1.1 Definitions of Statistics, Probability, and Key Terms
Learning Objectives
- Recognize the difference between descriptive and inferential statistics.
- Describe how probability measures the chance that an event happens.
- Identify a population, a sample, a parameter, a statistic, a variable, and data in a study.
- Classify a variable as numerical (discrete or continuous) or categorical.
In this section, we get our bearings. We pin down what the word statistics actually means, see how probability gives us a way to talk about chance, and lock in the core vocabulary — population, sample, parameter, statistic, variable, and data — that every later chapter leans on. None of this is heavy math yet; it is the language you will speak for the rest of the course.
1.1.1 The Science of Statistics
The science of statistics deals with the collection, analysis, interpretation, and presentation of data. We see and use data in our everyday lives — sleep schedules, test scores, prices at the store, weather forecasts.
In your classroom, try this exercise. Have class members write down the average time (in hours, to the nearest half-hour) they sleep per night. Your instructor will record the data. Then create a simple graph (called a dot plot) of the data. A dot plot consists of a number line and dots (or points) positioned above the number line. For example, consider the following data:
5; 5.5; 6; 6; 6; 6.5; 6.5; 6.5; 6.5; 7; 7; 8; 8; 9
The dot plot for this data appears in Figure 1.1.1 below. Does your dot plot look the same as or different from the example? Why? If you ran the same exercise in an English class with the same number of students, do you think the results would match? Why or why not? Where do your data appear to cluster, and how might you interpret that clustering?
Figure 1.1.1 — Frequency of average time (in hours) spent sleeping per night.
The questions above ask you to analyze and interpret your data. With this small example, you have already begun your study of statistics.
Here is the one-sentence version of the whole course: descriptive statistics describes the data you actually have, while inferential statistics uses that data to make a careful guess about a much larger group you could never measure all of. Describe first, then infer.
In this course, you will learn how to organize and summarize data. Organizing and summarizing data is called descriptive statistics. 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. Those formal methods are called inferential statistics — and statistical inference uses probability to measure how confident we can be that our conclusions are correct.
Effective interpretation of data (inference) rests on two things: good procedures for producing data, and thoughtful examination of the data you produced. Along the way you will meet what can feel like too many formulas. But the goal of statistics is not to grind through calculations — a calculator or computer can do those. The goal is to understand your data, and that understanding has to come from you. If you genuinely grasp the basics, you can be far more confident in the decisions you make in life.
A researcher records the number of hours of sleep for every student in a single dorm and then makes a dot plot and computes the average. Later, she uses that average to estimate the typical nightly sleep of all students at the university.
Which part of her work is descriptive statistics, and which part is inferential statistics?
Solution
Descriptive statistics: making the dot plot and computing the average for the students in that one dorm. She is describing the data she actually collected.
Inferential statistics: using that average to estimate the typical sleep of all students at the university — a group far larger than the one she measured. She is inferring beyond her data.
Answer: the dot plot and average are descriptive; the university-wide estimate is inferential.
1.1.2 Probability
This "few tosses are wild, many tosses settle down" idea is the quiet engine behind everything to come. Insurance companies, casinos, and medical researchers cannot predict any single outcome — but over thousands of cases the long-run proportions become remarkably predictable. That predictability is exactly what probability lets us pin a number on.
Probability is a mathematical tool used to study randomness. It measures the chance (the likelihood) of an event occurring. For example, if you toss a fair coin four times, the outcome may not be exactly two heads and two tails. However, if you toss that same coin 4,000 times, the results will land close to half heads and half tails. The expected theoretical probability of heads on any single toss is \(\frac{1}{2}\), or \(0.5\). Even though the outcome of a few tosses is uncertain, a regular pattern emerges over many repetitions.
History backs this up. The English statistician Karl Pearson once tossed a coin 24,000 times and got 12,012 heads. One of the original authors tossed a coin 2,000 times and got 996 heads. The fraction \(\frac{996}{2000} = 0.498\) is very close to \(0.5\), the expected probability.
The theory of probability began with the study of games of chance, such as poker. Today, predictions of all kinds take the form of probabilities. To estimate the chance of an earthquake, of rain, or of getting an A in this course, we use probabilities. Doctors use probability to weigh the chance a vaccination causes the very disease it prevents. A stockbroker uses probability to estimate the rate of return on a client's investments. You might use probability to decide whether to buy a lottery ticket. Throughout your study of statistics, you will use the power of probability to analyze and interpret your data.
A weather service says there is a \(70\%\) chance of rain tomorrow. A friend objects: "It either rains or it doesn't — so the chance is really just 50/50." Explain why your friend is wrong, using what you know about how probability behaves over many repetitions.
Solution
Probability is not about a single yes/no outcome being "even." It is a long-run measure: of all the days the weather service has called "70% chance of rain" under similar conditions, rain has actually occurred on about \(70\%\) of them.
Saying "it rains or it doesn't, so 50/50" ignores all the information packed into the forecast. A fair coin is genuinely 50/50; tomorrow's weather, given the atmospheric data, is not.
Answer: the two outcomes (rain / no rain) are not equally likely, so the chance is \(70\%\), not \(50\%\). "Either/or" tells you the possible outcomes, not how likely each one is.
1.1.3 Populations, Samples, and Variables
A population is the entire collection of persons, things, or objects under study. It is the whole group we ultimately want to draw a conclusion about.
A sample is a portion (a subset) of the population that we actually examine. We study the sample to gain information about the larger population. Data are the result of sampling from a population.
A statistic is a number that represents a property of the sample — for example, the average score of the students in one class we surveyed. A statistic is an estimate of a population parameter.
A memory trick that pays off all term: the two p's go together, and the two s's go together. Parameter describes a Population; Statistic describes a Sample. Whenever you are unsure which word to use, check whether you are talking about the whole group (population → parameter) or just the part you measured (sample → statistic).
A parameter is a numerical characteristic of the whole population — for example, the average score of every student in every class. We can rarely measure a parameter directly, so we estimate it with a statistic.
A variable, usually written with a capital letter such as \(X\) or \(Y\), is a characteristic or measurement that can be determined for each member of a population. Variables are either numerical (values with equal units, such as weight in pounds or time in hours) or categorical (values that place each member into a category).
Data are the actual values of the variable. They may be numbers or words. A single value is called a datum.
We generally want to study a population — a collection of persons, things, or objects. Because examining an entire population usually costs too much time and money, we instead select a sample: a manageable subset that we study to learn about the whole. If you wanted the overall grade point average at your school, you would not track down every student — you would survey a sample, and the GPAs you collect would be your data. In presidential elections, opinion polls sample just 1,000–2,000 people yet aim to represent the views of an entire country. Manufacturers sample cans off the line to check whether a 16-ounce can really holds 16 ounces.
From the sample data we compute a statistic — a number summarizing the sample. The statistic estimates a parameter, the matching number for the whole population. One of the central concerns of statistics is how accurately a statistic estimates a parameter, and that accuracy depends on how well the sample represents the population. A good sample carries the characteristics of the population it came from. Figure 1.1.2 ties these four ideas together in one picture.
Figure 1.1.2 — A sample is drawn from a population; a statistic (about the sample) estimates a parameter (about the whole population).
A bit more vocabulary you will reach for constantly. To see how a variable works, suppose 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 instead we let \(Y\) be a person's party affiliation, then values of \(Y\) include Republican, Democrat, and Independent, so \(Y\) is a categorical variable. We can do arithmetic with the values of \(X\) (we can average points earned), but it makes no sense to do arithmetic with the values of \(Y\) (there is no "average party affiliation").
Two more words show up often: mean and proportion. If you scored 86, 75, and 92 on three exams, your mean score is \(\frac{86 + 75 + 92}{3} = 84.3\) (to one decimal place). A proportion is the fraction of a group with some characteristic. For instance, if your math class has 40 students and 18 of them are women, then the proportion of women students is \(\frac{18}{40} = 0.45\). Mean and proportion both get a much fuller treatment in later chapters.
The words "mean" and "average" are often used interchangeably, and that substitution is common practice. The precise technical term is arithmetic mean; "average" technically refers to a center location. Among non-statisticians, though, "average" is widely accepted as standing in for "arithmetic mean."
Determine what the key terms refer to in the following study. We want to know the average (mean) amount of money spent on school uniforms each year by families with children at Knoll Academy. We randomly survey 100 families with children in the school. Three of the families spent $65, $75, and $95, respectively.
Solution
- Population: all families with children attending Knoll Academy.
- Sample: the 100 families randomly surveyed.
- Parameter: the average (mean) yearly amount spent on uniforms by all Knoll Academy families.
- Statistic: the average (mean) yearly amount spent on uniforms by the 100 families in the sample.
- Variable: \(X\) = the amount one surveyed family spends on uniforms in a year.
- Data: the dollar amounts spent, such as $65, $75, and $95.
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
Population: all first-year students attending ABC College this term.
Sample: the group we actually surveyed — for instance, all students enrolled in one section of a beginning statistics course at ABC College (note that such a sample may not perfectly represent the whole population).
Parameter: the average (mean) amount of money spent on supplies (excluding books) by all first-year students at ABC College this term.
Statistic: the average (mean) amount spent (excluding books) by the first-year students in the sample.
Variable: let \(X\) = the amount of money spent (excluding books) by one first-year student attending ABC College.
Data: the dollar amounts spent by the surveyed first-year students — for example, $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 of students who graduated last year. Match the letter of the phrase that best describes each item.
- Population _____
- Statistic _____
- Parameter _____
- Sample _____
- Variable _____
- Data _____
Solution
- Population → f (all students who graduated last year)
- Statistic → g (average GPA of the students in the study)
- Parameter → e (average GPA of all graduates)
- Sample → d (the randomly selected group of graduates)
- Variable → b (the GPA of one graduate)
- Data → c (the actual GPA values: 3.65, 2.80, 1.50, 3.90)
Answer: 1‑f, 2‑g, 3‑e, 4‑d, 5‑b, 6‑c.
Determine what the key terms refer to in the following study. A survey of athletes at a university was conducted to study the heights of athletes, in meters. Match the letter of the phrase that best describes each item.
- Population
- Statistic
- Parameter
- Sample
- Variable
- Data
Solution
- Population → c (all athletes in the university — not d, which is all students)
- Statistic → b (average height of athletes in the survey)
- Parameter → a (average height of all athletes in the university)
- Sample → f (the randomly selected group of athletes)
- Variable → e (the height of one athlete)
- Data → g (the actual heights: 1.82, 1.76, 1.69, 1.93)
Answer: 1‑c, 2‑b, 3‑a, 4‑f, 5‑e, 6‑g.
Determine what the key terms refer to in the following study. A survey checks the time a mobile phone takes to charge its battery from 50% to 100%. The criteria used to collect the data are:
| Wattage of charger used | Type of mobile used |
|---|---|
| 30 W | Android |
We want to know the proportion of Android mobiles that charge to 100% within 30 minutes. We start with a simple random sample of 200 mobiles.
Solution
- Population: all Android mobiles (charged with a 30 W charger) that the study is about.
- Sample: the 200 mobiles selected by simple random sampling.
- Parameter: the proportion of all such Android mobiles that reach 100% within 30 minutes.
- Statistic: the proportion of the 200 sampled mobiles that reach 100% within 30 minutes.
- Variable: \(X\) = whether a given mobile reaches 100% within 30 minutes (yes or no).
- Data: the yes/no results — yes, charged within 30 minutes, or no, did not.
Determine what the key terms refer to in the following study. As part of a study designed to test the safety of electric automobiles, the National Transportation Safety Board collected and reviewed data about the effects of a crash on test dummies. Here is the criterion they used:
| Speed at which cars crashed | Location of "drivers" (i.e., dummies) |
|---|---|
| 35 miles/hour | Front seat |
Cars with dummies in the front seats were crashed into a wall at 35 miles per hour. We want to know the proportion of dummies in the driver's seat that would have had head injuries, had they been actual drivers. We start with a simple random sample of 75 cars.
Solution
Population: all cars containing dummies in the front seat.
Sample: the 75 cars selected by a simple random sample.
Parameter: the proportion of driver dummies (if they had been real people) who would have suffered head injuries, across the whole population.
Statistic: the proportion of driver dummies who would have suffered head injuries, within the sample of 75 cars.
Variable: \(X\) = whether a dummy (if it had been a real person) would have suffered a head injury.
Data: either yes, had a head injury or no, did not.
Determine what the key terms refer to in the following study. A news agency wants to find the proportion of all truck drivers who have no points on their license. The agency randomly selects 1,000 truck drivers from the directory of truck drivers and determines how many in the sample have no points.
Solution
- Population: all truck drivers in the directory.
- Sample: the 1,000 truck drivers randomly selected.
- Parameter: the proportion of all truck drivers in the directory with no points on their license.
- Statistic: the proportion of the 1,000 sampled drivers with no points.
- Variable: \(X\) = whether a given truck driver has no points on their license (yes or no).
- Data: the yes/no results for the sampled drivers.
Determine what the key terms refer to in the following study. An insurance company wants 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 how many in the sample have been involved in a malpractice lawsuit.
Solution
Population: all medical doctors listed in the professional directory.
Sample: the 500 doctors selected at random from the directory.
Parameter: the proportion of all listed doctors who have been involved in one or more malpractice suits.
Statistic: the proportion of the 500 sampled doctors who have been involved in one or more malpractice suits.
Variable: \(X\) = whether an individual doctor has been involved in a malpractice suit.
Data: either yes, was involved in one or more malpractice lawsuits or no, was not.
Work in groups of up to four. Find a population, a sample, the parameter, the statistic, a variable, and the data for the following study: you want to determine the average (mean) number of glasses of milk college students drink per day. Suppose yesterday, in your English class, you asked five students how many glasses of milk they drank the day before, and the answers were 1, 0, 1, 3, and 4 glasses of milk.
1.1.4 Types of Variables
We just saw that variables come in two broad flavors — numerical and categorical. Numerical variables split one step further. A discrete numerical variable comes from counting (whole-number results like the number of siblings), while a continuous numerical variable comes from measuring on a scale that can take any value in a range (like height, which can be 1.72 m or 1.7236 m). Categorical variables, by contrast, sort each individual into a group rather than attaching a number.
For each pair of variables, decide whether you would expect them to be associated or independent:
- A person's height and their shoe size.
- The last digit of a person's phone number and their favorite color.
Solution
- Associated — taller people tend to have larger feet, so knowing height gives you some information about shoe size.
- Independent — there is no reason the last digit of a phone number would tell you anything about a favorite color; knowing one gives no information about the other.
Answer: (a) associated; (b) independent.
To close the loop on vocabulary: we collect a sample of data to better understand a population. A variable is the characteristic we measure for each individual or case. The overall quantity of interest — the mean, the median, a proportion, or some other summary of the whole population — is a parameter. We estimate that parameter by taking a sample and computing a matching numerical summary, the statistic. Remember the mnemonic: the two p's (population, parameter) go together, and the two s's (sample, statistic) go together.
Data were collected about students in a statistics course. Three variables were recorded for each student: number of siblings, student height, and whether the student had previously taken a statistics course. Classify each variable as continuous numerical, discrete numerical, or categorical.
Solution
Number of siblings — this is a count, and counts are whole numbers, so it is a discrete numerical variable.
Student height — height varies smoothly along a scale and can take any value in a range, so it is a continuous numerical variable.
Previously taken a statistics course? — this sorts each student into one of two groups (yes or no) rather than attaching a number, so it is a categorical variable.
Answer: siblings = discrete numerical; height = continuous numerical; prior course = categorical.
Associated or Independent, Not Both
A pair of variables are either related in some way (associated) or not (independent). No pair of variables is both associated and independent at the same time.
Problem Set 1.1
Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new drug is currently under study to address a respiratory virus. It is given to patients once the patient exhibits symptoms of the virus. Of interest is the average (mean) length of time in days from the time the patient starts the treatment until the symptoms are alleviated. Two researchers each follow a different set of 40 patients with the respiratory virus from the start of treatment until the symptoms are alleviated. The following data (in days) are collected.
Researcher A: 3; 4; 11; 15; 16; 17; 22; 44; 37; 16; 14; 24; 25; 15; 26; 27; 33; 29; 35; 44; 13; 21; 22; 10; 12; 8; 40; 32; 26; 27; 31; 34; 29; 17; 8; 24; 18; 47; 33; 34
Researcher B: 3; 14; 11; 5; 16; 17; 28; 41; 31; 18; 14; 14; 26; 25; 21; 22; 31; 2; 35; 44; 23; 21; 21; 16; 12; 18; 41; 22; 16; 25; 33; 34; 29; 13; 18; 24; 23; 42; 33; 29
For problems 1 through 5, determine what the key terms refer to in the example for Researcher A.
Problem 1. population
Solution
Step 1 — Recall the definition: The population is the entire collection of individuals or objects we want to draw a conclusion about.
Step 2 — Apply it to Researcher A's study: The goal is to learn about the recovery time for all patients who take the new drug for this respiratory virus, not just the 40 who were followed.
Answer: The population is all patients with the respiratory virus who take the new drug (every person to whom the treatment could apply).
Problem 2. sample
Solution
Step 1 — Recall the definition: A sample is the subset of the population that is actually observed and measured.
Step 2 — Apply it to Researcher A's study: Researcher A did not measure every drug-taking patient — only the specific 40 patients he followed from treatment start to relief.
Answer: The sample is the 40 patients with the respiratory virus that Researcher A followed.
Problem 3. parameter
Solution
Step 1 — Recall the definition: A parameter is a numerical summary that describes a characteristic of the entire population.
Step 2 — Apply it to Researcher A's study: The quantity of interest is the average recovery time. Computed over the whole population, that mean is a parameter.
Answer: The parameter is the (unknown) mean number of days to symptom relief for the entire population of patients taking the drug.
Problem 4. statistic
Solution
Step 1 — Recall the definition: A statistic is a numerical summary computed from the sample; it estimates the corresponding parameter.
Step 2 — Apply it to Researcher A's study: The mean recovery time calculated from Researcher A's 40 patients is a statistic.
Answer: The statistic is the mean number of days to symptom relief computed from Researcher A's sample of 40 patients.
Problem 5. variable
Solution
Step 1 — Recall the definition: A variable is the characteristic being measured on each member of the sample; its value changes from patient to patient.
Step 2 — Apply it to Researcher A's study: The measured characteristic is the length of time, in days, from the start of treatment until symptoms are relieved.
Answer: The variable is \(X =\) the number of days from the start of treatment until the patient's symptoms are alleviated.
For each of problems 6 through 13, identify: a) the population, b) the sample, c) the parameter, d) the statistic, e) the variable, and f) the data. Give examples where appropriate.
Problem 6. A fitness center is interested in the mean amount of time a client exercises in the center each week.
Solution
Step 1 — Apply the definitions: Population = the entire group of interest; sample = the subset actually studied; parameter = a numerical fact about the population; statistic = a numerical fact computed from the sample; variable = the characteristic being measured; data = the actual values recorded.
Step 2 — Map them to this scenario (mean weekly exercise time):
a) Population: all clients of the fitness center.
b) Sample: the particular group of clients whose exercise time is actually measured.
c) Parameter: the mean weekly exercise time of all clients.
d) Statistic: the mean weekly exercise time of the sampled clients.
e) Variable: \(X =\) the amount of time (e.g., hours) a client exercises in the center per week.
f) Data: the individual recorded times, e.g., 2 hours, 5 hours, 3.5 hours, ….
Answer: a) all fitness-center clients; b) the clients actually surveyed; c) mean weekly exercise time of all clients; d) mean weekly exercise time of the sampled clients; e) weekly exercise time per client; f) the specific time values recorded (e.g., 2, 5, 3.5 hours).
Problem 7. Ski resorts are interested in the mean age that children take their first ski and snowboard lessons. They need this information to plan their ski classes optimally.
Solution
Step 1 — Recall the six definitions: population (whole group), sample (subset studied), parameter (population number), statistic (sample number), variable (measured characteristic), data (the recorded values).
Step 2 — Map them to this scenario (mean age of first ski/snowboard lesson):
a) Population: all children who take ski and snowboard lessons (at the resorts of interest).
b) Sample: the particular group of children whose first-lesson ages are recorded.
c) Parameter: the mean first-lesson age of all such children.
d) Statistic: the mean first-lesson age of the sampled children.
e) Variable: \(X =\) the age at which a child takes their first ski/snowboard lesson.
f) Data: the individual ages, e.g., 4 years, 6 years, 5 years, ….
Answer: a) all children taking ski/snowboard lessons; b) the children actually surveyed; c) mean first-lesson age of all such children; d) mean first-lesson age of the sample; e) age at first lesson; f) the specific ages recorded (e.g., 4, 6, 5 years).
Problem 8. A cardiologist is interested in the mean recovery period of their patients who have had heart attacks.
Solution
Step 1 — Recall the six definitions: population, sample, parameter (population number), statistic (sample number), variable (measured trait), data (recorded values).
Step 2 — Map them to this scenario (mean recovery period after heart attack):
a) Population: all of the cardiologist's patients who have had heart attacks.
b) Sample: the particular group of those patients whose recovery time is measured.
c) Parameter: the mean recovery period of all the cardiologist's heart-attack patients.
d) Statistic: the mean recovery period of the sampled patients.
e) Variable: \(X =\) the length of the recovery period (e.g., in days) for a heart-attack patient.
f) Data: the individual recovery times, e.g., 30 days, 45 days, 28 days, ….
Answer: a) all of the cardiologist's heart-attack patients; b) the patients actually measured; c) mean recovery period of all such patients; d) mean recovery period of the sample; e) recovery-period length per patient; f) the specific recovery times recorded (e.g., 30, 45, 28 days).
Problem 9. Insurance companies are interested in the mean health costs each year of their clients, so that they can determine the costs of health insurance.
Solution
Step 1 — Recall the six definitions: population, sample, parameter (population number), statistic (sample number), variable (measured trait), data (recorded values).
Step 2 — Map them to this scenario (mean annual health costs of clients):
a) Population: all clients of the insurance company.
b) Sample: the particular group of clients whose annual health costs are examined.
c) Parameter: the mean annual health cost of all clients.
d) Statistic: the mean annual health cost of the sampled clients.
e) Variable: \(X =\) the annual health cost of a client (in dollars).
f) Data: the individual cost values, e.g., $2,000, $5,500, $1,200, ….
Answer: a) all clients of the insurance company; b) the clients actually sampled; c) mean annual health cost of all clients; d) mean annual health cost of the sample; e) annual health cost per client; f) the specific cost values recorded (e.g., $2,000, $5,500, $1,200).
Problem 10. A politician is interested in the proportion of voters in their district who think the politician is doing a good job.
Solution
Step 1 — Recall the six definitions: population, sample, parameter (population number), statistic (sample number), variable (measured trait), data (recorded values).
Step 2 — Map them to this scenario (proportion of voters who approve):
a) Population: all voters in the politician's district.
b) Sample: the particular group of voters who are actually surveyed.
c) Parameter: the proportion of all district voters who think the politician is doing a good job.
d) Statistic: the proportion of the sampled voters who think so.
e) Variable: \(X =\) whether a given voter thinks the politician is doing a good job (yes/no).
f) Data: the individual yes/no responses, e.g., yes, no, yes, yes, no, ….
Answer: a) all voters in the district; b) the voters actually surveyed; c) the true proportion of all district voters who approve; d) the proportion of surveyed voters who approve; e) each voter's approval (yes/no); f) the specific yes/no responses recorded.
Problem 11. A marriage counselor is interested in the proportion of clients they counsel who stay married.
Solution
Step 1 — Identify the population: The population is the entire group the counselor wants to draw conclusions about — here, all of the counselor's clients (those they counsel).
Step 2 — Identify the parameter of interest: The proportion of all clients who stay married. This is the population proportion, the numerical fact the counselor wants to learn.
Step 3 — Identify the variable and data: The variable is whether a given client stays married (yes/no). The data are the actual yes/no responses recorded for the clients studied.
Answer: Population = all clients the counselor counsels; parameter = the (true) proportion of those clients who stay married; variable = whether a client stays married; data = the yes/no marital-status outcomes collected.
Problem 12. Political pollsters may be interested in the proportion of people who will vote for a particular cause.
Solution
Step 1 — Identify the population: The population is everyone whose vote the pollsters care about — all eligible voters (the people who could vote on the cause).
Step 2 — Identify the parameter of interest: The proportion of all those people who will vote for the particular cause. This is the population proportion.
Step 3 — Identify the variable and data: The variable is whether a given person will vote for the cause (yes/no). The data are the recorded yes/no voting intentions.
Answer: Population = all eligible voters; parameter = the proportion who will vote for the cause; variable = whether a person votes for the cause; data = the yes/no responses collected.
Problem 13. A marketing company is interested in the proportion of people who will buy a particular product.
Solution
Step 1 — Identify the population: The population is all the people the marketing company is interested in — the potential buyers of the product.
Step 2 — Identify the parameter of interest: The proportion of all those people who will buy the particular product. This is the population proportion.
Step 3 — Identify the variable and data: The variable is whether a given person will buy the product (yes/no). The data are the recorded yes/no purchase responses.
Answer: Population = all potential buyers (people the company is interested in); parameter = the proportion who will buy the product; variable = whether a person buys the product; data = the yes/no purchase outcomes collected.
For problems 14 through 16, a Lake Tahoe Community College instructor is interested in the mean number of days Lake Tahoe Community College math students are absent from class during a quarter.
Problem 14. What is the population she is interested in?
a) all Lake Tahoe Community College students
b) all Lake Tahoe Community College English students
c) all Lake Tahoe Community College students in her classes
d) all Lake Tahoe Community College math students
Solution
Step 1 — Read the question carefully: The instructor studies the mean days absent of math students, so the population must be exactly the group she wants conclusions about.
Step 2 — Eliminate options that are too broad or wrong: "All Lake Tahoe Community College students" (a) and "English students" (b) are not her group; "students in her classes" (c) is only a subset she might sample, not the full target group.
Step 3 — Select the matching group: The full target group is all Lake Tahoe Community College math students.
Answer: d) all Lake Tahoe Community College math students.
Problem 15. Consider the following: \(X\) = number of days a Lake Tahoe Community College math student is absent. In this case, \(X\) is an example of a:
a) variable.
b) population.
c) statistic.
d) data.
Solution
Step 1 — Recognize what \(X\) measures: \(X\) = the number of days a math student is absent. Its value changes from student to student, so it is a characteristic that varies across individuals.
Step 2 — Match to the term: A characteristic that takes different values for different members of the population is a variable. It is not the population (the students themselves), not a statistic (a single computed summary), and not the data (the actual recorded values).
Answer: a) variable.
Problem 16. The instructor's sample produces a mean number of days absent of 3.5 days. This value is an example of a:
a) parameter.
b) data.
c) statistic.
d) variable.
Solution
Step 1 — Identify the source of the number: The value 3.5 days is the mean computed from the instructor's sample, not from the entire population of math students.
Step 2 — Match to the term: A numerical summary computed from a sample is a statistic. (A parameter would summarize the whole population; data are the raw values; a variable is the characteristic measured.)
Answer: c) statistic.
Key Terms
statistics — the science of collecting, analyzing, interpreting, and presenting data.
descriptive statistics — organizing and summarizing data, for example with a graph or an average.
inferential statistics — using sample data, together with probability, to draw conclusions about a larger population.
probability — a mathematical measure of the chance that an event occurs; stable over many repetitions.
population — the entire collection of persons, things, or objects under study.
sample — a subset of the population that is actually examined.
parameter — a numerical characteristic of the whole population (e.g., a population mean).
statistic — a numerical summary of a sample, used to estimate a parameter.
variable — a characteristic or measurement recorded for each member of a population, usually written \(X\) or \(Y\).
numerical variable — a variable whose values are numbers with equal units (e.g., weight, time).
categorical variable — a variable that places each member into a category (e.g., party affiliation).
discrete variable — a numerical variable that comes from counting (whole-number values).
continuous variable — a numerical variable that comes from measuring on a scale (any value in a range).
data — the actual recorded values of a variable; a single value is a datum.
mean — the arithmetic average of a set of numerical values.
proportion — the fraction of a group that has a given characteristic.
associated / independent — two variables are associated if knowing one gives information about the other, and independent if it does not; no pair is both.