Elementary Statistics Week 1 Intro to the Practice of Statistics

What is a statistic?

A numerical summary of a sample. Like a mean, median, range... of a sample. E.g. In a bag of M&Ms can we calculate how many red, yellow, and green M&Ms are there?

What indicates that a design experiment is being conducted?

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.

Types of Levels of Measurement of a Variable

Nominal, Ordinal, Interval, Ratio

Case-control Studies (retrospective)

Observational studies that are retrospective, meaning that they require individuals to look back in time or require the researcher to look at existing records. In case-control studies, individuals who have certain characteristics are matched with those that do not.

Types of Response Bias

Types of Response Bias: 1.Interviewer error 2.Misrepresented answers 3.Words used in survey question 4.Order of the questions or words within the question

What is a type of sampling bias?

Undercoverage

multistage sampling

Use a variety of sampling methods to create successively smaller groups at each stage. The final sample consists of clusters.

census

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

individual

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

When does a sample has bias?

a sample has bias if the results of the sample are not representative of the population

Undercoverage

a typle of sampling bias that occurs when the proportion of one segment of the population is lower in a sample than it is in the population.

What are the two types of quantitative variables?

discrete variables and continuous variables

Continuous data

observations corresponding to a continuous variable

population

the entire group of individuals to be studied

data

the list of observations a variable assumes

Random sampling

the process of using chance to select individuals from a population to be included in the sample.

Is gender data or a variable?

variable

EXAMPLE Obtaining a Stratified Sample

-In 2008, the United States Senate had 47 Republicans, 51 Democrats, and 2 Independents. The president wants to have a luncheon with 4 Republicans, 4 Democrats and 1 Other. Obtain a stratified sample in order to select members who will attend the luncheon. -To obtain the stratified sample, conduct a simple random sample within each group. That is, obtain a simple random sample of 4 Republicans (from the 47), a simple random sample of 4 Democrats (from the 51), and a simple random sample of 1 Other from the 100. Be sure to use a different seed for each stratum.

EXAMPLE Multistage Sampling

-In practice, most large-scale surveys obtain samples using a combination of the techniques just presented. -As an example of multistage sampling, consider Nielsen Media Research. Nielsen randomly selects households and monitors the television programs these households are watching through a People Meter. The meter is an electronic box placed on each TV within the household. The People Meter measures what program is being watched and who is watching it. Nielsen selects the households with the use of a two-stage sampling process. -Stage 1 Using U.S. Census data, Nielsen divides the country into geographic areas (strata). The strata are typically city blocks in urban areas and geographic regions in rural areas. About 6000 strata are randomly selected. Stage 2 Nielsen sends representatives to the selected strata and lists the households within the strata. The households are then randomly selected through a simple random sample. Nielsen sells the information obtained to television stations and companies. These results are used to help determine prices for commercials.

Define Statistics

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

Convenience Sampling

-the process of using easy to get results (internet polls, voluntary polls) -If convenience is used to obtain a sample, the results of the survey are meaningless.

continuous variable

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. e.g. height and weight

Discrete Variable (Quantitative)

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.

sample

a subset of the population that is being studied

What is data?

facts or propositions used to draw a conclusion or to make a decision

What is a key aspect of data?

Data varies. Eg. There are variables in the characteristics of individuals. IOW. I can run a study multiple times and may not ever get the same results.

What two things can be created after obtaining a data of an individual from a sample?

Descriptive statistics and a statistic

Discrete data

observations corresponding to a discrete variable.

Qualitative data

observations corresponding to a qualitative variable.

Quantitative data

observations corresponding to a quantitative variable.

inferential statistics

uses methods that take results from a sample, extends them to the population, and measures the reliability of the result. iow: Based on my sample, what do I think the entire population would be?

While gender is a variable, the observations, male or female, are data.

variable; data

Chapter 1 objectives

•Define statistics and statistical thinking •Explain the Process of Statistics •Distinguish between Qualitative and Quantitative Variables •Distinguish between Discrete and Continuous Variables •Determine the Level of Measurement of a Variable •Distinguish between an Observational Study and a designed experiment •Explain the Various Types of Observational Studies •Obtain a Simple Random Sample .Obtain a Stratified Sample .Obtain a Systematic Sample .Obtain a Cluster Sample .Explain the Sources of Bias in Sampling

EXAMPLE Obtaining a Systematic Sample

-A quality control engineer wants to obtain a systematic sample of 25 bottles coming off a filling machine to verify the machine is working properly. Design a sampling technique that can be used to obtain a sample of 25 bottles. -STEPS IN SYSTEMATIC SAMPLING, POPULATION SIZE KNOWN Step 1: Determine the population size, N. Step 2: Determine the sample size desired, n. Step 3: Compute N/n and round down to the nearest integer. This value is k. Step 4: Randomly select a number between 1 and k. Call this number p. Step 5: The sample will consist of the following individuals: p, p + k, p + 2k,..., p + (n - 1)k

convenience sample

-A sample is one in which the individuals in the sample are easily obtained -Caution: Any studies that use this type of sampling generally have results that are suspect. Results should be looked upon with extreme skepticism.

ordinal level of measurement

-Data that can be arranged in an order, but differences between data values is meaningless -E.g. Grades, rankings

ratio level of measurement

-Data that can be arranged, differences are meaningful and there is a zero starting point -e.g. lengths, heights, distances

interval level of measurement

-Data that can be arranged, differences are meaningful, but NO TRUE ZERO -e.g.Temperature, years

nominal level of measurement (qualitative)

-Data that consists of names, labels and categories -E.g. Yes/no, color, identification numbers

EXAMPLE Obtaining a Cluster Sample

-E.g. A school administrator wants to obtain a sample of students in order to conduct a survey. She randomly selects 10 classes and administers the survey to all the students in the class.

Is male or female data or a variable?

it is observation data of a variable

Eg of Parameter versus statistic

-Suppose the percentage of all the students on campus who have a job is 84.9%. This value represents a parameter b/c it is a numerical summary of a population. So, if given a numeric value as a percentage, it is a parameter. Mnemonic: p with p (parameter/population) -Suppose a sample of 250 students is obtained, and from this sample 84.9% have a job. This value represents a statistic b/c it is a numerical summary of a sample of the population. Mnemonic: s with s (statistic/sample) e.g. a news report that list statistics based on a sample taken and are giving you what was found out of the number of people surveyed. Note: The majority of surveys in this class will be conducted as a statistic b/c it is very difficult to survey an entire population)

EXAMPLE of Obtaining a Simple Random Sample

-The 112th Congress of the United States had 435 members in the House of Representatives. Explain how to conduct a simple random sample of 5 members to attend a Presidential luncheon. Then obtain the sample. -*Step 1 Put the members in alphabetical order. Number the members from 1 - 435. *Step 2 Randomly select five numbers using a random number generator. First, set the seed. The seed is 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. Then generate the random numbers. *Step 3 Match the generated random numbers to the corresponding Representatives.

Sampling bias

-a bias that exists when the technique used to obtain the individuals to be in the sample tends to favor one part of the population over another. -Undercoverage is a type of sampling bias. Undercoverage occurs when the proportion of one segment of the population is lower in a sample than it is in the population.

Nonresponse bias

-a type of bias that 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 be improved through the use of callbacks or rewards/incentives.

Response bias

-a type of bias that exists when the answers on a survey do not reflect the true feelings of the respondent. -Types of Response Bias: 1.Interviewer error 2.Misrepresented answers 3.Words used in survey question 4.Order of the questions or words within the question

qualitative variable (categorical)

-allow for classification of individuals based on some attribute or characteristic - ex. gender, political affiliation, color of car, zip code, etc. -words that describe a person is a qualitative type data

descriptive statistics

-consist of organizing and summarizing data -describe data through numerical summaries, tables, and graphs. i.o.w. Things we can calculate on what we collect

What does an observational study measure?

-measures the value of the response variable without attempting to influence the value of either the response or explanatory variables. 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. -Note: Observational studies do not allow a researcher to claim causation, only association.

parameter

-numerical summary of a population -iow a numerical measurement describing some characteristic of a population

quantitative variable

-provide numerical measures of individuals -Arithmetic operations, such as addition and subtraction, can be performed on the values of the qualitative variables and provide meaningful results

variables

-the characteristics of the individuals within the population (e.g. age, height, marital status, income, etc.) -Variables vary and change and are different for each individual (sample conducted) -Note: Before I look at a sample I have to know what is being studied

What is the goal of statistics?

-to draw conclusions about a population based on what is observed from a sample taken from that population. -One goal of statistics is to describe and understand sources of variability - IOW Find out something about the population, pull a sample of the population, and ask things of each individual within the sample. -Note: A sample taken of each individual in a population is extremely rare because there is just too many individuals!!!

Steps for Obtaining a Simple Random Sample

1) Obtain a frame that lists all the individuals in the population of interest. Number the individuals in the frame 1 - N. 2) Use a random number table, graphing calculator, or statistical software to randomly generate n numbers where n is the desired sample size.

Distinguishing between Discrete and Continuous Variables: 1. Goals scored in a season by a soccer player 2. Volume of water lost in leaky faucet 3. Length of a rope 4. Number of roses on a rose bush 5. Temperature

1. Discrete Variable 2. Continuous Variable 3. Continuous Variable 4. Discrete Variable 5. Continuous

The Process of Statistics

1. Identify the research objective 2. Collect the data needed to answer the question(s) posed in (1). 3. Describe the data. 4. Perform inference make an analysis and see what you come up with

Distinguishing between qualitative and quantitative variables: Classify each variable as qualitative of quantitative: 1. Nationality 2. Number of children 3. Household income in the previous year 4. Level of education 5. Daily intake of whole grains (measured in grams per day) 6. Nation of origin 7. Number of sibling 8. Grams of carbohydrates in a doughnut 9. Number on football jersey 10. Number of unpopped popcorn kernels 11. Assessed value of a house 12. phone number 12.

1. Qualitative 2. Quantitative 3. Quantitative 4. Qualitative 5. Quantitative 6. Qualitative 7. Quantitative 8. Quantitative 9. Qualitative 10. Quantitative 11. Quantitative 12. Qualitative

three sources of bias in sampling

1. Sampling bias 2. Nonresponse bias 3. Response bias

Identify the sampling method used: 1.Intel selects every 8th chip that comes off the assembly line to check for quality. 2.Southwest Airlines randomly selects 60 flights during a certain week and surveys all passengers on those flights 3.A radio station asks listeners to call in for a poll 4.To determine internet speed, Shawn divides the day into 3 parts: morning, afternoon, and evening. He then measures the internet speed at 5 randomly selected times during each part of the day 5.24 hour fitness wants to administer a satisfaction survey to list current members. Using the membership roster, they randomly select 40 members and ask them about their satisfaction with the health club.

1. Systemic 2. Cluster 3. Convenience 4. Stratified 5. Simple Random

Types of sampling methods

1. Systemic 2. Cluster 3. Convenience 4. Stratified 5. Simple Random

Determining the Level of Measurement of a Variable: 1. Number of snack and soft drink vending machines in the school 2. Whether or not the school has a closed campus policy during lunch 3. Class rank (Freshman, Sophomore, Junior, Senior) 4. Number of days per week a student eats school lunch 5. Nation of origin 6. Movie rating (based on five stars) 7. Volume of water used daily 8. Eye color 9. Assessed value of a house 10. Time of day (military time)

1. ratio 2. nominal 3. ordinal 4. ratio 5. nominal 6. ordinal 7. ratio 8. nominal 9. ratio 10. interval

cluster sample

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

systematic sample

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

stratified sample

A sample obtained by separating the population into homogeneous, nonoverlapping 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.

What are the Various Types of Observational Studies?

Cross-sectional Studies, Case-control Studies, Cohort Studies

Cross-sectional Studies

Observational studies that collect information about individuals at a specific point in time, or over a very short period of time.

Cohort Studies (prospective/longitudinal)

Observational studies that first identifies a group of individuals to participate in the study (the cohort). The cohort is then observed over a long period of time. Over this time period, characteristics about the individuals are recorded. Because the data is collected over time, cohort studies are prospective.

STEPS IN SYSTEMATIC SAMPLING, POPULATION SIZE KNOWN

STEPS IN SYSTEMATIC SAMPLING, POPULATION SIZE KNOWN Step 1: Determine the population size, N. Step 2: Determine the sample size desired, n. Step 3: Compute N/n and round down to the nearest integer. This value is k. Step 4: Randomly select a number between 1 and k. Call this number p. Step 5: The sample will consist of the following individuals: p, p + k, p + 2k,..., p + (n - 1)k

continuous variable example

height and weight

simple random sample

in a sample of size n from a population of size N, every possible sample of size n has an equally likely chance of occurring.

What do data do?

it describes the characteristics of an individual