Intro to Statistics

The Science of Statistics

• Statistics is a valuable science applicable to business, government, and various sciences.
• Statistics can be misleading if misapplied.
• Statisticians are trained in data collection, evaluation, and conclusion derivation.
• They determine relevant information and assess the trustworthiness of study conclusions.

Types of Statistical Applications

• Statistics involves describing data sets and drawing conclusions through sampling.
• Applications are divided into descriptive and inferential statistics.
• Focus for the semester is on descriptive statistics and probability, with an overview of statistical inference.
• A graph describing the various categories of boxes of Girl Scout cookies illustrates descriptive statistics.

Fundamental Elements of Statistics

• Statistical methods aid in studying populations of experimental units.
• Experimental or observational unit is an object about which data is collected.
• Population refers to a set of all units of interest, whether people, objects, or events.
• Examples of population are employed workers in the US or registered voters in California.
• Variable is a characteristic or property of population units.
• Variable examples include age, gender, and education years in unemployed individuals.
• Census involves measuring a variable for every unit of a population.
• It is viable if the population to be studied is small.
• Sample refers to a subset of population units.
• Instead of polling 145 million voters, a pollster questions a sample of 1,500.
• After measuring variables in a sample, data is analyzed using descriptive or inferential methods.
• Pollsters may describe voting patterns within a 1,500-voter sample.
• Inferences allow drawing conclusions about the whole 145 million voter population.
• Statistical inference involves estimates or generalizations about a population from a sample.
• It involves the use of smaller sample data to learn about a larger population.
• Consider a Broadway executive hypothesizing ticket-buyer average age is less than 42.5 years
• The population of interest is all ticket-buyers for the theatre's plays.
• The variable of interest is age in years of each ticket buyer.
• The sample is a subset of the population, specifically 200 ticket-buyers.
• Inference generalizes sample information to the population, estimating the average age of all ticket-buyers.
• Making the inference necessitates that reliability will signify the quality of the inference.
• Reliability is the fifth element of inferential statistical problems and indicates confidence in the inference.
• It is important to realize the incompleteness of an inference without a measure of its reliability.

Summary - Descriptive Statistical Problems

• Identify the population or sample of concern.
• Determine the variables such as the characteristics.
• Employ tables, graphs and summary tools.
• Recognize patterns in the data.

Summary - Inferential Statistical Problems

• Define the population of interest.
• Determine which population variable must be investigated.
• Establish the sample population units.
• Make inferences.
• Measure the reliability.

Types of Data

• Quantitative data are measurements recorded on a naturally occurring numerical scale.
• Examples include temperature, the unemployment rate in states, applicant score on the LSAT or the number of convicted murders.
• Qualitative (or categorical) data are measurements not on a natural numerical scale but are classified into categories.
• Examples include political affiliation, defective status of computer chips, car size, or ranking barbecue sauce.

Collecting Data: Sampling and Related Issues

• Data can be obtained from a published source or a designed experiment.
• The designed experiment can also be sourced from an observational study.
• Published sources include the Turkish Statistical Institute, The Wall Street Journal, and Kaggle.com.
• In a designed experiment the researcher exerts full control over sampled experimental units.
• Experimental units are assigned to treatment and control groups.
• The Observational study involves observing experimental units in their natural setting without manipulation; surveys are an example of this.
• It is important that data from any collection method remains a sample from a specified population, even for inferential statistics.
• A representative sample exhibits characteristics typical of those possessed by the target population.
• Simple random sample means every different sample size has an equal chance of selection.
• If every subset of 1,500 voters is equally likely to be selected from 145 million voters, a random sample exists.
• You may wish to gauge opinions on a new high school, by using a random number generator of 20 households from the 711
• All households will be assigned a number from 1 to 711.
• Requesting that 20 households be selected without replacement ensures that the sample output is correct.
• Stratified random sampling is used when a population can be separated into strata where characteristics are more similar within than across.
• These could be representative samples of both Republicans and Democrats.
• Cluster sampling samples natural groups of experimental units, such as sampling 10 restaurant locations out of 150.
• Systematic sampling involves selecting every kth experimental unit from a list.
• Every fifth person into a shopping mall could be asked whether they own a smart phone.
• Randomized response sampling is useful when questions may elicit false answers.
• People receive 2 questions, where one being the survey is innocuous.
• For instance ask if they cheated on a federal income tax return of of they drank coffee.
• A researcher wants to learn more about the amount of people who have shoplifted.
• Flip a coin, if it lands on heads, answer all the questions truthfully.
• If tails, answer "Yes" no matter what.
• If 70% of respondents answer "yes", 50% of those answers were due to truthful responses. True proportion = (observed proportion - probability of random yes) / probability of truthful response
• It makes approximately 40% of the respondents have actually shoplifted.

Randomized Response Sampling - Advantages

• Anonymity increases since true responses are hidden.
• Reduced bias since respondents are not compelled to lie about sensitive data.

Randomized Response Sampling - Limitations

• The analysis is more complex.
• Requires larger sample sizes to achieve precision.
• Selection bias occurs when some experimental units have little chance of selection, leading to unrepresentative samples.
• Nonresponse bias is a selection bias when data on experimental units in a sample are not obtained; such as telephone and mail surveys.
• Measurement error involves inaccuracies from ambiguous or leading questions.

Example

• What percentage of Web users are addicted to the Internet?
• Use a survey involving 10 questions.
• Conduct the survey on ABCNews.com.
• Internet users responded to the questions.
• The 17,251 Internet users form a subset.
• This implies the data is a sample of the entire internet user population.

The Role of Statistics in Critical Thinking and Ethics

• Statistical thinking involves rational thought and the science of statistics to critically assess data and inferences.
• Quantitative literacy helps make informed decisions and generalizations, enhancing critical thinking and includes comparison of fatality rates.