.

Descriptive vs Inferential Statistics

• (^) Descriptive Statistics is the art of describing important aspects of a set of measurements.  (^) Describing data through graphs and numerical summaries is important.
• (^) Inferential Statistics is the science of using a sample of measurements to draw conclusions about a population.  (^) More important however is using this data to draw educated conclusions about a population.

Sampling Schemes: Simple Random Sample (SRS)

• (^) In a simple random sample, on each selection from the population, every unit remaining in the population has the same chance of being chosen next.  (^) This is the most basic method of random sampling.  (^) A SRS of size n has the property that each group of size n that can be formed with objects in the population is equally likely to be the sample.

Sampling Schemes: (SRS)

• For example:

 (^) Suppose that a class consists of 5 people and I wish to do a SRS of n = 3 people.  (^) If I cannot pick the same person twice, then there are 10 possible groups of 3 people: A,B,C,D,E represent the 5 people in the class. The 10 possible groups are {A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {A,C,E}, {A,D,E}, {B,C,D}, {B,C,E}, {B,D,E}, {C,D,E}.  (^) With a SRS, each possible group therefore must have a 1 in 10 chance of being the sample I pick.

Simple Random Sample (SRS) With or Without Replacement?

• (^) In a SRS with replacement, after each selection from the population, the object is returned to the population for the remaining draws.  (^) De facto when sampling from theoretical populations – manufacturing processes and games of chance.

SRS or not?

• (^) Example: A class consists of 20 people: 10 male

and 10 female. The teacher wishes to sample 10

students. The teacher will flip a coin. If the coin

lands heads, the teacher will take all the male

students as the sample. If tails, the teacher will take

all the female students as the sample. SRS or not?

 (^) In order to be a SRS, it must be like drawing names out of a hat containing the names of all students.  (^) There are actually 184,756 different groups of 10 that can be formed from the class. Do each of these groups have the same chance?

Sampling Schemes: Convenience Sample

• (^) A convenience sample is a non-random sampling method that chooses members of the population that are easiest to observe.  (^) Great danger that the sample is not representative of the population.  (^) Example: using this class to represent a sample of all ECU students.

Sampling Schemes: Systematic Sample  (^) The members of the population are in some order.

• (^) A systematic sample observes every k th value starting with a random starting point.  (^) Example: Selecting every 100th (^) item from a production line.  (^) Example: Selecting every 50th (^) name from an ECU directory.

Sampling Schemes: Cluster Sample  (^) The population is broken into groups called clusters.  (^) Some clusters are randomly chosen.

• (^) All the members of the chosen clusters are combined to create a cluster sample.  (^) Useful when each cluster is filled with similar individuals.  (^) Example: Taking 25 consecutive items off a production line every 10 minutes.

Example

• (^) I wish to obtain a sample of first graders in

Wisconsin.

 (^) If I randomly select 850 names from a list of all first graders in Wisconsin, then my sample will be a ____________________ of size n = _____?  (^) If I randomly select 2 first graders from each of the 697 elementary schools in Wisconsin, then my sample would be a ______________ of size n = ___________?  (^) If I were to randomly select 4 counties from Wisconsin’s 72 counties, and the take every first grader in the 4 selected counties to be my sample, then my sample would be a _____________ of an undetermined size.