SAT-60: Sample Surveys and Random Sampling

SAT-60: Sample Surveys and Random Sampling

Description: When you can't ask everyone, you survey a sample and use it to estimate facts about the whole population. The SAT tests whether a sample was chosen properly — because only a random sample supports trustworthy conclusions.

Population vs sample

• Population: the entire group you want to learn about (e.g. all students in a school).
• Sample: the smaller group you actually survey (e.g. 100 of those students).
• Parameter vs statistic: a fact about the whole population is a parameter; the same fact measured from the sample is a statistic used to estimate it.

(Oʻzbekcha: populyatsiya — butun guruh; namuna (sample) — biz soʻraydigan kichik qism.)

Why random matters

In a random sample every member of the population has an equal chance of being chosen. Randomness keeps the sample from systematically favoring one type of person, so the sample tends to look like the population in miniature. Only then can results be generalized. (Oʻzbekcha: tasodifiy namunada har bir aʼzo tanlanish ehtimoli teng boʻladi.)

Worked Example 1 — identify population and sample

A company surveys 200 of its 5,000 employees about lunch options. Name the population and the sample.

• Population = all 5,000 employees. Sample = the 200 surveyed.

Worked Example 2 — is the sample random?

To learn what all students think of the cafeteria, a researcher surveys only students already eating in the cafeteria. Is this a good random sample?

• No. Students who dislike the cafeteria are less likely to be there, so they are under-represented.
• This is not random, and the result would be biased toward positive opinions.

(Oʻzbekcha: bu tasodifiy emas — oshxonani yoqtirmaydiganlar deyarli qatnashmaydi.)

Worked Example 3 — a proper method

How could the school choose 100 students randomly?

• Assign every student a number, then use a random number generator (or draw numbers from a hat) to pick 100.
• Now every student had an equal chance, so the sample is random and the results can be generalized.

Rule: valid generalization to a population requires a random sample from that population. (Oʻzbekcha: xulosani butun guruhga umumlashtirish uchun tasodifiy namuna shart.)

Sample size vs sample method

Students often think a bigger sample automatically fixes everything. It doesn't. A large sample chosen badly is still misleading — surveying 10,000 people leaving a stadium still tells you nothing about non-fans. The method (was it random?) decides whether the sample is trustworthy; the size only affects how precise the estimate is once the method is sound. So judge randomness first, size second. (Oʻzbekcha: katta namuna oʻzi yetarli emas — avvalo tanlash usuli tasodifiy boʻlishi kerak, keyin hajm aniqlikka taʼsir qiladi.)

Practice 1

A TV station asks viewers to call in and vote. Why might this sample be unreliable for "what all citizens think"?

Practice 2

A principal wants to estimate the average study time of all 1,200 students. She numbers them 1–1,200 and uses a random generator to pick 80. Is this valid, and what is the population?

Key words — Kalit soʻzlar

• Population — populyatsiya (butun guruh)
• Sample — namuna
• Random sample — tasodifiy namuna
• Survey — soʻrovnoma
• Equal chance — teng imkoniyat
• Representative — vakil boʻla oladigan (ifodalovchi)
• Generalize — umumlashtirish
• Parameter / Statistic — parametr / statistik koʻrsatkich
• Self-selected — oʻzini-oʻzi tanlagan

Summary

• Population = the whole group; sample = the part you survey.
• A random sample gives every member an equal chance and represents the population.
• Only random samples support generalizing results to the population.