Chapter 7 — Statistics and Probability

Answer: B

Explanation

Both sets have a mean of 20.

Set P is every value identical. Nothing deviates from the mean at all, so its standard deviation is exactly zero — the only time that happens.

Set Q spreads from 10 to 30, so its values sit well away from 20.

Q therefore has the greater standard deviation.

Why each wrong option is wrong:

• A, Set P has the greater standard deviation — has it backwards. A set of identical values cannot be the more spread out.
• C, The two standard deviations are equal — equal means do not imply equal spreads. That is exactly what standard deviation exists to distinguish.
• D, Standard deviation cannot be compared without calculating it — no calculation is needed. One set has zero spread, and no set can have less than that.

Takeaway: standard deviation is about spread, not centre. Identical values give zero. Two sets can share a mean and be nothing alike.

Answer: J

Explanation

Work out both, before and after.

Before: the five values total 70, so the mean is 14. Sorted they are $12, 13, 14, 15, 16\mathclose{,}$ so the median is 14 as well.

After: the six values total 130, so the mean is $130 \div 6 = 21.67\mathclose{.}$ Sorted they are $12, 13, 14, 15, 16, 60\mathclose{,}$ so the median is $\frac{14 + 15}{2} = 14.5\mathclose{.}$

The mean moved nearly 8; the median moved a half.

The reason is structural. The mean uses the size of every value, so one huge number drags it a long way. The median only cares about position, and 60 counts for no more than 17 would.

Why each wrong option is wrong:

• G, The median rises far more than the mean does — exactly backwards.
• H, Both rise by the same amount — they move by very different amounts, which is the point.
• F, Neither the mean nor the median changes — both change.

Takeaway: an outlier drags the mean and barely moves the median. That is why incomes and house prices are reported as medians.

Section 2 — Reading Tables

Most ACT statistics questions are a table and a careful sentence. The method is always the same three steps:

1. Find the group the question restricts you to — that is your denominator.
2. Find the count it asks about — that is your numerator.
3. Check the totals add up before you trust the table.

The word to watch is "of". "Of the students who cycle" restricts you to the cyclists' row; the grand total is then irrelevant.

Answer: D

Explanation

"A student who studies Spanish is chosen" restricts the pool to the Spanish row. That row has 70 students, so 70 is the denominator — not 120.

Of those 70, the number who also study history is 45.

$$P = \frac{45}{70} = \frac{9}{14}$$

Why each wrong option is wrong:

• A, $\frac{5}{14}$ — gave the probability of not studying history within the same row. Note it and the answer add to 1, as they must.
• B, $\frac{3}{8}$ — used 120, the whole school. That would be the answer to "a student is chosen at random from the school; what is the probability they study both?" — a different question.
• C, $\frac{7}{12}$ — gave the probability that a random student studies Spanish.

Takeaway: the group the question restricts you to is the denominator. Underline the "of the …" phrase before you look at the table.

Answer: H

Explanation

The percentages are a shorthand. Think of 100 tourists: 20 of them each visited 1 city, 40 each visited 2, 30 each visited 3 and 10 each visited 4. To find the mean, add up the total cities visited and divide by the total tourists.

$$\text{total cities} = 1(20) + 2(40) + 3(30) + 4(10) = 20 + 80 + 90 + 40 = 230$$

$$\text{total tourists} = 20 + 40 + 30 + 10 = 100$$

$$\text{mean} = \frac{230}{100} = 2.3$$

Sanity check: the answer must lie between 1 and 4, and nearer the middle where most tourists are. It does.

Why each wrong option is wrong:

• F, $25$ — averaged the four percentages instead, which answers nothing.
• G, $2.5$ — averaged $1, 2, 3, 4\mathclose{,}$ ignoring how many tourists gave each answer. That treats ten tourists and forty tourists as equally important.
• J, $2$ — gave 2, the mode — the most common answer, not the mean.

Takeaway: whether the frequencies come from a table or a chart, multiply each value by its frequency, add, then divide by the total frequency. Never average the values by themselves.

Answer: B

Explanation

Draw two overlapping circles. The 12 farms raising both sit in the overlap, and they have been counted twice — once in the 34 and once in the 28. Subtract them once to put that right:

$$34 + 28 - 12 = 50$$

Check by parts: cows only is $34 - 12 = 22\mathclose{;}$ sheep only is $28 - 12 = 16\mathclose{;}$ both is 12. And $22 + 16 + 12 = 50\mathclose{.}$ Correct.

Why each wrong option is wrong:

• A, $38$ — subtracted it twice, which removes the "both" farms from the survey altogether.
• C, $62$ — added without removing the double count.
• D, $74$ — added the overlap a third time.

Takeaway: for two overlapping groups, total $= A + B - \text{both}\mathclose{.}$ Subtract the overlap once, because it was counted twice. Then check by adding the three separate regions.

Section 3 — Probability

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

Every probability is between 0 and 1. An answer outside that range is wrong without further thought.

Three ideas the ACT tests:

• the complement — $P(\text{not } A) = 1 - P(A)\mathclose{.}$ Often far quicker than counting what you want directly.
• mutually exclusive events cannot both happen, so $P(A \text{ or } B) = P(A) + P(B)\mathclose{.}$
• independent events do not affect each other, so $P(A \text{ and } B) = P(A) \times P(B)\mathclose{.}$

"Or" tends to add; "and" tends to multiply.

One more, for when the outcomes are numbers: the expected value is the long-run average, each outcome multiplied by its probability and the results added. A game paying 10 with probability $0.3$ and nothing otherwise has an expected value of $10 \times 0.3 = 3\mathclose{.}$

Answer: H

Explanation

Two routes, and they agree.

Count directly: not green means red or blue, which is $5 + 7 = 12$ counters out of 20:

$$\frac{12}{20} = \frac{3}{5}$$

Use the complement: green is $\frac{8}{20}\mathclose{,}$ so not green is

$$1 - \frac{8}{20} = \frac{12}{20} = \frac{3}{5}$$

Why each wrong option is wrong:

• F, $\frac{1}{3}$ — assumed the three colours are equally likely because there are three of them. They are not; the counts differ.
• G, $\frac{2}{5}$ — answered the opposite question. Note it and the correct answer add to 1.
• J, $\frac{2}{3}$ — compared green with the other two colours rather than with the whole bag. The denominator of a probability is everything possible.

Takeaway: the denominator is everything possible. And when a question says "not", the complement is usually the faster route: $1 - P(A)\mathclose{.}$

Answer: B

Explanation

Mutually exclusive means the two events never overlap, so nothing gets counted twice and the probabilities simply add:

$$P(A \text{ or } B) = 0.3 + 0.45 = 0.75$$

Note the answer is below 1, as it must be. And it does not have to equal 1 — there is a $0.25$ chance that neither happens.

Why each wrong option is wrong:

• A, $1$ — assumed the two events cover every possibility.
• C, $0.15$ — subtracted.
• D, $0.135$ — multiplied. Multiplying is for "and" with independent events, and here "and" is impossible: mutually exclusive events have $P(A \text{ and } B) = 0\mathclose{.}$

Takeaway: mutually exclusive means add for "or". Independent means multiply for "and". Mutually exclusive is not the same as covering everything.

Answer: J

Explanation

The first draw: 3 red out of 8 counters.

$$P(\text{first red}) = \frac{3}{8}$$

Now both totals have changed. One red has gone, so there are 2 reds left out of 7 counters.

$$P(\text{second red}) = \frac{2}{7}$$

"And" with these two steps means multiply:

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

Why each wrong option is wrong:

• F, $\frac{3}{4}$ — added the counts.
• G, $\frac{3}{8}$ — gave the first draw only.
• H, $\frac{9}{64}$ — used $\frac{3}{8}$ twice, which is the answer for drawing with replacement.

Takeaway: without replacement, both the numerator and the denominator drop by one for the second draw. Write the two fractions side by side before multiplying.

Section 4 — Samples and Studies

A short but reliable topic. Two rules cover almost every question:

• A conclusion may only be drawn about the population that was sampled, and only if the sample was random.
• An observational study can show that two things go together. Only a controlled experiment, with random assignment, can show that one causes the other.

Answer: C

Explanation

Two limits apply, and both matter.

No cause. Nobody was assigned an amount of sleep — the researchers only watched. So this is an observational study, and it can establish that two things move together, not that one produces the other. Something else could explain both: a student who is organised may sleep more and study more.

No wider population. The sample was random, but only within one university. The conclusion can extend to that university's students and no further.

Option C claims association, and only for students at this university. Both limits respected.

Why each wrong option is wrong:

• B, Sleeping longer causes students to score higher and A, Higher scores cause students to sleep longer — claim causation in one direction or the other. Neither is available from an observational study, and the fact that the two opposite claims are equally consistent with the data is itself the argument.
• D, More sleep is associated with higher scores among all adults — extends the finding to all adults. The sample was random within one university, so that is as far as it reaches.

Takeaway: observational studies give association; only a randomised experiment gives causation. And a conclusion never travels beyond the population that was actually sampled.

Answer: J

Explanation

Each bread can go with each filling, and each of those pairs with each sauce, so the choices multiply:

$$4 \times 3 \times 2 = 24$$

Why it multiplies rather than adds: for each of the 4 breads there are 3 fillings, giving 12 combinations; for each of those 12 there are 2 sauces, giving 24. Adding would answer "how many ingredients are there altogether", which is a different question.

Why each wrong option is wrong:

• F, $9$ — added the three counts.
• G, $12$ — multiplied two of the three and forgot the sauces.
• H, $14$ — mixed multiplying and adding.

Takeaway: independent choices multiply. If the question says "one of each", multiply the counts together.

Answer: D

Explanation

A box plot is five numbers drawn on a number line. The two ends of the whiskers are the lowest and the highest value. The two ends of the box are the quartiles, and the line inside the box is the median.

The interquartile range is the width of the box. For set X that is $45 - 35 = 10\mathclose{;}$ for set Y it is $60 - 25 = 35\mathclose{.}$ Set Y has the greater interquartile range.

Why each wrong option is wrong:

• A — has the sets the wrong way round. The range is the whole width, whisker to whisker: $50 - 30 = 20$ for X against $75 - 10 = 65$ for Y.
• B — the two boxes are plainly not the same width.
• C — reads a wider plot as a higher one. The median is the line inside the box, and both lines stand at 40.

Takeaway: on a box plot, width is spread and position is centre. The box's width is the interquartile range, the whole width is the range, and the line inside the box is the median.

Answer: F

Explanation

Three limits, and each rules out one option.

The population. The sample was random within one university, so the conclusion reaches that university's students and no further.

Estimate, not certainty. A sample gives an estimate with a margin of error around it. "About 42%" is right; "exactly 42%" is not.

Opinion is not outcome. The survey asked what students support. It says nothing about whether the change would work.

Why each wrong option is wrong:

• G, About 42% of students nationwide support the change — extends beyond the population sampled.
• H, Exactly 42% of students at that university support the change — claims exactness a sample cannot give.
• J, The survey shows the change will improve student results — turns a question about opinion into a claim about results.

Takeaway: a random sample supports an estimate about the population it was drawn from, and nothing else.

Answer: D

Explanation

Causation requires a controlled experiment: the researcher assigns the treatment, and assigns it at random.

Random assignment is what makes the two groups alike in every other respect — motivation, prior attainment, hours available — so that any difference in results can be attributed to the method itself.

Why each wrong option is wrong:

• B, Ask students which method they used and compare their results — students chose their own method, so the groups may differ in ways that also affect results. That is an observational study: it shows association, not cause.
• C, Compare results at a school that adopted the method with one that did not — the schools differ in many ways besides the method.
• A, Survey teachers about which method they believe works better — measures belief, not effect.

Takeaway: assigned at random means an experiment, and only an experiment supports a claim about cause. Anything self-selected gives association only.

Answer: G

Explanation

In $y = mx + c$ the coefficient of $x$ is the slope: how much $y$ changes for each one-unit increase in $x\mathclose{.}$

Here one extra year of age corresponds to 6 more centimetres of height, on average.

Why each wrong option is wrong:

• F, The height of a child at birth, in centimetres — that is the 78, the value where the line meets the vertical axis.
• H, The age at which a child reaches 6 centimetres and J, The number of children in the study — neither quantity appears in the model.

Takeaway: in a line of best fit, the slope is a rate of change and the intercept is the value at zero. Say the units out loud: "6 centimetres per year".

Answer: C

Explanation

The line predicts

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

The actual value is 15, so the point sits 2 above the line:

$$15 - 13 = 2$$

That difference is called the residual. A positive residual means the line under-predicted.

Why each wrong option is wrong:

• A, $-2$ — subtracted actual from predicted, giving the residual's negative.
• B, $0$ — assumed the point lies exactly on the line. A line of best fit rarely passes through any particular point.
• D, $13$ — gave the prediction rather than the difference.

Takeaway: residual $=$ actual $-$ predicted. Positive means the point is above the line.

Answer: F

Explanation

An upward-sloping line of best fit shows association: the two quantities tend to rise together across these towns.

It cannot show cause, in either direction. Nobody opened bookshops experimentally to see what happened; the data was only observed. A third factor — population size, say — could easily drive both.

Why each wrong option is wrong:

• G, Opening more bookshops would raise a town's average income and H, Higher average income causes towns to open more bookshops — claim causation, in opposite directions. That both are equally consistent with the same scatterplot is the argument against either.
• J, Bookshop numbers and income are unrelated in these towns — contradicts the upward slope.

Takeaway: a scatterplot shows association. Only a randomised experiment shows cause, and "these two rise together" never says which one moves the other.

Answer: A

Explanation

Four choices for first place, then three left for second, two for third, and the last runner takes fourth:

$$4 \times 3 \times 2 \times 1 = 24$$

The counts shrink by one each time because a runner cannot finish in two places at once — unlike the sandwich question, where each choice was independent and the counts stayed the same.

Why each wrong option is wrong:

• B, $16$ — used $4^2\mathclose{,}$ which would allow repeats.
• C, $10$ — added the choices.
• D, $4$ — gave the number of runners.

Takeaway: arrangements of $n$ distinct things in order come to $n \times (n-1) \times \dots \times 1\mathclose{.}$ When the same item cannot be reused, the count drops by one at every step.

Answer: G

Explanation

"All customers" means the grand total, 300, goes underneath. The snack column totals 162.

$$\frac{162}{300} = \frac{27}{50}$$

Why each wrong option is wrong:

• F, $\frac{7}{15}$ and H, $\frac{13}{20}$ — restricted the question to one row. Neither was asked for.
• J, $\frac{27}{23}$ — compared the two column totals with each other rather than with the whole.

Takeaway: the denominator is whatever group the question names. "All" means the grand total; "of the adults" would mean 180.

Answer: B

Explanation

Read the height of each bar: 6, 9, 4 and 1. That is 20 customers, so the median lies between the 10th and the 11th wait when they are lined up in order.

Count along the bars, keeping a running total:

The first bar reaches only the 6th customer; the second takes the count to the 15th. The 10th and the 11th are both in the second bar: 5 to 10 minutes.

Why each wrong option is wrong:

• A — the first bar holds only 6 of the 20 waits. The count has not reached the middle yet.
• C — the middle of the axis, not of the data. Ten minutes is halfway along the scale, but 15 of the 20 customers had been served before then.
• D — the last bar holds a single wait, the longest of all.

Takeaway: a histogram is a frequency table drawn as bars. Find which positions are the middle, then run the heights until you pass them. The median follows the customers, not the scale underneath them.

Answer: G

Explanation

Read the chart: 12 households own no car, 20 own one, 14 own two and 4 own three. Total cars owned:

$$0(12) + 1(20) + 2(14) + 3(4) = 0 + 20 + 28 + 12 = 60$$

Households: 50.

$$\text{mean} = \frac{60}{50} = 1.2$$

Why each wrong option is wrong:

• F, $1$ — gave the mode, the most common answer, which is 1.
• H, $1.5$ — averaged $0, 1, 2, 3$ without weighting by how many households gave each.
• J, $50$ — gave the number of households.

Takeaway: multiply each value by its frequency, add, then divide by the total frequency. A mean need not be a whole number even when every value is.

Answer: C

Explanation

Substitute $x = 15$ into the line:

$$y = 12(15) - 40 = 180 - 40 = 140$$

Why each wrong option is wrong:

• A, $-40$ — gave the intercept, which is the prediction at zero degrees — and note it is negative, which is why a line of best fit should not be trusted far outside the data it came from.
• B, $12$ — gave the slope.
• D, $180$ — multiplied and forgot to subtract.

Takeaway: a line of best fit predicts by substitution. The intercept is the prediction at zero, which is often physically meaningless.

Answer: F

Explanation

A negative slope means that as $x$ increases, $y$ decreases. Here $x$ is age and $y$ is value, so older cars tend to be worth less.

The word "tend" is doing real work. A line of best fit describes a trend across the whole scatterplot; individual points sit above and below it.

Why each wrong option is wrong:

• G, Older cars tend to be worth more — describes a positive slope.
• H, Age and value are unrelated — a slope of zero would mean no relationship; this one is negative.
• J, Every older car is worth less than every newer one — far too strong. A trend is not a rule about every pair of cars, and a scatterplot with a clear negative slope will still contain an old car worth more than some newer one.

Takeaway: the sign of the slope gives the direction of the trend. "Tends to" is right; "always" is not.

Answer: A

Explanation

The line predicts

$$y = 4(3) + 2 = 14$$

The actual value is 11, so

$$\text{residual} = 11 - 14 = -3$$

A negative residual means the point lies below the line — the model over-predicted.

Why each wrong option is wrong:

• B, $3$ — subtracted predicted from actual the wrong way round. The sign carries the meaning, so the order matters.
• D, $14$ and C, $11$ — gave one of the two values rather than their difference.

Takeaway: residual = actual − predicted. Negative means the point sits below the line.

Answer: H

Explanation

Nine digits for the first position; once one is used, eight remain for the second, then seven, then six:

$$9 \times 8 \times 7 \times 6 = 3024$$

Why each wrong option is wrong:

• F, $24$ — counted the orderings of four fixed items, ignoring the nine digits to choose from.
• G, $36$ — multiplied the digit count by the number of positions.
• J, $6561$ — $9^4$ allows repeats, which the question forbids.

Takeaway: when items cannot repeat, the count drops by one at each step. When they can, it stays the same.

Answer: D

Explanation

Every main can go with every side, and each of those pairs with every drink, so the counts multiply:

$$5 \times 4 \times 3 = 60$$

The counts stay the same at each step here, because choosing a main does not use up any side.

Why each wrong option is wrong:

• A, $12$ — added, which answers "how many items are on the menu".
• B, $15$ — left out one of the three choices.
• C, $23$ — mixed multiplying and adding.

Takeaway: independent choices multiply. Compare with the previous question: there the counts shrank, because a digit once used was gone.

Answer: H

Explanation

The far-left position is fixed, so there is only one choice for it. That leaves four books for four positions:

$$1 \times 4 \times 3 \times 2 \times 1 = 24$$

Why each wrong option is wrong:

• F, $5$ — gave the number of books.
• G, $20$ — counted only the first two positions.
• J, $120$ — $5! = 120$ counts every arrangement, including the ones with the wrong book on the left.

Takeaway: handle the restricted position first, then count the free ones. A fixed position contributes a factor of 1.

Answer: A

Explanation

A sample is representative when every member of the population has the same chance of being chosen. Only a random draw from the full list of 900 does that.

The other three are self-selected, and each in a way that is plausibly linked to the very thing being measured — which is what makes the bias serious rather than merely untidy.

Why each wrong option is wrong:

• B, Ask the first 100 students who arrive at school one morning — students who arrive early may be more organised, and so may do more homework.
• C, Ask 100 students in the library after school — students in the library after school are almost certainly doing homework.
• D, Ask for 100 volunteers to fill in a form — volunteers choose themselves, and people with strong feelings volunteer more.

Takeaway: representative means randomly selected from the whole population. Ask whether the way people ended up in the sample is connected to what is being measured.

Answer: G

Explanation

An expected value is a long-run average: each outcome multiplied by how often it happens, then added.

$$5(0.2) + 2(0.5) + 0(0.3) = 1 + 1 + 0 = 2$$

The third outcome matters even though it pays nothing: the probabilities must total 1, so "nothing" happens with probability $1 - 0.2 - 0.5 = 0.3\mathclose{.}$

Check the size. The answer must lie between the smallest outcome, 0, and the largest, 5 — and nearer the small end, because the small outcomes are the common ones. It does.

Why each wrong option is wrong:

• F, $0.7$ — added the two probabilities. That is the chance of winning something, not the amount won.
• H, $3.5$ — averaged the two prizes, treating them as equally likely and forgetting that most plays do not win 5.
• J, $7$ — added the two prizes. No single play can win both.

Takeaway: expected value is $\sum (\text{value} \times \text{probability})\mathclose{.}$ Multiply each outcome by its own probability, add, and check the answer sits between the smallest and largest outcomes.

Answer: A

Explanation

Scores rising after a course is consistent with the course working — and equally consistent with employees simply getting better at their jobs over the same period, or getting better at the test itself.

A comparison group of similar employees who did not take the course separates those explanations: if the course group improved more, the course is the plausible reason.

Why each wrong option is wrong:

• B, A larger number of employees taking the course — a bigger group makes the same flawed comparison more precisely. Size fixes noise, not bias.
• C, A second test of the same employees after the course — retesting the same people adds no comparison.
• D, A survey asking employees whether they enjoyed the course — enjoyment is not attainment.

Takeaway: to claim cause, you need something to compare against. "Before and after" on its own cannot rule out everything else that happened in between.