1.1 | Definitions of Statistics, Probability, and Key Terms

From the sample data, we can calculate a statistic.

A statistic is a number that represents a property of the sample. For example, if we consider one math class to be a sample of the population of all math classes, then the average number of points earned by students in that one math class at the end of the term is an example of a statistic. The statistic is an estimate of a population parameter.

A variable

A variable, usually notated by capital letters such as X and Y, is a characteristic or measurement that can be determined for each member of a population.

The idea of sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population.

Because it takes a lot of time and money to examine an entire population, sampling is a very practical technique. If you wished to compute the overall grade point average at your school, it would make sense to select a sample of students who attend the school. The data collected from the sample would be the students' grade point averages. In presidential elections, opinion poll samples of 1,000-2,000 people are taken. The opinion poll is supposed to represent the views of the people in the entire country. Manufacturers of canned carbonated drinks take samples to determine if a 16 ounce can contains 16 ounces of carbonated drink.

Data

Data are the actual values of the variable.

Variables may be numerical or categorical.

Numerical variables take on values with equal units such as weight in pounds and time in hours. Categorical variables place the person or thing into a category. If we let X equal the number of points earned by one math student at the end of a term, then X is a numerical variable. If we let Y be a person's party affiliation, then some examples of Y include Republican, Democrat, and Independent. Y is a categorical variable. We could do some math with values of X (calculate the average number of points earned, for example), but it makes no sense to do math with values of Y (calculating an average party affiliation makes no sense)

A parameter is a numerical characteristic of the whole population that can be estimated by a statistic. Since we considered all math classes to be the population, then the average number of points earned per student over all the math classes is an example of a parameter.

One of the main concerns in the field of statistics is how accurately a statistic estimates a parameter. The accuracy really depends on how well the sample represents the population.

Descriptive Statistics.

Organizing and summarizing data is called descriptive statistics.

Determine what the key terms refer to in the following study. We want to know the average (mean) amount of money first year college students spend at ABC College on school supplies that do not include books. We randomly surveyed 100 first year students at the college. Three of those students spent $150, $200, and $225, respectively.

Solution 1.1 The population is all first year students attending ABC College this term.The sample could be all students enrolled in one section of a beginning statistics course at ABC College (although this sample may not represent the entire population). The parameter is the average (mean) amount of money spent (excluding books) by first year college students at ABC College this term. The statistic is the average (mean) amount of money spent (excluding books) by first year college students in the sample. The variable could be the amount of money spent (excluding books) by one first year student. Let X = the amount of money spent (excluding books) by one first year student attending ABC College. The data are the dollar amounts spent by the first year students. Examples of the data are $150, $200, and $225.

Determine what the key terms refer to in the following study. A study was conducted at a local college to analyze the average cumulative GPA's of students who graduated last year. Fill in the letter of the phrase that best describes each of the items below. 1. Population_____ 2. Statistic _____ 3. Parameter _____ 4. Sample _____ 5. Variable _____ 6. Data _____ a) all students who attended the college last yearb) the cumulative GPA of one student who graduated from the college last yearc) 3.65, 2.80, 1.50, 3.90d) a group of students who graduated from the college last year, randomly selectede) the average cumulative GPA of students who graduated from the college last yearf) all students who graduated from the college last yearg) the average cumulative GPA of students in the study who graduated from the college last year

Solution 1.2 1. f; 2. g; 3. e; 4. d; 5. b; 6. c

Determine what the key terms refer to in the following study. As part of a study designed to test the safety of automobiles, the National Transportation Safety Board collected and reviewed data about the effects of an automobile crash on test dummies. Here is the criterion they used: Speed at which cars crashed 35 miles/hour Location of "drive" (i.e. dummies) Front Seat Table 1.1 Cars with dummies in the front seats were crashed into a wall at a speed of 35 miles per hour. We want to know the proportion of dummies in the driver's seat that would have had head injuries, if they had been actual drivers. We start with a simple random sample of 75 cars.

Solution 1.3 The population is all cars containing dummies in the front seat. The sample is the 75 cars, selected by a simple random sample. The parameter is the proportion of driver dummies (if they had been real people) who would have suffered head injuries in the population. The statistic is proportion of driver dummies (if they had been real people) who would have suffered head injuries in the sample. The variable X = the number of driver dummies (if they had been real people) who would have suffered head injuries. The data are either: yes, had head injury, or no, did not.

Determine what the key terms refer to in the following study. An insurance company would like to determine the proportion of all medical doctors who have been involved in one or more malpractice lawsuits. The company selects 500 doctors at random from a professional directory and determines the number in the sample who have been involved in a malpractice lawsuit.

Solution 1.4 The population is all medical doctors listed in the professional directory.The parameter is the proportion of medical doctors who have been involved in one or more malpractice suits in the population.The sample is the 500 doctors selected at random from the professional directory. The statistic is the proportion of medical doctors who have been involved in one or more malpractice suits in the sample. The variable X = the number of medical doctors who have been involved in one or more malpractice suits. The data are either: yes, was involved in one or more malpractice lawsuits, or no, was not.

Datum

They may be numbers or they may be words. Datum is a single value.

Inferential Statistics. Statistical inference uses probability to determine how confident we can be that our conclusions are correct.

Two ways to summarize data are by graphing and by using numbers (for example, finding an average). After you have studied probability and probability distributions, you will use formal methods for drawing conclusions from "good" data. The formal methods are called inferential statistics.

The sample must contain the characteristics of the population in order to be a representative sample.

We are interested in both the sample statistic and the population parameter in inferential statistics. In a later chapter, we will use the sample statistic to test the validity of the established population parameter.

Population In statistics, we generally want to study a population.

You can think of a population as a collection of persons, things, or objects under study. To study the population, we select a sample.

Effective interpretation of data (inference) is based on good procedures for producing data and thoughtful examination of the data.

You will encounter what will seem to be too many mathematical formulas for interpreting data. The goal of statistics is not to perform numerous calculations using the formulas, but to gain an understanding of your data. The calculations can be done using a calculator or a computer. The understanding must come from you. If you can thoroughly grasp the basics of statistics, you can be more confident in the decisions you make in life.

Probability - is a mathematical tool used to study randomness. It deals with the chance (the likelihood) of an event occurring.

example, if you toss a fair coin four times, the outcomes may not be two heads and two tails. However, if you toss the same coin 4,000 times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is 12 or 0.5. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin 24,000 times with a result of 12,012 heads, one of the authors tossed a coin 2,000 times. The results were 996 heads. The fraction 996 is equal to 0.498 which is very close to 0.5, the expected probability.