The Nature of Probability and Statistics Chapter 1 The Nature of Probability and Statistics WCB/McGraw-Hill The McGraw-Hill Companies, Inc., 1998

1-1 Descriptive & Inferential Statistics Statistics consists of conducting studies to collect, organize, summarize, analyze, and draw conclusions. A variable is a characteristic or attribute than can assume different values.

1-1 Descriptive and Inferential Statistics Variables whose values are determined by chance are called random variables. Data are the values (measurements or observations) that the variables can assume.

1-1 Descriptive and Inferential Statistics A collection of data values forms a data set. Each value in the data set is called a data value or a datum.

1-1 Descriptive and Inferential Statistics A population consists of all subjects (human or otherwise) that are being studied. When data are collected from every subject in the population, it’s called a census. A sample is a group of some subjects from the population.

1-1 Descriptive and Inferential Statistics Statistics is divided into two main areas, determined by the way the data is used. Descriptive statistics consists of the collection, organization, summation, and presentation of data. Describes a situation

1-1 Descriptive and Inferential Statistics Inferential statistics consists of generalizing from samples to populations, performing hypothesis testing, determining relationships among variables, and making predictions. Uses probability

1-1 Descriptive and Inferential Statistics Determine whether descriptive or inferential statistics were used: 1. In a weight loss study using teenagers at Boston University, 52% of the group said that they lost weight and kept it off by counting calories. Descriptive

1-1 Descriptive and Inferential Statistics Determine whether descriptive or inferential statistics were used: 2. Because of the current economy, 49% of 18-34 year olds have taken a job to pay the bills. Inferential

1-1 Descriptive and Inferential Statistics Determine whether descriptive or inferential statistics were used: 3. A recent article stated that over 38 million U.S. adults binge watch TV shows. Inferential

Example: Owners of comic book shops are concerned that sales of comic books are declining. A study revealed that women currently account for 53% of comic readers, (Graphic Policy & Facebook). Three years ago women accounted for 40% of readers. Over time, depiction of women in comics has changed. Further study to evaluate whether the perception of women is affected by the rise in strong, confident female comic characters depicted as heroes in movies. Also, currently, according to WSC professor Joe Salzman, fewer than 30% of comic characters and writers are female. (Fermoso, J. September 11, 2014, The Rise of the Woman Comic Buyer, OZY.com). It has been predicted that comic book publishers will focus on comics that lure the female comic book reader.

What are the variables under study? What data have been collected? Have descriptive, inferential, or both types of statistics used? What is the population under study? Was a sample collected? If so, from where? From the information given, comment on the relationship between the variables.

1-2 Variables and Types of Data Remember from 1-1, A variable is a ______________ or a(n) _______________ that can assume ____________ ________. Variables may be categorical, also called _________________ variables, or they may be numerical, called __________________ variables.

1-2 Variables and Types of Data Qualitative variables are variables that can be placed into distinct categories, according to some characteristic or attribute. For example, gender (male or female).

1-2 Variables and Types of Data Quantitative variables are numerical in nature and can be ordered or ranked. Example: temperature and age Quantitative variables can be classified in 2 groups: _____________ and _____________

1-2 Variables and Types of Data Discrete variables assume values that can be counted. EX. The number of phone calls received by a 911 call center.

1-2 Variables and Types of Data Continuous variables can assume all values between any two specific values. They are obtained by measuring. EX. Weights of suitcases on a flight

1-2 Variables and Types of Data Variables Qualitative Quantitative Discrete Continuous

1-2 Variables and Types of Data Qualitative and quantitative variables can also be classified, or categorized, using measurement scales. The four common types of scales for classifying variables are nominal, ordinal, interval, and ratio.

1-2 Variables and Types of Data The nominal level of measurement classifies data into mutually exclusive (nonoverlapping), exhausting categories in which no order or ranking can be imposed on the data. Male/female

1-2 Variables and Types of Data The ordinal level of measurement classifies data into categories that can be ranked; precise differences between the ranks do not exist. tall, grande, venti

1-2 Variables and Types of Data The interval level of measurement ranks data; precise differences between units of measure do exist; there is no meaningful zero. IQ tests- No true zero; doesn’t measure people with no intelligence.

1-2 Variables and Types of Data The ratio level of measurement possesses all the characteristics of interval measurement, and there exists a true zero. In addition, true ratios exist for the same variable. height, weight, area Weight 100 lbs to 50 lbs; ratio is 2 to 1

Classify each of the following variables first as qualitative or quantitative, then determine the level of measurement (nominal, ordinal, interval or ratio):   _______________ _______________ Eye color _______________ _______________ Number of years employed by a company _______________ _______________ Ranking of a tennis player _______________ _______________ The year you graduated from high school _______________ _______________ IQ _______________ _______________ Time elapsed _______________ _______________ Grade (A, B, C, D, F) _______________ _______________ Street address _______________ _______________ Items on a menu _______________ _______________ The number of pasta dishes on a menu _______________ _______________ The price of menu items

1-3 Data Collection and Sampling Techniques Data can be collected in a variety of ways. One of the most common methods is through the use of surveys. Surveys can be done by using a variety of methods - Examples are telephone, mail questionnaires, personal interviews, surveying records and direct observations.

1-3 Data Collection and Sampling Techniques Researchers use samples to collect data about variables from a large population. A sample is used to get information about a population because: It save the researcher time and money. It enables the researcher to get information that may not be easy to get otherwise. It enables the researcher to get more detailed information about a subject.

1-3 Data Collection and Sampling Techniques Biased samples have a systematic error that was made in the selection of the subjects. A) sampling bias- some subjects are more likely to be included in a study than others B) nonresponse bias- subjects that do not respond to a question would answer differently than those that did respond C) response bias- subject gives a different response than he/she truly believes D) volunteer bias – volunteers are used because they are more interested in the survey

1-3 Data Collection and Sampling Techniques To obtain samples that are unbiased, ie. that give each subject in the population an equal chance of being selected, statisticians use different methods of sampling.

1-3 Data Collection and Sampling Techniques Random samples are selected by using chance methods or random numbers. Every member has an equal chance of being selected.

1-3 Data Collection and Sampling Techniques Example: Use your TI-83/84 calculator, generate 10 random numbers between 1 and 85.

1-3 Data Collection and Sampling Techniques Example: Use the random number table to select 10 subjects.

1-3 Data Collection and Sampling Techniques Systematic samples are obtained by numbering each value in the population and then selecting the kth value. Suppose there is a list of 350 subjects and a sample of 35 is needed. 350 divided by 35 is 10, so each group includes 10 subjects. Use your calculator to randomly select a number between 1 and 10, ______. This is the first subject selected. The sample includes the first, ____, and each one following by adding 10 to each following. List your systematic sample:

1-3 Data Collection and Sampling Techniques Stratified samples are selected by dividing the population into groups (strata) according to some characteristic and then taking samples from each group. Ex.take a sample from Freshmen class by eye color

1-3 Data Collection and Sampling Techniques Cluster samples are selected by dividing the population into groups and then taking samples of the groups. Ex. Sample from a Freshmen class

1-3 Data Collection and Sampling Techniques Convenience samples uses subjects that are available or easy to find. Ex. Sample from people entering a store

1-3 Data Collection and Sampling Techniques Sequence sampling used in quality control, where successive units taken from production lines are sampled to ensure the products meet certain standards Ex. Blue Bell tests units #122-128 to ensure ingredients are distributed throughout the entire gallon

1-3 Data Collection and Sampling Techniques Sampling error Samples are not perfect representatives of the populations from which they are selected, therefore there is always some difference in the results. Other errors that exist in samples occur for other reasons.

1-3 Data Collection and Sampling Techniques Sampling error is the difference between the results obtained from a sample and the results obtained from the population from which the sample is selected. A sample of full-time students indicates that 56% of the student body is female. However, the admissions office states that of all full-time students, 54% are female. The difference of 2% is due to __________________.

1-3 Data Collection and Sampling Techniques Non-sampling error occurs when the data are obtained erroneously or the sample is biased or nonrepresentative.

1-3 Data Collection and Sampling Techniques A researcher interviews subjects who enter the mall through a particular entrance to determine the reason for their visit. This sample is most likely not representative of general customers for several reasons, such as, the sample was likely taken at a specific time of day when not all customers to the mall were shopping and the sample was taken at a particular entrance through which all customers do not enter. Therefore, all customers to the mall did not have an equal chance of being selected. The difference between the sample and the population is due to ______________________________.

1-4 Experimental Design Observational study – the researcher observes what is happening or what has happened in the past & tries to draw conclusions based on the observations. Experimental study– the researcher manipulates one of the variables and tries to determine how the manipulation influences other variables.

1-4 Experimental Design Independent variable– the variable that is manipulated by the researcher (also called explanatory variable). Dependent variable– the resultant variable (also called outcome variable).

Example: Identify the independent and dependent variable: A study was conducted to determine whether when a restaurant server drew a happy face on the check, that would increase the amount of the tip.

1-4 Experimental Design Given two study groups, the group that receives special treatment is called the treatment group and the group that receives no special treatment is called the control group.

1-4 Factors that influence experiments Hawthorne effect- subjects who know they are participating in an experiment change their behavior and therefore affect the results of the study. Placebo effect- results of a study obtained by subjects who improve but not due to the conditions of the study

1-4 Factors that influence experiments blinding – subjects don’t know if they are receiving treatment or a placebo Double blinding- subjects and researchers aren’t told who was given the placebo

1-4 Guidelines used to do statistical studies: Formulate the purpose of the study. Identify the variables for the study. Define the population. Decide what sampling method to use to collect data. Collect the data. Summarize the data & perform needed statistical calculations. Interpret the results.

1-4 Uses and misuses of statistics Suspect samples Ambiguous averages Detached statistics Implied connections Misleading graphs Faulty survey questions. Changing the subject

1-4 Ethics An Ethics problem is when you are considering an action that: benefits you or supports you in some way Hurts or reduces benefits to others Violates some rule

1-4 Example: A company gives you $10,000 to research a new drug with a promise of $100,000 more if it is successful You find the data is inconclusive: 20/100 deaths with the new drug, 21/102 deaths with another drug Should you: Look deeper to find evidence that the new drug is better? Do an analysis you suspect is wrong? Do an analysis you know is wrong? Fake the data?

1-4 Ethics Data Collection: data collection can be made inherently biased by posing the wrong questions that stimulate strong emotions rather than objective realities. This happens all the time when the survey is aimed to try and prove a viewpoint rather than find out the truth. Other unethical behaviors might include scientists not including data outliers in their report and analysis to validate their theory or viewpoint. By obscuring data or taking only the data points that reinforce a particular theory, scientists are indulging in unethical behavior. Data Representation: Numbers don't lie but their interpretation and representation can be misleading. For example, after a broad survey of many customers, a company might decide to publish and make available only the numbers and figures that reflect well on the company and either totally neglect or not give due importance to other figures. For example, a car might be ranked high on comfort but low on safety. By showing only the comfort figures for the car, the company is, in a way, misleading customers and shareholders about the real picture.

1-4 Ethics There are ethics in statistics that need to be followed by a researcher so that only the truth is reported and there is no misrepresentation of the data. It is relatively simple to manipulate and hide data, projecting only what one desires and not what the numbers actually speak. As a researcher it is important to be objective and provide the complete picture that has been obtained from the experiment without hiding any details or overemphasizing something for personal gain. Ethics in statistics are important to give the right direction to research so that it is objective and reflects the truth.

1-5 Computers and Calculators Computers and calculators make numerical computation easier. Many statistical packages are available. Two examples are MINITAB and Excel. The TI-84 calculator can also be used to do statistical calculations. A free online program called “R” is useful as well & may be used in other classes in college.