Name or pick six variables and you must mention the following for each:

1. Mention if it is a quantitative variable or a qualitative variable and WHY.

2. If the variable is a quantitative variable, say whether it is discrete or continuous and WHY. If it is a qualitative variable, say whether it is either nominal or ordinal and WHY.

Week

1 – **Data** Collection

**– Introduction to the Practice of Statistics**

**Objective**s

1. Define statistics and statistical thinking

2. Explain the process of statistics

3. Distinguish between qualitative and quantitative variables

4. Distinguish between discrete and continuous variables

5. Determine the level of measurement of a variable

**Objective 1** – Define statistics and statistical thinking

Statistics

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.

Data

· The information referred to in the definition is data.

· **Data **are a “fact or proposition used to draw a conclusion or make a decision.” Data describe characteristics of an individual.

· A key aspect of data is that they vary.

· One goal of statistics is to describe and understand sources of variability.

**Objective 2 – Explain the Process of Statistics
**

· A **Population** consists of the entire group of individuals to be studied.

· A **Sample**

is a subset of the population that is being studied.

· An **Individual** is a person or object that is a member of the population being studied.

· **Descriptive Statistics** consists of organizing and summarizing data and describing data through numerical summaries, tables, and graphs.

· A Statistic is a numerical summary based on a sample.

· **Inferential Statistics** uses methods that take results from a sample, extends them to the population,

and measures the reliability of the result.

· A **Parameter**

is a numerical summary of a population.

**Parameter versus Statistic**

Example: Suppose the percentage of all students on our campus who have a job is 84.9%.

a) What is the population?

** ** **All students on our campus. **

b) Does the value 84.9% represent a parameter or a statistic?

** This is a Parameter. **

c) Suppose a sample of 250 students is obtained, and from this sample we find that 86.4% have a job.

Does the value 86.4% represent a parameter or a statistic?

** This is a statistic. **

**Illustrating the process of Statistics**

**Step 1**: Identify the research objective.

**Step 2**: Collect the information needed to answer the question.

**Step 3:** Describe the data Organize and summarize the information.

**Step 4**: Draw conclusions from the data.

**Objective 3 – Distinguish between Qualitative and Quantitative Variables
**

· **Variables **are the characteristics of the individuals within the population.

· **Key Point: Variables vary.**

Consider the variable height. If all individuals had the same height, then obtaining the height of one individual would be sufficient in knowing the heights of all individuals. Of course, this is not the case. As researchers, we wish to identify the **factor**s that influence variability.

**Variables and Types of Data**

Variables can be classified as Qualitative or Quantitative.

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

· Quantitative variables provide numerical measures of individuals. Arithmetic operations such as addition and subtraction can be performed on the values of the quantitative variable and provide meaningful results.

**Objective 4 – Distinguish between Discrete and Continuous Variables**

**Quantitative variables can be further classified into two groups.**

· A

Discrete variable is a quantitative variable that has either a finite number of possible values or a countable number of possible values. The term “countable” means the values result from counting such as 0, 1, 2, 3, and so on. (e.g. # of books, # of desks)

· A Continuous variable is a quantitative variable that has an infinite number of possible values it can take on and can be measured to any desired level of accuracy.

**Classification of Variables**

Example: Classify each variable as qualitative or quantitative. If the variable is quantitative, further classify it as discrete or continuous.

a) The number of heads obtained after flipping a coin five times.

**This is Quantitative and Discrete since there is a finite number of heads. **

b) Weights of newborn babies in a hospital.

**This is Quantitative and Continuous since the value is measurable or has an infinite number of possibilities. **

c) Eye colors of students in Math 227.

**This Qualitative since it implies a category. **

**Objective 5 – Determine the Level of Measurement of a Variable**

· Variables can also be classified by how they are categorized, counted, or measured.

· The level of measurement of the data is useful in deciding what procedure to take to apply statistics to real problems.

**Four common types of measurement scales are used to classify variables:**

· **Nominal Level of Measurement**: 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.

· **Ordinal Level of Measurement**: it has the properties of the nominal level of measurement.

The naming scheme allows for the values of the variable to be arranged in a ranked, or specific, order.

· **Interval Level of Measurement**: it has the properties of the ordinal level of measurement.

The differences in the values of the variable have meaning. A value of zero in the interval level of measurement does not mean the absence of the quantity. Arithmetic operations such as addition and subtraction can be performed on values of the variable.

· **Ratio Level of Measurement**: it has the properties of the interval level of measurement.

The ratios of the values of the variable have meaning. A value of zero in the ratio level of measurement means the absence of the quantity. Arithmetic operations such as multiplication and division can be performed on the values of the variable.

Example: Classify each as nominal-level, ordinal-level, interval-level, or ratio level data.

a) Sizes of cars **(This is ordinal since it implies order or ranking)**

b) Nationality of each student **(This is nominal since you can name it)**

c) IQ of each student **( This is an interval since a value of ZERO does not mean the absence of the quantity, a person can take the **

** test and score zero )**

d) Weight

**( This is a ratio since a value of ZERO means the absence of the quantity, if there is not weight, there is nothing there)**

**– Data Collection**

**Observational Studies Versus Designed Experiments
**

Objective

1. Distinguish between an observational study and an experiment

Objective 1 – **Distinguish between an Observational Study and an Experiment
**

· An Observational Study measures the value of the **response variable** without attempting to influence the value of either the response or **explanatory variable**s. That is, in an observational study, the researcher observes the behavior of the individuals in the study without trying to influence the outcome of the study.

· If a researcher assigns the individuals in a study to a certain group, intentionally changes the value of the explanatory variable, and then records the value of the response variable for each group, the researcher is conducting a Designed Experiment.

Example: Cellular Phones and Brain Tumors

In both studies, the goal of the research was to determine if radio frequencies from cell phones increase the risk of contracting brain tumors. Whether or not brain cancer was contracted is the response variable (dependent variable). The level of cell phone usage is the explanatory variable (independent variable).

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

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

A **Lurking Variable** is an explanatory variable that was not considered in a study, but that affect the value of the response variable in the study. In addition, lurking variables are typically related to any explanatory variables considered in the study.

Example: Identify the explanatory variable and the response variable for the following studies.

a) Rats with cancer are divided into two groups. One group receives 5 mg of a medication that is used to fight cancer, and the other receives 10 mg. After 2 years, the spread of the cancer is measured.

**Explanatory variable – The Amount of Medication/Dosage**

** Response Variable – Measure of the Spread of Cancer**

b) A researcher wants to determine whether young couples who marry are more likely to gain weight than those who stay single.

**Explanatory variable – Marital Status**

** Response variable – Weight Gained **

A **
census
** is a list of all individuals in a population along with certain characteristics of each individual.

– Data Collection

**Simple Random Sampling**

Objective

1. Obtain a Simple Random Sample

**Objective 1 – Obtaining a Simple Random Sample**

· 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.

Example: Illustrating Simple Random Sampling

Suppose a study group of consists of 5 students: Bob, Patricia, Mike, Jan, and Maria

2 of the students must go to the board to demonstrate a homework problem.

List all possible samples of size 2 (without replacement).

**{ (Bob, Patricia) , (Bob, Mike) , (Bob, Jan) , (Bob, Maria) , (Patricia, Mike) , (Patricia, Jan) , (Patricia, Maria) , (Mike, Jan) , (Mike, Maria) , (Jan, Maria) }**

Data Collection

**Other Effective Sampling Methods**

**Objectives**

1. Obtain a **Stratified Sample**

2. Obtain a **Systematic Sample**

3. Obtain a **Cluster Sample**

**Objective 1 – Obtain a Stratified Sample**

· A Stratified Sample is one obtained by separating the population into homogeneous, 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.

Example: A school official divides the student population into five classes. Freshman, Sophomore, Junior, Senior, and graduate students. The official takes a simple random sample from each class and asks the member’s opinions regarding student services.

**Objective 2 – Obtain a Systematic Sample**

· A Systematic Sample is obtained by selecting every kth individual from the population. The first individual selected is a random number between 1 and k.

Example: Every 10th customer entering a grocery store is asked to select her or his favorite color.

**Objective 3 – Obtain a Cluster Sample**

· A Cluster Sample is obtained by selecting all individuals within a randomly selected collection or group of individuals.

Example: A farmer divides his orchard into 30 subsections, randomly selects 4 and sample all the trees within the 4 subsections to approximate the yield of his orchard.

**Note: We will cover Frequency Tables and Frequency Distributions in Week 2 on Descriptive Statistics !!! **

Data Collection

**– Bias in Sampling**

Objectives

1. Explain the Sources of Bias in Sampling

**Objective 1 – Explain the Sources of Bias in Sampling**

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

**Three Sources of Bias**

1. **Sampling Bias** occurs when the technique used to obtain the individuals to be in the sample tends to favor one part of the population over another.

2. **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.

3. **Response Bias** exists when the answers on a survey do not reflect the true feelings of the respondent.

Types of Bias:

1) Interviewer error 2) Misrepresented answers

3) Words used in survey question 4) Order of the questions or words within the question.

Data Collection

**The Design of Experiments**

Objectives

1. Describe the Characteristics of an Experiment

2. Explain the Steps in Designing an Experiment

3. Explain the **Completely Randomized Design**

4. Explain the **Matched-Pairs Design**

5. Explain the Randomized Block Design

**Objective 1 – Describe the Characteristics of an Experiment**

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

The **Experimental Unit** is a person, object or some other well-defined item upon which a treatment is applied.

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

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

**Blinding **refers to nondisclosure of the treatment an experimental unit is receiving.

A **single-blind** experiment is one in which the experimental unit (or subject) does not know which treatment he or she is receiving.

A **double-blind **experiment is one in which neither the experimental unit nor the researcher in contact with the experimental unit knows which treatment the experimental unit is receiving.

Example: The Characteristics of an Experiment

The English Department of a community college is considering adopting an online version of the freshman English course. To compare the new online course to the traditional course, an English Department faculty member randomly splits a section of her course. Half of the students receive the traditional course and the other half is given an online version. At the end of the semester, both groups will be given a test to determine which performed better.

(a) Who are the experimental units? **The students in the class**

(b) What is the population for which this study applies?** All students who enroll in the class**

(c) What are the treatments? **Traditional vs. online instruction**

(d) What is the response variable? **The exam score**

(e) Why can’t this experiment be conducted with blinding? **Both the student and instructor know which treatment they are receiving **

**Objective 2 – Explain the Steps in Designing an Experiment**

To Design an experiment means to describe the overall plan in conducting the experiment.

**Steps in Conducting an Experiment**

Step 1: Identify the problem to be solved.

· Should be explicit

· Should provide the experimenter direction

· Should identify the response variable and the population to be studied.

· Often referred to as the claim.

Step 2: Determine the factors that affect the response variable.

· Once the factors are identified, it must be determined which factors are to be fixed at some predetermined level (the control), which factors will be manipulated and which factors will be uncontrolled.

Step 3: Determine the number of experimental units.

Step 4: Determine the level of the predictor variables

**1. Control: **There are two ways to control the factors.

· Fix their level at one predetermined value throughout the experiment. These are variables whose effect on the response variable is not of interest.

· Set them at predetermined levels. These are the factors whose effect on the response variable interests us. The combinations of the levels of these factors represent the treatments in the experiment.

**2. Randomize: **Randomize the experimental units to various treatment groups so that the effects of variables whose level cannot be controlled is minimized. The idea is that randomization “averages out” the effect of uncontrolled predictor variables.

**
Step 5: **Conduct the Experiment

· **Replication **occurs when each treatment is applied to more than one experimental unit. This helps to assure that the effect of a treatment is not due to some characteristic of a single experimental unit. It is recommended that each treatment group have the same number of experimental units.

· Collect and process the data by measuring the value of the response variable for each replication. Any difference in the value of the response variable is a result of differences in the level of the treatment.

**Step 6:** Test the claim.

· This is the subject of inferential statistics.

· Inferential statistics is a process in which generalizations about a population are made on the basis of results obtained from a sample. Provide a statement regarding the level of confidence in the generalization. Methods of inferential statistics are presented later in the text.

**Objective 3 – Explain the Completely Randomized Design**

A Completely Randomized Design is one in which each experimental unit is randomly assigned to a treatment.

Example: Designing an Experiment

The octane of fuel is a measure of its resistance to detonation with a higher number indicating higher resistance. An engineer wants to know whether the level of octane in gasoline affects the gas mileage of an automobile. Assist the engineer in designing an experiment.

Step 1: The response variable is miles per gallon.

Step 2: Factors that affect miles per gallon:

Engine size, outside temperature, driving style, driving conditions, characteristics of car

Step 3: We will use 12 cars all of the same model and year.

Step 4: We list the variables and their level.

Octane level – manipulated at 3 levels. Treatment A: 87 octane, Treatment B: 89 octane, Treatment C: 92 octane Engine size – fixed

Temperature – uncontrolled, but will be the same for all 12 cars.

Driving style/conditions – all 12 cars will be driven under the same conditions on a closed track – fixed.

Other characteristics of car – all 12 cars will be the same model year, however, there is probably variation from car to car. To account for this, we randomly assign the cars to the octane level.

Step 5: Randomly assign 4 cars to the 87 octane, 4 cars to the 89 octane, and 4 cars to the 92 octane. Give each car 3 gallons of gasoline. Drive the cars until they run out of gas. Compute the miles per gallon.

Step 6: Determine whether any differences exist in miles per gallon.

Completely Randomized Design

**Objective 4. Explain the Matched-Pairs Design**

A Matched-Pairs Design is an experimental design in which the experimental units are paired up. The pairs are matched up so that they are somehow related (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.

Example: A Matched-Pairs Design

Xylitol has proven effective in preventing dental caries (cavities) when included in food or gum. A total of **75 Peruvian children** were given milk with and without Xylitol and were asked to evaluate the taste of each. The researchers measured the children’s ratings of the two types of milk. (Source: Castillo JL, et al (2005) Children’s acceptance of milk with Xylitol or Sorbitol for dental caries prevention. BMC Oral Health (5)6.)

(a) What is the response variable in this experiment?

**The Ratings **

(b) Think of some of the factors in the study. Which are controlled? Which factor is manipulated?

**Age and gender of the children are controlled. Milk with Xylitol is the factor manipulated. **

(c) What are the treatments? How many treatments are there?

**Milk with Xylitol and milk without xylitol, two treatments. **

(d) What type of experimental design is this? **Matched-pairs design**

(e) Identify the experimental units. 75 Peruvian children

(f) Why would it be a good idea to randomly assign whether the child drinks the milk with Xylitol first or second?

**Remove any effect due to order in which milk is drunk. **

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