Math250 Statistics I Chapter 1

Front 1

STATISTICS

Back 1

The science of collecting, organizing, summarizing, and analyzing information to draw conclusions or answer questions. In addition, statistics is about providing a measure of confidence in any conclusions.

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Lurking Variable

Back 2

Lurking variables must be considered because two variables are often influenced by a third variable.

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1. Population

2. Individual

3. Sample

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1. The entire group to be studied.

2. A person or object that is a member of the population being studied.

3. A subset of the population that is being studied.

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a Statistic & Descriptive Statistics

Back 4

a numerical summary of a sample.

Descriptive statistics consist of organizing and summarizing data. These describe data through numerical summaries, tables, and graphs.

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Parameter

Back 5

a numerical summary of a population.

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The Process of Statistics

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1. Identify the research objective.

2. Collect the data needed to answer to question(s) posed in (1).

3. Describe the data.

4. Perform inference.

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Variables

Back 7

the characteristics of the individuals within the population.

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Qualitative (or categorical) Variables

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allow for classification of individuals based on some attribute or characteristic.

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Quantitative Variables

Back 9

provide numerical measures of individuals. The values of a quantitative variable can be added or subtracted and provide meaningful results.

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Approach

Back 10

a way to look at and organize a problem so that it can be solved. The approach will be a suggested method of attack toward solving a problem.

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Discrete Variable

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a quantitative variable that has either a finite number of possible values or a countable number of possible values. The term countable means that the values result from counting, such as 0, 1, 2, 3, and so on. A discrete variable cannot take on every possible value between any two possible values.

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Continuous Variable

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a quantitative variable that has an infinite number of possible values that are not countable. A continuous variable may take on every possible value between any two values.

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Discrete vs. Continuous

Variables

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If you count to get the value of a quantitative variable, it is discrete. If you measure to get the value of a quantitative variable, it is continuous.

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DATA

Back 14

the list of observed values for a variable.

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Qualitative DATA

Back 15

observations corresponding to a qualitative variable.

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Quantitative DATA

Back 16

observations corresponding to a quantitative variable.

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Discrete DATA

Back 17

observations corresponding to a discrete variable.

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Continuous DATA

Back 18

observations corresponding to a continuous variable.

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NOMINAL

level of measurement

nominal: Latin, nomen for "name"

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A variable is at the nominal level of measurement if the values of the variable name, label, or categorize. In addition, the naming scheme does not allow for the values of the variable to be arranged in a ranked or specific order.

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ORDINAL

level of measurement

ordinal: think "order"

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A variable is at the ordinal level of measurement if it has the properties of the nominal level of measurement, however the naming scheme allows for the values of the variable to be arranged in a ranked or specific order.

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INTERVAL

level of measurement

Back 21

A variable is at the interval level of measurement if it has the properties of the ordinal level of measurement and the differences in the values of the variable have meaning. A value of zero does not mean the absence of the quantity. Arithmetic operations such as addition and subtraction can be performed on values of the variable.

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RATIO

level of measurement

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A variable is at the ratio level of measurement if it has the properties of the interval level of measurement and the ratios of the values of the variable have meaning. A value of zero means the absence of the quantity. Arithmetic operations such as multiplication and division can be performed on the values of the variable.

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Nominal & Ordinal vs. Interval & Ratio

(Qualitative vs. Quantitative)

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Variables that are nominal or ordinal are

qualitative variables.

Variables that are interval or ratio are

quantitative variables.

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Response vs. Explanatory

Variables

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Response Variable: End result.

Explanatory Variable: Affect leading to the end result.

"In research, we wish to determine how varying the amount of an explanatory variable affects the value of a response variable.

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Observational Study

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measures the value of the response variable without attempting to influence the value of either the response or explanatory variables.

The researcher observes the behavior of the individuals without trying to influence the outcome of the study.

Observational studies do not allow a researcher to claim causation, only association.

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Designed Experiment

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a researcher assigns the individuals in a study to a certain group, intentionally changes the value of an explanatory variable, and then records the value of the response variable for each group.

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Confounding

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occurs in a study when the effects of two or more explanatory variables are not separated. Therefore, any relation that may exist between an explanatory variable and the response variable may be due to some other variable or variables not accounted for in the study.

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Lurking Variable

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an explanatory variable that was not considered in a study, but that affects the value of the response variable in the study. In addition, lurking variables are typically related to explanatory variables considered in the study.

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Cross-sectional Studies

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These observational studies collect information about individuals at a specific point in time or over a very short period of time.

An advantage is that they are cheap and quick to do. However, limitations include affects that occur after data is collected, thus not giving the full picture.

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Case-control Studies

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These studies are retrospective, meaning that they require individuals to look back in time or require the researcher to look at existing records. Individuals who have a certain characteristic may be matched with those who do not.

A disadvantage is that it requires individuals to recall information from the past and to be truthful in their responses. An advantage is that they can be done relatively quickly and inexpensively.

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Cohort Studies

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A cohort study first identifies a group of individuals to participate in the study (the cohort). They are then observed over a long period of time. During this period, characteristics about them are recorded and some are exposed to certain factors (not intentionally) and others will not be. At the end, the value of the response variable is recorded for the individuals. Typically, this requires many individuals to participate over long periods of time. Because of this, cohort studies are prospective. Problem: individuals tend to drop out due to the long time frame. This could cause misleading results. Cohort studies are the most powerful of the observational studies.

Front 32

Census

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a list of all individuals in a population along with certain characteristics of each individual.

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Simple Random Sample

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A sample of size "n" from a population of size "N" is obtained through simple random sampling if every possible sample of size "n" has an equally likely chance of occurring. The sample is then called a simple random sample.

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Frame

Back 34

a list of all the individuals within the population.

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Sample without Replacement

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an individual who is selected is removed from the population and cannot be chosen again.

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Sample with Replacement

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a selected individual is placed back into the population and could be chosen a second time.

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SEED

Back 37

an initial point for the generator to start creating random numbers - like selecting the initial point in the table of random numbers. The seed can be any nonzero number.

Front 38

Stratified Sample

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A stratified sample is obtained by separating the population into non overlapping groups called strata and then obtaining a simple random sample from each stratum. The individuals within each stratum should be homogeneous (or similar) in some way.

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Stratum vs. Strata

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Stratum is singular, while strata is plural. The word strata means divisions. so a stratified sample is a simple random sample of different divisions of the population.

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Systematic Sample

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A systematic sample is obtained by selecting every kith individual from the population. The first individual selected corresponds to a random number between 1 and k.

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Cluster Sample

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A cluster sample is obtained by selecting all individuals within a randomly selected collection or group of individuals.

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Convenience Sample

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A convenience sample is a sample in which the individuals are easily obtained and not based on randomness.

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Self-Selected / Voluntary Response

Samples

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The most popular of the many types of convenience samples are those in which the individuals themselves decide to participate in a survey. There are also called voluntary samples.

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BIAS

Back 44

If the results of the sample are not representative of the population, then the sample has bias.

The word bias could mean to give preference to selecting some individuals over others; it could also mean that certain responses are more likely to occur in the sample than in the population.

Three sources of bias in sampling:

1. Sampling bias

2. Nonresponse bias

3. Response bias

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Sampling Bias

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Sampling bias means that the technique used to obtain the sample's individuals tends to favor on e part of the population over another. Any convenience sampling has sampling bias because the individuals are not chosen through a random sample.

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Undercoverage

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Sampling bias also results due to undercoverage, which occurs when the proportion of one segment of the population is lower in a sample than it is in the population. Undercoverage can result if the frame used to obtain the sample is incomplete or not representative of the population.

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Nonresponse Bias

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Nonresponse bias exists when individuals selected to be in the sample who do not respond to the survey have different opinions from those who do. Nonresponse can occur because individuals selected for the sample do not wish to respond or the interviewer was unable to contact them.

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Response Bias

Back 48

Response bias exists when the answers on a survey do not reflect the true feelings of the respondent. Response bias can occur in a number of ways:

1. Interviewer Error

2. Misrepresented Answers

3. Wording of Questions

4. Ordering of Questions or Words

5. Type of Question (open vs. closed)

6. Data-entry Error

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Non sampling Errors

and

Sampling Errors

Back 49

Non sampling errors result from undercoverage, nonresponse bias, response bias, or data-entry error. Such errors could also be present in a complete census of the population.

Sampling error results from using a sample to estimate information about a population. This type of error occurs because a sample gives incomplete information about a population.

Front 50

Experiment

Factors

Treatment

Back 50

An experiment is a controlled study conducted to determine the effect varying one or more explanatory variables or factors has on a response variable. Any combination of the values of the factors is called a treatment.

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Experimental Unit

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a person, object, or some other well-defined item upon which a treatment is applied. We often refer to the experimental unit as a subject when he or she is a person. The subject is analogous to the individual in a survey.

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Control Group

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A control group serves as a baseline treatment that can be used to compare to other treatments.

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Placebo

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A placebo is an innocuous medication, such as a sugar tablet, that looks, tastes, and smells like the experimental medication.

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Blinding

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refers to nondisclosure of the treatment an experimental unit is receiving. There are two types of blinding: single blinding and double blinding.

Front 55

Single-blind Experiments

vs.

Double-blind Experiments

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In single-blind experiments, the experimental unit (or subject) does not know which treatment he or she is receiving. In double-blind experiments, neither the experimental unit nor the researcher in contact with the experimental unit knows which treatment the experimental unit is receiving.

Front 56

DESIGN

Back 56

To design an experiment means to describe the overall plan in conducting the experiment. Conducting an experiment requires a series of steps:

1. Identify the problem to be solved.

2. Determine the Factors that affect the Response Variable

3. Determine the Number of Experimental Units

4. Determine the Level of each Factor

5. Conduct the Experiment

6. Test the Claim

Front 57

Randomize

Back 57

Randomly assign the experimental units to various treatment groups so that the effect of factors whose levels cannot be controlled is minimized. The idea is to average out the effects of uncontrolled factors (explanatory variables). It is difficult to identify all factors in an experiment, so randomization is important. It mutes the effect of variation attributable to factors not controlled.

Front 58

Replication

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occurs when each treatment is applied to more than one experimental unit.

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Inferential Statistics

Back 59

a process in which generalizations about a population are made on the basis of results obtained from a sample.

Front 60

Completely Randomized Design

Back 60

one in which each experimental unit is randomly assigned to a treatment.

Front 61

Matched-Pairs Design

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an experimental design in which the experimental units are paired up. The pairs are selected so that they are related in some way (that is, the same person before and after a treatment, twins, husband and wife, same geographical location, and so on). There are only two levels of treatment in a matched-pairs design.