An Overview of Statistics § 1.1 An Overview of Statistics

Data and Statistics Data consists of information coming from observations, counts, measurements, or responses. Statistics is the science of collecting, organizing, analyzing, and interpreting data in order to make decisions. A population is the collection of all outcomes, responses, measurement, or counts that are of interest. A sample is a subset of a population.

Populations & Samples Example: In a recent survey, 250 college students at Union College were asked if they smoked cigarettes regularly. 35 of the students said yes. Identify the population and the sample. Responses of all students at Union College (population) Responses of students in survey (sample)

Parameters & Statistics A parameter is a numerical description of a population characteristic. A statistic is a numerical description of a sample characteristic. Parameter Population Statistic Sample

Parameters & Statistics Example: Decide whether the numerical value describes a population parameter or a sample statistic. a.) A recent survey of a sample of 450 college students reported that the average weekly income for students is $325. Because the average of $325 is based on a sample, this is a sample statistic. b.) The average weekly income for all students is $405. Because the average of $405 is based on a population, this is a population parameter.

Branches of Statistics The study of statistics has two major branches: descriptive statistics and inferential statistics. Statistics Descriptive statistics Inferential statistics Involves the organization, summarization, and display of data. Involves using a sample to draw conclusions about a population.

Descriptive and Inferential Statistics Example: In a recent study, volunteers who had less than 6 hours of sleep were four times more likely to answer incorrectly on a science test than were participants who had at least 8 hours of sleep. Decide which part is the descriptive statistic and what conclusion might be drawn using inferential statistics. The statement “four times more likely to answer incorrectly” is a descriptive statistic. An inference drawn from the sample is that all individuals sleeping less than 6 hours are more likely to answer science question incorrectly than individuals who sleep at least 8 hours.

Note: The development of Inferential Statistics has occurred only since the early 1900’s. Examples: 1. The medical team that develops a new vaccine for a disease is interested in what would happen if the vaccine were administered to all people in the population. 2. The marketing expert may test a product in a few “representative” areas, from the resulting information, he/she will draw conclusion about what would happen if the product were made available to all potential customers.

Probability forms a bridge between the descriptive and inferential techniques and leads to a better understanding of statistical conclusions. Both Probability and Statistics deal with questions involving population and samples but do so in an “inverse manner” to one another

Probability (Properties of population are known) Sample Population Characteristics of the samples are known and you predict about whole population

Examples: Suppose you have a deck of cards and you select one card , what is the probability of selecting a king? Prob(king) = 4/52 = 1/13 Note: Here we know the population (deck of cards) and the sample is one card selected randomly → Probability

2. Every day, you see and hear public opinion polls (Harris poll, Gallup poll etc.). Even with most powerful computers and resources available, still pollsters can not find the opinions of more than 100 million Americans (population) in United States. Rather, they sample the opinions of a small number of voters (sample) and then use this information to make conclusion about the whole population → Inferential Statistics

The Essential Elements of a Statistical Problem The objective of statistics is to make inferences (predictions, and/or decisions) about a population based upon the information contained in a sample. A statistical problem involves the following 1. A clear definition of the objectives of the experiment and the pertinent population. For example, clear specification of the questions to be answered. 2 The design of experiment or sampling procedure. This element is important because data cost money and time.

3. The collection and analysis of data. 4. The procedure for making inferences about the population based upon the sample information. The provision of a measure of goodness (reliability) of the inference. The most important step, because without the reliability the inference has no meaning and is useless. Note, above steps to solve any statistical problem are sequential.

Population or Sample? A. Population B. Sample The age of each state governor A. Population B. Sample

Population or Sample? A. Population B. Sample The speed of every fifth car passing a police speed trap A. Population B. Sample

Population or Sample? A. Population B. Sample A survey of 500 students from a university with 10,000 students A. Population B. Sample

Population or Sample? A. Population B. Sample The annual salary for each employee at a company A. Population B. Sample

Population or Sample? A. Population B. Sample The cholesterol levels of 20 patients in a hospital with 100 patients A. Population B. Sample

Population or Sample? A. Population B. Sample The number of pets in each U.S. household A. Population B. Sample

Parameter or Statistic? The average annual salary for 35 of a company’s 1200 accountants is $68,000. A. Parameter B. Statistic

Parameter or Statistic? In a survey of a sample of high school students, 43% said that their mother has taught them the most about managing money. A. Parameter B. Statistic

Parameter or Statistic? In 2007, the interest category for 12% of all new magazines was sports. A. Parameter B. Statistic

Inferential or Descriptive Statistics? Fact: The following numbers of students passed this course in the past four years: 17, 19, 4, 20. Is the following conclusion obtained from purely descriptive measures or by inferential methods? The last four semesters the instructor taught Probability & Statistics, an average of 15 people passed the class. A. Inferential B. Descriptive

Inferential or Descriptive Statistics? Fact: The following numbers of students passed this course in the past four years: 17, 19, 4, 20. Is the following conclusion obtained from purely descriptive measures or by inferential methods? The next time the instructor teaches Probability & Statistics, we can expect approximately 15 people to pass the class. A. Inferential B. Descriptive

Inferential or Descriptive Statistics? Fact: The following numbers of students passed this course in the past four years: 17, 19, 4, 20. Is the following conclusion obtained from purely descriptive measures or by inferential methods? This instructor will never pass more than 20 people in a Probability & Statistics class. A. Inferential B. Descriptive

Inferential or Descriptive Statistics? Fact: The following numbers of students passed this course in the past four years: 17, 19, 4, 20. Is the following conclusion obtained from purely descriptive measures or by inferential methods? The last four semesters the instructor taught Probability & Statistics, no more than 20 people passed the class. A. Inferential B. Descriptive

Inferential or Descriptive Statistics? Fact: The following numbers of students passed this course in the past four years: 17, 19, 4, 20. Is the following conclusion obtained from purely descriptive measures or by inferential methods? Only 4 people passed one semester because the instructor was in a bad mood the entire semester. A. Inferential B. Descriptive

Inferential or Descriptive Statistics? Fact: The following numbers of students passed this course in the past four years: 17, 19, 4, 20. Is the following conclusion obtained from purely descriptive measures or by inferential methods? The instructor passes so few people in his Probability & Statistics classes because he doesn't like teaching that class. A. Inferential B. Descriptive