Two randomizations, two separate permissions
Every study description on the exam answers two questions, and the reader must keep them apart.
| Random selection | Random assignment | What may be concluded |
|---|---|---|
| Yes | Yes | Cause and effect, for the whole population sampled. |
| Yes | No | Association only, but for the whole population sampled. |
| No | Yes | Cause and effect, but only for the participants in the study. |
| No | No | Association only, and only among the participants. |
Random selection = who got into the study. It controls how far the conclusion travels.
Random assignment = who got which treatment. It controls whether the word “cause” is allowed.
Đây là chỗ mất điểm dễ nhất của cả cuốn, vì chương trình phổ thông Việt Nam không dạy phần này. Hai từ tiếng Anh nghe rất giống nhau: randomly selected (chọn ngẫu nhiên vào nghiên cứu) và randomly assigned (bốc thăm chia nhóm). Nhưng chúng cấp hai giấy phép khác nhau và không thay thế nhau được. Đọc đề, tìm đúng hai chữ đó, ghi ra lề “chọn: có/không” và “chia: có/không” trước khi đọc bốn phương án.
Observational study: association only
In an observational study the researcher records what people already do. Nobody decides who does what, so the two groups being compared may differ in many ways besides the one being measured. Any of those other differences could produce the gap that was observed.
A random sample of $350$ of the $12{,}000$ employees of a manufacturer was surveyed. Employees who cycle to work reported a mean of $2.1$ fewer sick days last year than employees who drive. Which conclusion is appropriate?
The sample was randomly selected from the manufacturer's employees, so the finding extends to all $12{,}000$ of them. But nobody assigned employees to cycle or to drive — employees chose. Cyclists may be younger, may live closer to work, or may exercise more in general, and any of those could explain the gap.
Appropriate: among this manufacturer's employees, cycling to work is associated with fewer sick days. Not appropriate: any statement that cycling causes fewer sick days.
Nhầm cỡ mẫu lớn với quyền nói nhân quả. Hỏi cả $12{,}000$ người thay vì $350$ vẫn không cho phép nói “gây ra” — thiếu sót không nằm ở số lượng người mà nằm ở chỗ ai quyết định người nào vào nhóm nào. Chỉ có phân nhóm ngẫu nhiên mới vá được chỗ đó.
A clinic compared recovery times for patients who chose a course of physiotherapy with recovery times for patients who chose rest. Patients made their own choice, and the clinic studied only its own patients. Which conclusion is appropriate?
Neither randomization is present. No random selection, so the finding covers nobody outside this clinic; no random assignment, so no causal claim. The only statement available is that, among these patients, choosing physiotherapy is associated with the observed difference in recovery time. Patients healthy enough to attend physiotherapy sessions were probably in better shape to begin with.
Experiment: random assignment buys the word “cause”
In an experiment the researcher assigns the treatment. When that assignment is made at random, the two groups start out alike on average in every respect — age, habits, motivation, everything measured and everything not measured. The treatment is then the only systematic difference left, so a difference in outcome may be attributed to it.
Researchers randomly selected $400$ of the $6{,}000$ employees of a company and randomly assigned each selected employee to one of two training programmes. Employees in Programme A completed a mean of $3.4$ more tasks per week than employees in Programme B. Which conclusion is appropriate?
Both randomizations are present. Random assignment licenses a causal statement; random selection from the company's employees lets that statement cover all $6{,}000$ of them.
Appropriate: Programme A caused an increase in tasks completed, and this is likely to hold for the employees of this company. Nothing may be said about employees of other companies.
Two hundred volunteers who responded to an advertisement were randomly assigned to a new exercise programme or to their usual routine. After ten weeks the programme group had a significantly greater mean improvement in endurance. Which conclusion is appropriate?
Random assignment is present, so the programme may be credited with causing the improvement. Random selection is not present — the volunteers put themselves forward and need not resemble adults in general.
Appropriate: the programme caused the improvement in endurance for the people in this study. Not appropriate: extending that to all adults.
Người tình nguyện là nhóm tự chọn: họ thường khoẻ hơn, rảnh hơn và có động lực hơn người trung bình. Vì thế thí nghiệm trên người tình nguyện vẫn nói được chữ “gây ra” (nhờ bốc thăm chia nhóm) nhưng chỉ nói cho đúng những người trong nghiên cứu. Muốn nói cho cả cộng đồng thì phải chọn người tham gia ngẫu nhiên từ cộng đồng đó ngay từ đầu.
In a trial, $600$ participants were randomly assigned to two groups, with $\tfrac35$ of them placed in the treatment group. Improvement was reported by $35\%$ of the treatment group and $20\%$ of the control group. How many participants reported improvement in total?
Treatment group: $\tfrac35\times600=360$, of whom $0.35\times360=126$ improved.
Control group: $600-360=240$, of whom $0.20\times240=48$ improved.
Total: $126+48=\boxed{174}$. The two percentages cannot be averaged, because the groups are unequal in size — rebuild the counts first, then add.
Samples that license nothing
Three collection methods that carry no permission at all: voluntary response (people choose to answer — those with strong opinions answer more), convenience (whoever is easy to reach), and heavy non-response (most of the people contacted never replied, and the ones who did may differ from the ones who did not).
A radio programme invited listeners to call in and say whether they support a new parking policy; $9{,}300$ listeners called and $68\%$ said they oppose it. Which conclusion is appropriate?
None about the city's residents. Listeners chose themselves into the sample, and people who feel strongly call far more often than people who do not. The only honest statement is about the people who chose to call. Note that $9{,}300$ is a large number and changes nothing: a large self-selected sample is still self-selected.
Đọc thấy con số lớn rồi kết luận mẫu “đủ đại diện”. Cỡ mẫu chỉ ảnh hưởng tới biên sai số. Nếu cách chọn người đã lệch thì tăng cỡ mẫu chỉ làm cái sai chính xác hơn, không làm nó đúng lên.
Reading the answer choices
The four choices in these questions are built to a pattern. Learn the pattern and most questions are decided before any thinking about the subject matter.
- Causal overreach — says “causes”, “leads to”, “results in” about an observational study.
- Population overreach — correct relationship, wrong group: it talks about all adults when the sample came from one school, one club, one company.
- Certainty overreach — “proves”, “all”, “every”, “guarantees”. Statistics supports plausible statements, never proof.
The surviving choice is usually the cautious-sounding one containing associated with, plausible, or a named group.
A random sample of $300$ of the $4{,}500$ households in a district was surveyed. Of those sampled, $40\%$ subscribe to a streaming service; among subscribers $65\%$ own a smart television, compared with $38\%$ among non-subscribers. Which statement is not supported?
Supported: subscribing and owning a smart television are associated among the households surveyed; an estimate of the number of subscribing households in the district is $0.40\times4{,}500=1{,}800$; it is plausible that fewer than half of the district's households subscribe.
Not supported: subscribing to a streaming service causes households to buy a smart television. Nobody assigned households to subscribe, and a single third factor — household income — comfortably explains both figures at once.
Chiến thuật đọc: quét bốn phương án tìm động từ nhân quả (causes, leads to, results in, increases) và tìm tên nhóm (all adults, everyone, people in general). Gạch bỏ những phương án vượt quyền trước, rồi mới đọc kỹ cái còn lại. Phần lớn câu chỉ còn đúng một phương án sau bước gạch.