Lesson 1.1 · Unit 1 · CED topic 1.1
Statistical studies: asking questions with data
Statistics is the science of learning from data. A statistical study starts with a question about a large group, collects information from a small part of that group, and uses what it finds to answer the question. Before you can graph, summarize or test anything, you have to know what the study is about: who or what is being studied, what is being measured, and what the answer is supposed to tell you. This first lesson gives you the vocabulary for those parts and the habit of connecting every number back to the real situation it came from.
Learning objectives.
- 1.1.A. Identify components within a statistical study.
- 1.1.B. Determine an investigative question within a statistical study.
Essential knowledge.
- 1.1.A.1. A statistical study is a study in which data are collected from a sample to answer an investigative question about a larger population.
- 1.1.A.2. Statistical studies are necessary when the population is too large or it is too difficult to collect data from every item or individual in the population.
- 1.1.A.3. A datum (singular form of data) is a piece of information about an item or individual. A collection of data is called a data set.
- 1.1.A.4. A population consists of all items or individuals of interest. The population size is represented by the symbol N.
- 1.1.A.5. A sample selected for study is a subset of the population from which data are obtained. The number of items in the sample, called the sample size, is represented by the symbol n.
- 1.1.A.6. Each component of a statistical study and the resulting calculations can be related to an aspect of the corresponding real-world context from which the components were derived. This identification of a statistical result with the corresponding contextual component is what is meant by “in context.”
- 1.1.B.1. An investigative question for a specific study should have a defined purpose and should not be changed based on the data analysis or results.
- 1.1.B.2. An investigative question should be posed so that the required data can be collected and analyzed.
The parts of a statistical study
A statistical study collects data from a sample to answer an investigative question about a larger population. Each part has a name and a symbol.
- A datum (the singular of data) is one piece of information about one item or individual. A collection of data is a data set.
- The population is all the items or individuals of interest. Its size is written \(N\).
- The sample is the subset of the population from which data are actually obtained. Its size, the sample size, is written \(n\).
Notice what the arrows in Figure 1.1 mean. A sample is drawn from the population, and the information in the sample is used to say something about the population. The sample is almost never the whole story, so a statistic computed from a sample usually differs a little from the population value it estimates. Learning how much it can differ, and what to conclude anyway, is what most of this course is about.
Why we sample
You would not need statistics if you could measure everyone. A statistical study is necessary when the population is too large, or when it is too difficult to collect data from every item or individual in it.
- Too large. Asking every teenager in a country about their daily screen time is not realistic.
- Too difficult or destructive. A battery maker who wants to know how long its batteries last cannot run all of them until they die; nothing would be left to sell.
- Too slow or expensive. A hospital that wants this year's average waiting time cannot wait until every patient has been seen to decide whether to change a policy.
In each case a well-chosen sample lets us answer the question with far less effort, provided we are honest about the uncertainty.
"In context" means attaching every number to the real situation
The most common way to lose points on an AP Statistics free-response question is to answer correctly but without context. Each component of a study, and every calculation made from it, can be tied to an aspect of the real-world situation it came from. Doing that is what the exam means by in context. Compare:
| Statement | Not in context | In context |
|---|---|---|
| A mean | "The mean is 7.1." | "The 400 sampled students slept an average of 7.1 hours on school nights." |
| The population | "All of them." | "All 6,200 high school students in the district." |
| A proportion | "The proportion is 0.35." | "About 35% of the sampled shoppers said they prefer the store brand." |
A good in-context sentence names the variable, the group it describes and the units.
A school district has 6,200 high school students. The district wants to know how many hours the typical student sleeps on a school night. It randomly selects 400 students and asks each of them how many hours they slept last night. Identify the population, the sample, \(N\) and \(n\), and describe the data.
Population: all high school students in the district, so \(N=6{,}200\). Sample: the 400 students selected, so \(n=400\). Data: each of the 400 students supplies one datum, the number of hours slept, so the data set contains 400 values measured in hours.
Suppose the sample mean is 7.1 hours. In context: "The 400 students in the sample slept an average of 7.1 hours last night." It would be a mistake to write "Students sleep 7.1 hours," because that claims the sample result is exactly true for all 6,200 students, and the sample alone cannot show that.
A library lets patrons borrow videos either in person or through an online system. A librarian randomly selects patrons who borrow videos and asks how they feel about the two methods. What are the sample and the population?
The sample is the patrons the librarian selected. The population is every patron who borrows videos from this library, because that is the group the librarian wants to learn about. It is not "all patrons of the library," because someone who never borrows a video has no opinion about the two borrowing methods, and it is not only the in-person borrowers or only the online borrowers, because the question concerns both methods. Describing the population precisely is half of identifying it.
The investigative question
Every statistical study is organized around an investigative question. It must meet two requirements.
- It should have a defined purpose and should not be changed based on the data analysis or results. Deciding what you are asking after you have looked at the data lets you hunt through the numbers until something looks interesting, which makes the conclusion unreliable.
- It should be posed so that the required data can be collected and analyzed. A question that cannot be answered with measurable data is not a statistical question.
A student council wants to study the new late start time. Decide whether each question is a good investigative question for a statistical study, and fix any that are not.
- (a) "Is the late start better?"
- (b) "Is the mean number of hours of sleep per school night of students at our school greater than 7?"
- (c) "The mean sleep turned out to be 7.4 hours, so was our question really about lunch periods?"
(a) is too vague: "better" is not defined, so it is unclear what to measure. A fix is to name a variable and a parameter, as in (b). (b) is a good question: the variable (hours of sleep per school night), the population (students at this school) and the parameter (the mean) are clear, and the data can be collected with a survey. (c) changes the question after seeing the results, which defeats its purpose. The purpose of the study should be set before the data are collected and kept through the analysis.
A researcher wants to know the average daily screen time of teenagers 13 to 17 years old in a large country. Explain why a statistical study is appropriate.
The population is all teenagers in that age range in the country. Surveying every one of them would be far too difficult, so the researcher collects data from a sample and uses the sample to learn about the population. The reason a statistical study is needed is that surveying every member of the population is not practical, not that statistics is "always required" and not that it guarantees everyone participates.
- A statistical study collects data from a sample (size \(n\)) to answer an investigative question about a larger population (size \(N\)).
- Studies are needed when the population is too large or too difficult to measure completely.
- Always state the population precisely and answer in context: name the variable, the group and the units.
- An investigative question has a defined purpose, is set before the analysis and cannot be changed after seeing the results, and must be answerable with data that can be collected and analyzed.
- A city has 48,000 registered voters. A polling firm selects 1,200 of them and asks whom each one plans to vote for. Identify the population, the sample, \(N\) and \(n\).
Show answer
Population: the 48,000 registered voters in the city, so \(N=48{,}000\). Sample: the 1,200 voters selected, so \(n=1{,}200\).
- A quality inspector tests how much weight each of 25 steel cables can hold before breaking, out of a shipment of 800 cables. Explain why the inspector samples instead of testing the whole shipment.
Show answer
Testing a cable to failure destroys it, so testing all 800 cables would leave nothing to use. Collecting data from every item in the population is too difficult (and wasteful), so a sample of 25 is tested and used to learn about the shipment.
- A student writes: "The mean score was 82." Rewrite the sentence in context for a random sample of 60 students who took a chemistry exam.
Show answer
"The 60 students in the sample had a mean exam score of 82 points." (It names the group, the variable and the units; a sample mean describes the sample, not automatically all students.)
- A researcher collects data on 300 randomly chosen teenagers, finds that average screen time is not far from 5 hours, and then decides the real question was how much time is spent on video games. Explain what is wrong with this approach.
Show answer
The investigative question was changed after looking at the data. The purpose of a study should be defined before data collection and not be altered because of the results; otherwise the analysis can be bent to find something that looks interesting.
- Which is a better investigative question for a sample of recent graduates: "Are graduates doing well?" or "Is the mean starting salary of graduates of this college greater than \$50,000?" Explain.
Show answer
The second. It names the variable (starting salary), the parameter (the mean), the population (graduates of this college) and a direction, so the data can be collected and analyzed. "Doing well" is undefined and cannot be measured as stated.
- In a study of 150 randomly selected apartments in a city of 90,000 apartments, the mean monthly rent is \$1,640. Write one sentence interpreting this number in context without claiming it is the exact value for all apartments.
Show answer
"The 150 sampled apartments had a mean monthly rent of \$1,640, which we can use as an estimate of the mean monthly rent for all 90,000 apartments in the city."