Chapter 6 — Ratios, Percentages, Data and Probability

Answer: D

Explanation

The ratio has $4 + 7 = 11$ parts in total. So red marbles are $\frac{4}{11}$ of everything:

$$\frac{4}{11} \times 44 = 4 \times 4 = 16$$

Check: red 16, blue $\frac{7}{11} \times 44 = 28$, and $16 + 28 = 44$ ✓ And $16 : 28$ simplifies to $4 : 7$ ✓

The quick way: 11 parts share 44 marbles, so one part is 4 marbles. Red is 4 parts, so $4 \times 4 = 16$. Finding the value of one part first works on every ratio question and is worth making automatic.

Why each wrong option is wrong:

• A, $4$ — the ratio number, not a count of marbles.
• C, $28$ — the blue marbles.
• B, $11$ — divided the total by 4 instead of by 11.

Takeaway: Add the ratio parts to get the whole, then find the value of one part. Everything else is multiplication after that.

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Answer: D

Explanation

Keep the same quantity on top of both fractions — cups over biscuits, both sides:

$$\frac{3 \text{ cups}}{8 \text{ biscuits}} = \frac{c}{20 \text{ biscuits}}$$

Cross-multiply:

$$8c = 60 \quad\Longrightarrow\quad c = \frac{60}{8} = \frac{15}{2} = 7.5$$

Check the size before accepting it. 20 biscuits is a bit more than twice 8, so the flour should be a bit more than twice 3 — around 7 or 8. $7.5$ fits ✓

Why each wrong option is wrong:

• A, $\frac{160}{3}$ — set the fractions up upside down on one side.
• B, $\frac{8}{3}$ — the rate of biscuits per cup, which is not what was asked.
• C, $15$ — doubled 3 to get 6 and then adjusted wrongly, or stopped at 60 divided by 4.

Takeaway: Same units on top on both sides, then cross-multiply. Estimate the answer's size first — it eliminates the inverted-ratio option immediately.

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Answer: 60

Explanation

Two conversions, done one at a time.

First, minutes to seconds, because the rate is per second:

$$4 \text{ minutes} = 4 \times 60 = 240 \text{ seconds}$$

Now the volume in millilitres:

$$240 \times 250 = 60\,000 \text{ ml}$$

Finally millilitres to litres:

$$\frac{60\,000}{1000} = 60 \text{ litres}$$

Check the direction of each conversion by asking whether the number should get bigger or smaller. Seconds are smaller than minutes, so there are more of them — multiply. Litres are bigger than millilitres, so there are fewer of them — divide. Getting a conversion upside down changes the answer by a factor of 3600 here, so it is never a close call.

This is a grid-in. Enter 60.

Takeaway: Convert one unit at a time and check each direction by asking whether the count should grow or shrink. Smaller unit, bigger number.

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Section 2 — Percentages

Every percentage question becomes easy once you turn the percentage into a multiplier:

The words	The multiplier
20% of	$\times 0.20$
increase by 20%	$\times 1.20$
decrease by 20%	$\times 0.80$
what percent is $a$ of $b$	$\dfrac{a}{b} \times 100$

Three things the test relies on:

• Increase then decrease by the same percentage does not return you to the start. Up 10% then down 10% is $\times 1.1 \times 0.9 = \times 0.99$ — a 1% loss overall.
• Percentage change is always measured against the original, not the new value.
• Working backwards from a final amount means dividing by the multiplier, not subtracting the percentage.

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Answer: D

Explanation

An increase of 15% means the new price is 115% of the old:

$$80 \times 1.15 = 92$$

Or find the increase and add it: $15\%$ of $80$ is $12$, and $80 + 12 = 92$ ✓

Why each wrong option is wrong:

• A, $12$ — the size of the increase, not the new price.
• B, $68$ — decreased by 15%.
• C, $95$ — added $15. A percentage of 80 is not 15.

Takeaway: "Increase by $r$%" means multiply by $1 + \frac{r}{100}$. One multiplication, no addition step to forget.

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Answer: D

Explanation

The discount means the customer paid 80% of the original:

$$0.80 \times \text{original} = 96$$

To undo a multiplication, divide:

$$\text{original} = \frac{96}{0.80} = 120$$

Check: 20% of 120 is 24, and $120 - 24 = 96$ ✓

The tempting move is to add 20% to $96, and it does not work. 20% of 96 is 19.20, not 24, because the discount was 20% of the larger original price. That is why working backwards requires division, not a second percentage.

Why each wrong option is wrong:

• A, $115.2$ — added 20% to the sale price, which is the mistake just described.
• B, $116$ — added $20.
• C, $76.8$ — took another 20% off.

Takeaway: To undo a percentage change, divide by the multiplier. Adding the same percentage back always undershoots, because it is a percentage of a smaller number.

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Answer: A

The options give the overall multiplier: 1 means the price is unchanged.

Explanation

Multiply the two multipliers together. Do not add the percentages.

$$1.25 \times 0.80 = 1.00$$

The price ends exactly where it started.

Follow $100 through to see why. Up 25%: $100 \to 125$. Down 20%: 20% of 125 is 25, so $125 \to 100$. The same percentage applied to a bigger number is a bigger amount, which is exactly what makes the two changes cancel here.

Why each wrong option is wrong:

• B, $\frac{21}{20}$ — added the percentages, $+25 - 20 = +5$. Percentage changes never add.
• C, $\frac{19}{20}$ — subtracted in the other direction.
• D, $\frac{9}{20}$ — treated the two as a combined 55% loss.

Takeaway: Successive percentage changes multiply. Test with $100 — it takes ten seconds and settles the question completely.

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Section 3 — One-Variable Data

Measure	What it is	Careful of
Mean	total ÷ how many	dragged by extreme values
Median	the middle value, once sorted	sort first
Mode	the most common value	there can be several, or none
Range	largest − smallest	a spread, not a centre
Standard deviation	how spread out the values are	never negative

Two facts the SAT tests repeatedly:

An outlier moves the mean far more than the median. A single very large value drags the mean up but shifts the median by at most one place. So if a question says a distribution is skewed, or mentions one unusually large value, the median is the more representative measure.

Standard deviation measures spread only. Two data sets can have the same mean and completely different standard deviations. Values bunched close together give a small standard deviation; values spread far apart give a large one. You will never be asked to calculate one — only to compare two sets and say which is larger.

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Answer: B

Explanation

Sort the values first — this is the step people skip:

$$3,\ 3,\ 5,\ 7,\ 9,\ 11$$

There are six values, an even number, so there is no single middle one. The median is the average of the middle two, which are 5 and 7:

$$\frac{5 + 7}{2} = 6$$

Why each wrong option is wrong:

• A, $\frac{19}{3}$ — the mean, $\frac{38}{6} = \frac{19}{3}$.
• C, $3$ — the mode, the most frequent value.
• D, $8$ — averaged the two middle values of the unsorted list.

Takeaway: Sort, then find the middle. Even count → average the middle two. And check which measure the question named; all four are offered.

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Answer: B

Explanation

Work out both, before and after.

Before: the values sorted are $12, 13, 14, 15, 16$. The mean is $\frac{70}{5} = 14$ and the median is the middle value, 14.

After adding 60: the values are $12, 13, 14, 15, 16, 60$. The mean is

$$\frac{130}{6} \approx 21.7$$

and the median is the average of the middle two, $\frac{14 + 15}{2} = 14.5$.

So the mean jumped by about 7.7 while the median moved by 0.5.

The reason is structural. The mean uses the size of every value, so one huge number pulls hard. The median only uses position, so an extra value at the top shifts the middle by half a place regardless of how extreme it is. Replace the 60 with 6000 and the median would still be 14.5.

Why each wrong option is wrong:

• A — they change by very different amounts.
• C — backwards. The median is the resistant measure.
• D — both change; the median just changes very little.

Takeaway: Outliers drag the mean and barely move the median. When a question mentions one extreme value, or calls a distribution skewed, the median is the measure that represents it fairly.

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Answer: C

Explanation

Both sets have a mean of 20. But standard deviation is not about the centre — it is about how far the values sit from the centre.

Every value in P is 20, so nothing deviates at all. P's standard deviation is zero — the smallest it can ever be.

Q's values sit 10, 5, 0, 5 and 10 away from the mean. There is real spread, so its standard deviation is clearly greater.

You never have to compute one on this test. Look at how tightly the values cluster: bunched means small, spread out means large.

A has it backwards. B ignores the spread entirely. D is wrong because comparison by inspection is exactly what the test wants here.

Takeaway: Standard deviation measures spread, not centre. Identical values give zero. Two sets can share a mean and differ completely in standard deviation.

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Section 4 — Two-Variable Data and Probability

Scatterplots and lines of best fit. A line of best fit is a linear model drawn through a cloud of points. Its slope is a rate of change in context, and its $y$-intercept is the predicted value at $x = 0$. Reading a value inside the data range is reliable; predicting far outside it is not, and the test sometimes asks you to say so.

Probability on this test is almost always counting:

$$P(\text{event}) = \frac{\text{outcomes you want}}{\text{outcomes altogether}}$$

The work is deciding what "altogether" means. In a two-way table, the denominator is whichever total the question restricts you to — the grand total, a row total, or a column total. Getting that wrong is the mistake the wrong options are built from.

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Answer: B

Explanation

Read the restriction: "from those who cycle". That fixes the pool. You are no longer choosing from all 200 people — only from the 80 who cycle.

Of those 80, the number who own a car is 30:

$$P = \frac{30}{80} = \frac{3}{8}$$

The phrase that sets the denominator is worth hunting for in every one of these questions. "Of those who cycle", "among car owners", "given that" — each one narrows the pool to a single row or column.

Why each wrong option is wrong:

• A, $\frac{3}{20}$ — used the grand total, 200. That answers "what is the probability a randomly chosen person both cycles and owns a car".
• C, $\frac{1}{4}$ — used 120, the car-owner total. That answers the reversed question: given they own a car, do they cycle?
• D, $\frac{2}{5}$ — the probability of being a cyclist at all.

Takeaway: Find the phrase that restricts the group; its total is your denominator. "Of those who X" means the X row, not the whole table.

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Answer: A

Explanation

The 8 multiplies $x$, so it is the slope — the change in $y$ for each 1 added to $x$. In context: each extra hour of study predicts 8 more points.

Check: $f(3) = 66$ and $f(4) = 74$, a rise of 8 ✓

B describes the 42, the value at $x = 0$. C and D are quantities the model says nothing about — a line of best fit describes a relationship, not the size of the sample.

Note the word predicted. A line of best fit does not promise that an extra hour gives exactly 8 more marks; it describes the trend across the data.

Takeaway: In $y = mx + b$ fitted to data, $m$ is "per one more $x$" and $b$ is "at $x = 0$". Check the units: a score per hour cannot be a number of students.

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Answer: C

Explanation

Two rules decide every question of this kind.

Who was sampled? The survey drew from students at that university. A random sample supports conclusions about the population it was drawn from, and no further. Nothing here says anything about students elsewhere.

How precise can it be? A sample gives an estimate, not an exact figure. The right language is "approximately", "it is plausible that", "close to" — never "exactly".

Option C respects both. It stays inside the university and it hedges the number.

Why each wrong option is wrong:

• A — says exactly. A sample of 300 cannot pin down the whole university's opinion to the decimal.
• B — generalises beyond the population sampled. Students at one university are not a random sample of the country.
• D — too pessimistic. 300 randomly selected people support a reasonable estimate; that is the entire purpose of random sampling.

Takeaway: A random sample supports an estimate about the population it came from. Reject options that say "exactly", and reject options that reach beyond the group actually sampled.

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Section 5 — Mixed Practice

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Answer: A

Explanation

"What percent is $a$ of $b$" means $\frac{a}{b} \times 100$, with the part on top:

$$\frac{40}{250} \times 100 = 16$$

Check the size: 40 is well under half of 250, so the answer must be well under 50% ✓

B divided the wrong way round. C subtracted. D is the inverted division scaled up.

Takeaway: Part over whole, then times 100. If the answer comes out above 100% when the part is smaller than the whole, the division is upside down.

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Answer: 45

Explanation

Find the rate per second first:

$$\frac{3 \text{ pages}}{4 \text{ seconds}} = 0.75 \text{ pages per second}$$

Then a minute is 60 seconds:

$$0.75 \times 60 = 45$$

Check by scaling instead: 60 seconds is 15 lots of 4 seconds, and each lot gives 3 pages, so $15 \times 3 = 45$ ✓ Two routes, same answer.

This is a grid-in. Enter 45.

Takeaway: Reduce to a rate per one unit, then scale up. Or find how many whole blocks of the given time fit — often cleaner, and it avoids decimals.

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Answer: C

Explanation

Turn the mean into a total. Five numbers with mean 18 must add to

$$5 \times 18 = 90$$

The four given numbers add to $12 + 20 + 15 + 25 = 72$. So the fifth is

$$90 - 72 = 18$$

Check: $\frac{12 + 20 + 15 + 25 + 18}{5} = \frac{90}{5} = 18$ ✓

That the answer happens to equal the mean is a coincidence of these numbers, not a rule.

A subtracted wrongly. B is the total of all five. D is the total of the four.

Takeaway: Mean questions become easy the moment you convert to a total: total $=$ mean $\times$ count.

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Answer: B

Explanation

The 25% applies only to bicycle owners, not to everyone. So take a percentage of a percentage — which means multiply:

$$0.60 \times 0.25 = 0.15 = 15\%$$

Follow 1000 residents through to see it. 60% own bicycles, so 600 people. A quarter of those ride daily: $0.25 \times 600 = 150$. And 150 out of 1000 is 15% ✓

A reported the 25% as though it applied to everyone. C added the percentages, D subtracted them — neither operation means anything here.

Takeaway: "Of those…" signals a percentage of a subgroup, so multiply the two. Testing with a round population of 100 or 1000 makes it concrete in seconds.

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Answer: B

Explanation

$$\frac{96}{400} \times 100 = 24$$

Check the size: 96 is a bit under a quarter of 400 ✓ And a quick sanity check on the whole table — $180 + 96 + 84 + 40 = 400$, so nothing is missing.

A gave the raw count. D answered the opposite question.

Takeaway: Part over whole, times 100. Add the table up first; if it does not reach the stated total, you have misread a row.

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Answer: D

Explanation

Every 50 km uses 5 litres, so per kilometre:

$$\frac{5 \text{ litres}}{50 \text{ km}} = 0.1 \text{ litres per km}$$

Check the units the question asked for — litres per kilometre. Option B, 10, is kilometres per litre: the same relationship the other way up, and a perfectly sensible number that answers a different question.

A gave the fuel per 50 km without dividing.

Takeaway: Write the units on your answer and compare them with the units the question named. "Per" tells you which quantity goes on the bottom.

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Answer: C

Explanation

The first draw: 3 red out of 8, so $\frac38$.

The counter is not replaced, so for the second draw there are only 2 reds left out of 7 counters:

$$\frac{3}{8} \times \frac{2}{7} = \frac{6}{56} = \frac{3}{28}$$

The words "not replaced" are doing all the work. A ignored them and used $\frac38$ twice, which would be right with replacement.

Takeaway: "And then" means multiply. Check whether the first item goes back: without replacement, both the top and the bottom of the second fraction drop.

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Answer: A

Explanation

You cannot average two averages unless the groups are the same size — and here they are not. Go back to totals.

$$20 \times 68 = 1360 \qquad 30 \times 78 = 2340$$

$$\text{combined average} = \frac{1360 + 2340}{50} = \frac{3700}{50} = 74$$

The plain average of 68 and 78 would be 73. The true answer is pulled towards 78, because more students sat in the class that scored 78. That pull is the whole point of the question.

Takeaway: Averages combine through totals, never by averaging the averages. The answer always leans towards the larger group.

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Answer: 360

Explanation

Convert to the unit the rate uses. Two hours is $2 \times 60 = 120$ minutes, so

$$3 \times 120 = 360 \text{ litres}$$

Check the direction: minutes are smaller than hours, so there are more of them — multiply. Enter 360.

Takeaway: Convert time into the rate's own unit first. Smaller unit, bigger number.

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Answer: D

Explanation

The problem is who was asked, not how many. A basketball team is not a random slice of the school: its members train, may keep different hours, and are selected for something. Conclusions from it apply to the team, not the school.

Forty people chosen at random from the whole school would support a reasonable estimate. Forty chosen from one team does not, however carefully the average is worked out.

A blames the size, which is not the fault here. C invents a magic number; there is no required sample size.

Takeaway: A sample supports conclusions about the group it was randomly drawn from. Ask who was left out before you ask how many were asked.

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Answer: B

Explanation

Multiply the multipliers:

$$0.80 \times 0.90 = 0.72$$

So 72% of the price remains, and the reduction is

$$100 - 72 = 28\%$$

Follow $100: down 20% to $80, then 10% off $80 is $8, leaving $72. A $28 reduction ✓

A added the percentages, which is the mistake the question exists for — the second 10% is taken off a smaller price, so it is worth less than 10% of the original. C gave what remains rather than what was taken off.

Takeaway: Successive percentage changes multiply. Then read whether the question wants the amount remaining or the amount removed.

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Answer: C

Explanation

A slope is a rate: for each 1 added to $x$, the fitted $y$ changes by $-2$. Because this is a line of best fit and not an exact rule, the honest wording is "on average".

A overstates it — a line of best fit passes near the points, not through them; if it did, it would not need to be a line of best fit. B describes $y = x - 2$, a different relationship entirely. D confuses the intercept with a maximum: 30 is the fitted value at $x = 0$, and the line goes above 30 for negative $x$.

Takeaway: For a line of best fit, the slope is an average rate of change and the intercept is the fitted value at zero. Neither is a promise about any individual data point.

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Answer: 42

Explanation

Sand is 2 parts and weighs 12 kg, so one part is 6 kg.

The whole mixture is $1 + 2 + 4 = 7$ parts:

$$7 \times 6 = 42 \text{ kg}$$

Check: cement 6, sand 12, stone 24, total 42 ✓ and $6 : 12 : 24$ simplifies to $1 : 2 : 4$ ✓ Enter 42.

Takeaway: Find the value of one part first. Every ratio question becomes multiplication after that.

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Answer: A

Explanation

There are 21 students, so the median is the 11th value when they are lined up in order.

Count along: the first 4 students report 0 siblings (positions 1–4), the next 9 report 1 (positions 5–13). Position 11 falls inside that block, so the median is 1.

B is the mean, $\frac{27}{21}$. C took the middle column of the table rather than the middle student — the table's columns are values, not people. D halved the number of students, which finds the position, not the value at it.

Takeaway: With a frequency table, find which position is the middle, then count along the frequencies to see which value sits there. The middle column is not the median.

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Section 6 — Samples, Margins of Error and Claims

The test asks a small number of questions about what a study is entitled to conclude. They involve almost no calculation, and they are among the easiest marks on the paper once you know the three rules.

1. A margin of error gives a range. If a survey estimates 64% with a margin of error of 4%, the plausible values run from $64 - 4 = 60\%$ to $64 + 4 = 68\%$. The estimate sits in the middle; the margin reaches out both ways.

The wording matters: it is likely that the true value lies in that range — never certain, and never exactly the estimate.

2. A bigger random sample gives a smaller margin of error. Asking more people does not change the estimate's direction, it narrows the range around it. Nothing else on this test changes the margin.

3. A conclusion may only reach as far as the sampling did. Two separate limits, and questions test both:

What was done	What may be concluded
Randomly sampled from group X	something about group X, and nothing beyond it
Observed a relationship	that the two things are associated
Randomly assigned a treatment	that the treatment caused the difference

That last row is the whole of "observational study versus experiment". If nobody assigned anything, no cause may be claimed — however strong the pattern.

Every option that says "proves", "exactly", or "causes" is suspect. The right answers use "likely", "plausible", "suggests" and "approximately".

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Answer: B

Explanation

The margin of error reaches out both ways from the estimate:

$$47 - 3 = 44 \qquad 47 + 3 = 50$$

So the plausible range is 44% to 50%, and the conclusion is about all voters, not just the ones polled — that is the whole purpose of taking a random sample.

Why each wrong option is wrong:

• A — says exactly. A sample never pins down a population precisely; that is why a margin of error is quoted at all.
• C — applies the range to the people polled. Their support is a measured fact, not an estimate; the uncertainty is about everyone else.
• D — 50% is the extreme top of the range, not a floor. The poll is at least as consistent with 44%.

Takeaway: Estimate ± margin gives the range, and the claim is about the whole population. Reject any option saying "exactly" — and check whether a boundary value is being quoted as a minimum.

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Answer: C

Explanation

A larger random sample gives a better estimate, and "better" here means a narrower range around it — a smaller margin of error.

A has it backwards. B ignores the sample size, which is the one thing that changed. D overreaches: only surveying every single person would remove the uncertainty, and even a sample of 4,000 leaves some.

Note what does not change the margin: the estimate itself, the wording of the question, or how the results are presented. On this test, sample size is the lever.

Takeaway: Bigger random sample, smaller margin of error. Never zero unless everyone was asked.

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Answer: A

Explanation

Nobody assigned anyone an amount of sleep. The researchers recorded what students already did, which makes this an observational study, and an observational study can establish that two things go together — an association — but not that one causes the other.

Both B and C claim a direction of cause, and the study cannot separate them. Nor can it rule out some third thing affecting both: a student who is ill, or working long hours, might sleep less and score lower for reasons that have nothing to do with sleep itself.

D contradicts the finding. A relationship was observed; it just cannot be called causal.

To support a causal claim, the researchers would have had to randomly assign students to different amounts of sleep. That is what makes something an experiment rather than an observation, and it is the only design on this test that licenses the word "cause".

Takeaway: Observed → association. Randomly assigned → cause. If nobody assigned anything, reject every option containing "causes".

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Answer: D

Explanation

The sample was drawn at random from one gym's members, so it supports conclusions about that gym's members and no further.

A blames the size. Two hundred people chosen at random support a perfectly reasonable estimate — the number is not the problem here.

B is the error the question is testing: stretching the conclusion past the population that was sampled.

C has it backwards. Random selection is what makes the estimate trustworthy; a sample of whoever happened to be there on a Monday morning would not be.

Takeaway: A random sample supports conclusions about the group it was drawn from. Ask "who could have been chosen?" — that group, and only that group, is what the estimate describes.

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Answer: A

Explanation

Look at the differences between consecutive $y$ values:

$$7, \quad 10, \quad 12, \quad 11$$

They rise, and they are roughly the same size each time — so the relationship increases at a roughly constant rate. That is linear, allowing for the scatter real data always has.

B has the direction wrong; $y$ climbs throughout. C would need the values to stay put. D would mean multiplying by 2 each step: $12, 24, 48, 96, 192$. The actual value at $x = 5$ is 52, nowhere near 192.

Remember the distinction from Chapter 5: constant differences mean linear, constant ratios mean exponential. Here $\frac{19}{12} \approx 1.6$ but $\frac{52}{41} \approx 1.3$, so the ratios are not constant.

Takeaway: Differences roughly constant → linear. Ratios roughly constant → exponential. Real data will never be exact; look for which is closer to constant.

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Answer: D

Explanation

Every one of these four numbers is what the model gives — check them and they all come out right. So the arithmetic is not what separates them.

What separates them is where the prediction sits. The data covers houses from 50 to 200 square metres:

• A, B and C ask about 60, 120 and 200 — all inside that range, or at its edge. Predicting inside the data is what a line of best fit is for.
• D asks about 900 square metres, four and a half times the largest house ever measured. Nothing in the data says the relationship continues out there, and in reality it almost certainly does not.

Predicting beyond the data is called extrapolation, and it is the least reliable thing you can do with a model. The test asks about it directly.

Takeaway: A model is trustworthy inside the range of the data it was built from. A prediction far outside that range is the least reliable one, even when the arithmetic is perfect.

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Answer: B

Explanation

The model predicts

$$y = 2(4) + 5 = 13$$

The actual value is 15, so the actual is 2 above the prediction:

$$15 - 13 = 2$$

A point sitting above the line of best fit means the model under-predicted it. Below the line means it over-predicted.

A subtracted predicted minus actual, which reverses the sign and so reverses the meaning. C and D each give one of the two values rather than the gap between them.

Takeaway: Substitute the $x$ into the model to get the prediction, then subtract it from the actual value. Above the line is positive; below is negative.

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Answer: C

Explanation

This is the other side of Q29, and the difference is one word: assigned.

Because the researchers randomly assigned who got the painkiller and who got the placebo, the two groups were alike in every other respect on average. So when one group did better, the treatment is what explains it. That is an experiment, and an experiment does license a causal claim.

Why the other options fail:

• A would be right for an observational study — if the researchers had merely recorded which volunteers happened to take a painkiller. They did not; they assigned it.
• B overreaches in the other direction. "Significantly more often" is not "every time"; the study compares rates, not guarantees.
• D dismisses a perfectly workable study. Two hundred randomly assigned participants support a conclusion.

Note the careful wording of the correct option: "for people like these volunteers". The volunteers were not a random sample of everybody, so the causal claim holds for the kind of people who took part. Random assignment licenses cause; random selection licenses generalising to a wider population. This study had the first and not the second.

Takeaway: Random assignment → cause may be claimed. Random selection → the result may be generalised to the population sampled. They are different things, and a question will give you one, the other, or neither.

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Answer: .008 or 0.008

Explanation

The unit is metres per second squared — metres divided by (seconds × seconds). Only the metres are being changed; the seconds are the same on both sides, so they are left alone.

$$\frac{8 \text{ m}}{\text{s}^2} \times \frac{1 \text{ km}}{1000 \text{ m}} = \frac{8}{1000} = 0.008 \text{ km/s}^2$$

Check the direction: a kilometre is bigger than a metre, so the same rate is a smaller number of kilometres ✓

The "squared" is what makes this look harder than it is. It attaches to the seconds, and the seconds are not being converted — so it never enters the arithmetic at all. Had the question asked for metres per minute squared, the squared would matter and you would multiply by $60^2 = 3600$, not by 60.

This is a grid-in. Enter .008 or 0.008. Both fit the five-character limit; 0.008 uses exactly five.

Takeaway: In a compound unit, convert one part at a time and leave the rest alone. A squared unit is squared only when that unit is the one being changed.

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Answer: D

Explanation

Two conversions, one at a time.

Minutes to hours:

$$3 \times 60 = 180 \text{ litres per hour}$$

Hours to a day:

$$180 \times 24 = 4320 \text{ litres per day}$$

Check the direction at each step: a day is much longer than a minute, so the number must get much bigger ✓

A stopped after one step, B used 24 without the 60, and C worked out how many minutes are in a day and forgot to multiply by the leak rate — a number that is correct about time and says nothing about litres.

Takeaway: Chain conversions one unit at a time, checking after each step whether the number should grow or shrink. Most wrong answers here are correct partial results.

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Section 7 — Comparing and Transforming Data Sets

The test shows two distributions — as dot plots, histograms or lists — and asks you to compare them, or it changes every value in a set and asks what happened. Both are answered by thinking about centre and spread separately.

Adding the same number to every value slides the whole distribution along. Every measure of centre moves by that amount; every measure of spread stays exactly the same, because nothing has moved relative to anything else.

Measure	Add $k$ to every value	Multiply every value by $k$
Mean	increases by $k$	multiplied by $k$
Median	increases by $k$	multiplied by $k$
Range	unchanged	multiplied by $k$
Standard deviation	unchanged	multiplied by $k$

That "unchanged" is the whole point, and it is worth seeing why: if every value moves 56 to the right, the gaps between them are exactly what they were.

Comparing two spreads by eye. You are never asked to calculate a standard deviation. You are asked which set is more spread out — so look at how tightly the values cluster around the middle. Values bunched near the centre give a small standard deviation; values pushed out to the extremes give a large one.

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Answer: B

Explanation

Adding the same amount to every value slides the whole distribution. The middle value slides with it, so the median rises by 56.

The range is largest minus smallest. Both ends move up by 56, so the difference between them does not change at all:

$$(\text{largest} + 56) - (\text{smallest} + 56) = \text{largest} - \text{smallest}$$

Try it on five values: $3, 5, 8, 9, 14$ has median 8 and range 11. Adding 56 gives $59, 61, 64, 65, 70$ — median 64, which is $8 + 56$ ✓ and range $70 - 59 = 11$, unchanged ✓

The same argument covers the standard deviation, which is also a measure of spread: sliding everything changes no gap between any two values.

Takeaway: Adding a constant shifts every measure of centre and leaves every measure of spread alone. Multiplying by a constant scales both.

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Answer: C

Explanation

Same mean, different spread — which is exactly what standard deviation is for.

Class A's values run from 6 to 10, a span of 4, and most sit at 8 or 9. Class B runs from 1 to 15, a span of 14, with values pushed out to both extremes.

B is far more spread out, so B has the greater standard deviation.

B's option confuses centre with spread: equal means say nothing about how tightly the values cluster. D is wrong because comparing by inspection is precisely what the test expects — you are never asked to compute one.

Takeaway: Look at how tightly the values cluster around the middle. Equal means with different clustering is the standard setup for this question.

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Answer: C

Explanation

With 23 values, the median is the 12th when they are lined up in order.

Count along the intervals, keeping a running total:

Interval	Frequency	Running total
10–19	4	4
20–29	7	11
30–39	9	20
40–49	3	23

After two intervals you have reached only the 11th value, so the 12th is in the next one: 30–39.

B is the trap: 20–29 is the interval where the count is closest to halfway, but it stops one short. Run the total to 12, not to "about half".

Takeaway: Find which position is the middle, then run the frequencies until you pass it. The interval you are in when you pass it holds the median.

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Answer: A

Explanation

Multiplying stretches the distribution rather than sliding it, so both the centre and the spread scale.

Try $2, 4, 5, 9$: the mean is 5 and the range is 7. Tripling gives $6, 12, 15, 27$ — mean 15, which is $3 \times 5$ ✓ and range $27 - 6 = 21$, which is $3 \times 7$ ✓

Contrast this with Q37. Adding slides everything and leaves the gaps alone, so spread is unchanged. Multiplying stretches the gaps too, so spread scales with everything else. Knowing which operation was applied decides the answer.

Takeaway: Add → centre moves, spread unchanged. Multiply → both scale. The two questions look alike and have different answers.

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