Estimating from statistical data, and scaling the estimate up

What to assume they know

• Experimental probability: relative frequency over many trials — relative frequency, and why a count over a total is a probability estimate
• Finding a fraction of a whole number, and reading a percentage as a fraction
• Reading a frequency table and totalling its entries
• What "rational" means, and why the denominator cannot be zero — a fraction as an exact object, so that scaling by 7/30 rather than by 0.23 is a meaningful choice

What they should be able to do

• Compute a probability estimate from survey counts as a relative frequency
• Explain why a randomly chosen respondent is what makes that relative frequency a probability
• Scale a sample proportion up to a population and state the estimate as a count
• Show that the scaled estimates across all categories must total the population, and use that as a check
• Distinguish increasing a sample's size from making it representative, and say what each fixes
• Use the words population, sample and sampling correctly for a described survey
• Explain why scaling with the exact fraction is better than scaling with a rounded decimal, with a worked case
• Read probabilities directly off a table of a thousand recorded cases
• State what a scaled estimate is not — a count, a guarantee, or a measurement of the population

Where it usually goes wrong

• "The sample proportion is the population proportion." It is an estimate of it. The chapter's word for 600 mangoes is approximately, and the word is carrying the whole argument.
• "A bigger sample fixes a biased one." It does not. Ask the same class two hundred times over and you learn about that class with great precision. Size and representativeness are separate repairs, and the chapter's own box names them separately.
• "600 is how many mangoes the school will want." It is what the estimate suggests. A real school of 1500 will not divide 40 : 30 : 20 : 10.
• "Probability from data is a different kind of probability." It is a relative frequency, exactly as in §7.2.1, with respondents in place of trials.
• "Round the decimal, then multiply." 0.23 × 600 = 138 where 7/30 × 600 = 140. Cancel first.
• "Just survey everyone." The chapter's own reason for sampling is that collecting from the whole population is usually impractical. That is a fact about cost and access, not laziness.
• "Anonymous means random." The fruit survey is anonymous, which protects the answers. What makes the estimate a probability is that the student is picked at random, and that is a different property.

Questions to check understanding

• Compute a probability from survey counts and express it as a fraction and a decimal
• Estimate the number in a population who fall into a given category, from sample counts and a stated population size — the form both parts (ii) of Exercise Set 7.2 Q1 and Q2 take
• Check a set of scaled estimates by confirming they total the population
• Read probabilities off a grouped-frequency table, including a range spanning two columns
• Explain why a stated sample might not be representative, and what would improve it
• Distinguish, in a described survey, the population from the sample
• Reasoning question: two students scale the same sample proportion up and get 138 and 140. Which method is right, and why?

Examples worth working on the board

Inputs, not answers, except where the chapter prints the result itself. Values marked Verified are worked out here; the chapter prints no answers and this volume has no appended answer key.

• Where the method is used (p. 162). The chapter's own list: marketing, forecasting sales, insurance, and research in science and social science. Worth naming at the top, because it is the only place in the chapter where probability is shown earning a living.
• Example 5, the fruit survey (p. 162). Fifty students in one class are asked anonymously for their favourite fruit. The counts: mango 20, apples 15, bananas 10, grapes 5. Verified: the four counts total 50, so nobody has been left out and nobody double-counted.
• The probability the chapter computes (p. 162). Pick one student from the class at random and guess their favourite fruit. Mango's estimate is 20 out of 50 = 0.4, and the chapter reads it as a 40% chance. Section 3's point: the phrase at random is what makes the relative frequency serve as a probability. Pick the student sitting nearest the fruit bowl and the same table supports no such claim.
• The scale-up (pp. 162–163). The school has 1500 students; the sample was 50 from one class. The chapter estimates approximately 600 mangoes needed, being 40% of 1500, and says the rest would be the other three fruits.
• The other three, which the chapter leaves undone. Verified: apples 15/50 = 0.3 → 450; bananas 10/50 = 0.2 → 300; grapes 5/50 = 0.1 → 150. And 600 + 450 + 300 + 150 = 1500 exactly. That total is the check worth teaching: the four estimates must account for the whole school, because the four proportions add to 1. It also exposes how strong the assumption is — the estimate asserts that one class's tastes tile the entire school in those exact ratios.
• The chapter's own remedy (p. 163). To be more confident, take a larger sample — it suggests 100 students — and make it more representative, for instance by including students from different classes or grades. Section 7's job is to pull those apart:
  • Bigger reduces the wobble. Ask 100 instead of 50 and the same underlying preference shows through more steadily.
  • Representative changes what the estimate is about. A hundred students all from the same class is a bigger sample of that class, and still not a sample of the school.
  • The chapter presents both in one sentence with the word "and", which is correct and easy to read as one idea. It is two.
• The LEARN MORE ABOUT SAMPLING box (p. 163). It names the two questions explicitly — how big the sample should be, and how to be sure it represents the whole population — says that size and freedom from bias both make results more accurate, and points the reader to further reading. So the chapter itself flags both repairs; it just does not work either one.
• Exercise Set 7.2 Q1 (p. 165). A teacher mixes a large bag of coloured sweets and takes a random sample of 30. The counts: red 10, green 8, yellow 7, blue 5. Verified: they total 30. Asked: first, the chance that a sweet taken at random out of that sample is a green one; and then, given 600 sweets in the bag altogether, an estimate of how many of them are yellow. Verified: green is 8/30 = 4/15 ≈ 0.267; yellow is 7/30, and 600 × 7/30 = 140 exactly.
• Section 9's worked case, and it uses those very numbers. 7/30 = 0.2333…, so a student who rounds to 0.23 and multiplies gets 138, and one who rounds to 0.233 gets 139.8. The exact fraction gives 140. Verified. Do the same with green: 8/30 = 0.2666…, and 600 × 8/30 = 160, while 0.27 × 600 = 162. The lesson is one line — cancel first, divide last — and this question is where it costs marks.
• Exercise Set 7.2 Q2 (p. 165). A random sample of 40 students at a school is asked for a favourite club. The counts: Science 14, Arts 11, Sports 9, Debate 6. Verified: they total 40. The school has 800 students. Asked: the probability that a sampled student prefers Arts, and an estimate of how many in the school prefer Sports. Verified: Arts is 11/40 = 0.275; Sports is 800 × 9/40 = 180. Note this survey is described as a random sample of the school, unlike the fruit survey's one class — so its scale-up rests on a better-founded assumption. The two questions are a matched pair on exactly the point section 7 makes.
• Section 10's distinction, which is easy to miss. In Q1 the sample was drawn from the very bag being estimated, and the bag's total is known. In the fruit survey the 1500 students are a different set of people from the 50 asked. Sampling from what you are estimating and sampling one part and generalising to another are two different situations wearing the same arithmetic.
• End-of-Chapter Q7 (p. 171). A tyre company's record of distances before replacement, over 1000 cases, printed as a two-row table. Its four columns: less than 4000 km, 20 cases; 4001 to 9000 km, 210; 9001 to 14000 km, 325; more than 14000 km, 445. Verified: the counts total 1000. Asked for the probability that a randomly chosen tyre lasts less than 4000 km, between 4000 and 14000 km, and more than 14000 km. Verified: 20/1000 = 0.02; (210 + 325)/1000 = 535/1000 = 0.535; 445/1000 = 0.445. And 0.02 + 0.535 + 0.445 = 1, which is the check again. Here no scaling is needed at all — the population is the thousand cases, so the relative frequencies are the answers.
• The boundary problem in that table, read on the printed page. The four bins are described as under 4000, then 4001 to 9000, then 9001 to 14000, then over 14000. Of the three round boundary values, only 4000 falls outside every column: 9000 sits inside the second bin and 14000 sits inside the third, because both of those bins are written inclusively. What the labels genuinely miss is narrower and stranger — the slivers between 4000 and 4001, and between 9000 and 9001, where a whole-number scheme has been laid over a quantity that is not whole. The printed counts still total 1000, so the data is internally consistent and the answers are unaffected; but a student who reads the boundaries strictly is right about 4000. The middle part's phrase must also be read as the two middle columns together.

Figures to have open

• A fifty-student icon grid, colour-coded by fruit, that can be re-grouped and scaled up to a fifteen-hundred-student block. Not in the book; §7.2.3 carries no figure at all.
• A one-class-versus-whole-school panel showing the sample sitting inside the population, with the class boundary drawn. This is the topic's central image and the chapter never draws it.
• A stacked bar of 1500 filling with the four scaled estimates, for the total check. Standard schematic.
• The tyre table (End-of-Chapter Q7, p. 171) redrawn as a table plus a bar of a thousand cases split into four blocks, so the middle answer is visibly two blocks joined.
• A side-by-side of the rounded and exact scale-ups for section 9. Standard schematic.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics (NCF-SE 2023), Chapter 7, §7.2.3 "Analysing Statistical Data Using Probability" (pp. 162–163), comprising the uses paragraph and Example 5 on p. 162 and the scale-up and sampling paragraph on p. 163.
• The boxed panel LEARN MORE ABOUT SAMPLING (p. 163), which sits immediately after §7.2.3 and names both repairs.
• §7.2's first numbered route (p. 159), where analysing past observations is placed alongside running an experiment as one method.
• Exercise Set 7.2, Q1 and Q2 (p. 165).
• End-of-Chapter Q7 (p. 171), with its four-column table.
• The Chapter Summary (p. 173) carries none of this section's ideas: no sampling, no population, no notion of a sample standing in for a larger group. The word sample does appear there, but only inside the phrase sample space, which is a different idea entirely — a list of possible outcomes, not a drawn subset of a population. Checked on the printed page. So this section's vocabulary is examinable from the body of the chapter only.