1. A comparison of two quantities

2. A type of fraction

3. An equation that states two ratios are equal ✓

4. A method for simplifying fractions

1. b and c

2. a and d ✓

3. a and b

4. c and d

1. a and d

2. b and c ✓

3. a and c

4. b and d

1. In a proportion, the sum of extremes equals the sum of means

2. In a proportion, divide the extremes by the means

3. In a proportion, multiply all four terms together

4. In a proportion, the product of extremes equals the product of means ✓

1. Yes, because 3 + 9 = 12

2. No, because the numerators are different

3. Yes, because both ratios simplify to 3/4 ✓

4. No, because the denominators are different

1. Yes, because 5 × 21 = 7 × 15 = 105 ✓

2. No, because 5 × 21 ≠ 7 × 15

3. Yes, because 5 + 15 = 20

4. Cannot determine

1. 6/8 and 9/12 ✓

2. 2/3 and 4/5

3. 5/7 and 10/15

4. 3/4 and 7/9

1. Add 3 and 20

2. Cross multiply: 3 × 20 = 5 × x ✓

3. Divide 20 by 5

4. Subtract 5 from 20

1. 4/10 = 2/x

2. 2/4 = x/10

3. 4/2 = 10/x

4. Both A and B are correct ✓

1. To make the problem easier to read

2. To make cross multiplication work

3. It doesn't matter where units are placed

4. To ensure corresponding quantities are being compared correctly ✓

1. x = 10

2. x = 12 ✓

3. x = 14

4. x = 16

1. n = 18

2. n = 20

3. n = 24 ✓

4. n = 27

1. y = 25 ✓

2. y = 22

3. y = 20

4. y = 30

1. $18.00

2. $22.00

3. $20.00 ✓

4. $25.00

1. 300 km

2. 350 km

3. 400 km

4. 375 km ✓

1. 30 km

2. 40 km

3. 45 km ✓

4. 50 km

1. 6 cups ✓

2. 5.5 cups

3. 5 cups

4. 6.5 cups

1. a = 7.0

2. a = 8.5

3. a = 8.0

4. a = 7.5 ✓

1. 28/7 = 20/x; Cost = $5

2. 7/28 = 20/x; Cost = $80

3. 7/28 = x/20; Cost = $5

4. 28/7 = x/20; Cost = $80 ✓

1. 12 minutes

2. 18 minutes

3. 16 minutes

4. 14 minutes ✓

1. 25 teachers

2. 30 teachers ✓

3. 35 teachers

4. 40 teachers

1. 6/15

2. 8/15 ✓

3. 10/15

4. 2/5

1. 90 m ✓

2. 85 m

3. 80 m

4. 95 m

1. Store A, cheaper by $0.20 per item ✓

2. Store B, cheaper by $0.20 per item

3. Store A, cheaper by $0.30 per item

4. Store B, cheaper by $0.30 per item

1. First student; 5 × 32 = 8 × 20

2. Both are wrong; x should be 24

3. First student; cross multiply 5 × 32 = 8x gives 160 = 8x, so x = 160 ÷ 8 = 20 ✓

4. Second student; 5 × 32 = 160 and 8 × 20 = 160, but cross multiply gives 8x = 160, so x = 20

1. Adding all the numbers

2. Multiplying all ratio numbers

3. Adding ratio numbers to get total parts, then dividing total amount by parts ✓

4. Subtracting the ratio numbers

1. 3

2. 8 ✓

3. 2

4. 15

1. 20 - 4

2. 20 ÷ 4 = 5 (the multiplier) ✓

3. 4 × 7

4. 7 + 20

1. The sum of ratios

2. A percentage

3. A type of fraction

4. An equation stating two ratios are equal ✓

1. ₩30,000

2. ₩15,000

3. ₩12,000 ✓

4. ₩20,000

1. The factor by which you scale the ratio to match given values ✓

2. The sum of parts

3. The ratio itself

4. The total amount

1. Nothing - you're done

2. Convert to percentages

3. Check that amounts add to total and ratio is preserved ✓

4. Simplify the ratio

1. It's just a trick

2. It treats each ratio number as equal-sized portions, ensuring fair division according to the ratio ✓

3. It makes numbers smaller

4. It's the only method

1. Because 3 × 5 = 15

2. It's not correct

3. Total parts = 8, one part = ₩5,000, so first person (3 parts) gets 3 × ₩5,000 = ₩15,000 ✓

4. Because ₩15,000 is less than ₩40,000

1. Random guess

2. It's not 30

3. Because 24+5=29, but rounded

4. Multiplier is 24÷4=6, so second value = 5×6=30, maintaining the proportional relationship ✓

1. Because 3 is a good number

2. Because 18 people is 3 times 6 people (18÷6=3), so all ingredients must triple to maintain proportions ✓

3. Because the ratio has 3 numbers

4. To make it bigger

1. Parts are always 10

2. It's a mistake

3. Total parts = 3+7=10 ratio units; these 10 parts equal the sum of 50, so each part = 50÷10=5 ✓

4. Because 50÷7=10

1. Checking sum ensures correct total; checking ratio ensures proportional relationship is maintained; both needed to confirm full correctness ✓

2. It's not important

3. Only ratio matters

4. Only sum matters

1. Random amounts

2. Guess and check

3. Total parts = 1+2+3=6, one part = 24÷6=4m³, then multiply: 1×4=4, 2×4=8, 3×4=12 ✓

4. Just divide 24 by 3

1. ₩32,000

2. ₩45,000

3. ₩36,000

4. ₩40,000 ✓

1. 49

2. 20

3. 25 ✓

4. 30

1. ₩24,000 ✓

2. ₩36,000

3. ₩20,000

4. ₩60,000

1. 49 liters

2. 7 liters

3. 15 liters

4. 9 liters ✓

1. 90cm ✓

2. 75cm

3. 60cm

4. 100cm

1. 11

2. 45

3. 5

4. 10 ✓

1. Teacher is wrong

2. Student forgot to find total parts (should be 5, not 2); correct division: ₩36,000 and ₩54,000 ✓

3. ₩45,000 each is correct

4. Student should have used multiplication

1. Correct

2. Wrong - initial boys=16, girls=20; after 3 boys join: 19 boys, 20 girls, new ratio is 19:20, not 4:5 ✓

3. Correct ratio, wrong total

4. Cannot determine

1. Large partner gets ₩100,000 (5×₩20,000 per part), which is ₩40,000 MORE than small partner's ₩60,000; complaint shows misunderstanding of ratios (3:5 means one gets 5 parts to other's 3 parts, not 5 total parts) ✓

2. Both wrong

3. Cannot calculate

4. Large partner correct

1. Teacher correct - both methods mathematically equivalent and reach same correct answer; parts method more intuitive, equation method more algebraic, both valid ✓

2. Both students wrong

3. Only B correct

4. Only A correct

1. C is completely correct

2. C gets less than deserved

3. Ratios don't apply here

4. C misunderstands - actual ratio IS 2:5:8 (not simplified to 4:5); C DOES get 4× what A gets (8÷2=4); C's complaint suggests confusion about what the numbers mean ✓

1. Ratios that add up to the same number

2. Ratios with the same numbers

3. Different ratios that express the same relationship ✓

4. Ratios with equal parts

1. Multiply by different numbers

2. Multiply or divide all parts by the same number ✓

3. Add the same number to all parts

4. Subtract the same number from all parts

1. Adding up all the numbers

2. Reducing it to the smallest whole numbers that express the same relationship ✓

3. Converting to a fraction

4. Making the numbers bigger

1. The average of the numbers

2. The sum of all factors

3. The smallest number that divides into all numbers

4. The largest number that divides evenly into all given numbers ✓

1. 3:4 and 4:5

2. 4:5 and 6:7

3. 4:6 and 6:9 ✓

4. 2:6 and 3:9

1. 3

2. 2

3. 6 (the GCF) ✓

4. 9

1. Multiply by the Least Common Denominator (LCD) to get whole numbers ✓

2. Add the fractions

3. Divide by the GCF

4. Convert to decimals

1. You don't have to - any number works

2. Because using different multipliers changes the proportional relationship between quantities ✓

3. It's just a rule

4. To make the math easier

1. Both give the same numerical value

2. They're not similar

3. Both involve dividing by the GCF to reduce to lowest terms ✓

4. Both involve multiplication only

1. They're not equivalent

2. They have the same sum

3. Because they use the same digits

4. They express the same proportional relationship - the second quantity is always 1.5× the first ✓

1. Adding is faster

2. Adding is wrong - must multiply to maintain proportions ✓

3. Both work equally well

4. Multiplication is a rule

1. 6+10 and 9+15

2. 6×9 and 10×15

3. 6×15 and 10×9 ✓

4. 6÷10 and 9÷15

1. Both are prime numbers, so GCF=1 ✓

2. 7 is prime but 11 is not

3. They're already small

4. You can simplify it

1. Converted to fractions

2. Rounded the decimals

3. Multiplied by 2 to remove decimals, then simplified ✓

4. Added 1.5 to both sides

1. 10:17

2. 5:8

3. 2:3

4. 4:7 ✓

1. 15:30

2. 12:24

3. 9:21 ✓

4. 6:12

1. 2:3:4 ✓

2. 3:4:5

3. 6:9:12

4. 4:6:8

1. No, they simplify to different ratios

2. No, different numbers

3. Cannot determine

4. Yes, both simplify to 2:3 ✓

1. 8:12:20 ✓

2. 4:6:10

3. 2:3:5

4. 1:2:3

1. 8:12:4

2. 10:15:5

3. 4:6:2

4. 6:9:3 ✓

1. Nothing wrong - it's simplified

2. Student only divided by 2; should divide by GCF=12 to get 2:3 for fully simplified form ✓

3. Should multiply instead

4. Should have divided by 6

1. Cannot verify

2. Correct - both simplify to 3:4, representing the same proportion ✓

3. Wrong - different numbers

4. Wrong - Class A has more students

1. Store correct (6:9 = 4:6 = 2:3, same color); Owner claim misleading (proportions same, but Mix B could be smaller total batch, not 'less' ratio-wise) ✓

2. Both claims wrong

3. Owner correct - definitely uses less

4. Store wrong about color

1. Partially correct: 3:15 is wrong (×5 on one side only); only valid equivalent with 15 is 9:15 (×3 both sides) or 15:25 (×5 both sides) ✓

2. Correct - both valid

3. Completely wrong

4. Correct for 15:5, wrong for 3:15

1. Design A always better

2. Cannot compare

3. Identical in all ways

4. Mathematically equivalent (both 1:4 proportion), but practically different if absolute numbers matter (Design A: 10 total units vs B: 15 total units in stated form) ✓

1. A type of fraction

2. A multiplication operation

3. A comparison of two or more quantities ✓

4. The sum of two numbers

1. Percentage notation (35%)

2. Colon notation (3:5) ✓

3. Decimal notation (3.5)

4. Scientific notation (3.5×10¹)

1. The second quantity

2. The first quantity being compared ✓

3. The total

4. The difference

1. A ratio with fractions

2. A ratio with only whole numbers

3. A ratio comparing one part to the total

4. A ratio comparing one part to another part ✓

1. A ratio with three numbers

2. A ratio using percentages

3. A ratio comparing one part to the total ✓

4. A ratio comparing two parts

1. Three ways (colon, fraction, words) ✓

2. Only one way

3. Five ways

4. Two ways

1. Yes, order matters - 3:5 and 5:3 are different ✓

2. Only in word problems

3. Only for large numbers

4. No, 3:5 and 5:3 are the same

1. To make calculation easier

2. To ensure we're comparing like quantities; comparing 2 meters to 150 cm requires converting to same unit first ✓

3. Only for mathematical elegance

4. They don't need the same units

1. 15:12 (girls to boys)

2. 12:15 (boys to girls)

3. 12:27 (boys to total) ✓

4. 27:12 (total to boys)

1. They're not similar at all

2. They use the same notation

3. Only in division

4. Both compare a part to the total; part-to-whole ratio 3:8 means 3/8 of the whole ✓

1. To make more servings

2. It ensures consistent taste and texture regardless of batch size ✓

3. Only for appearance

4. It's not important

1. 5:3 is bigger

2. No mathematical difference

3. 5:3 compares two separate groups; 5/8 represents apples as part of total (5+3=8) ✓

4. They're exactly the same

1. 3 liters

2. 6 liters

3. 4 liters ✓

4. 9 liters

1. Only works with three numbers

2. You can always add them

3. Only works for part-to-part ratios to find total; for 5:3 apples to oranges, total is 5+3=8, but doesn't work for all ratio types ✓

4. Never works

1. 15:9

2. 6:9

3. 9:15

4. 9:6 ✓

1. 28:16

2. 16:28

3. 12:16 ✓

4. 16:12

1. 10:50 ✓

2. 10:15

3. 10:25

4. 15:50

1. 10:15:20

2. 45:20:15

3. 20:10:15

4. 20:15:10 ✓

1. 4/8 = 1/2 ✓

2. 4/3

3. 4/4 = 1

4. 4/7

1. 30:450

2. 450:30

3. 45:3

4. 15:1 ✓

1. True if we simplify

2. False - 16 > 12; the claim is incorrect ✓

3. Cannot determine from ratios alone

4. True - 12 > 16

1. Double everything: 4:6:2

2. Multiply each by 3 (since 12÷4=3): ratio becomes 6:9:3 ✓

3. Keep ratio same: 2:3:1

4. Add 8 to each number: 10:11:9

1. Incorrect - ratios 100:80 and 5:4 are equivalent (both simplify to 5:4); they represent same proportion, just different quantities; claim confuses ratio with absolute amount ✓

2. Correct - Store A has more variety

3. Correct - 100 and 80 are bigger than 5 and 4

4. Cannot compare different stores

1. Student B correct - ratios matter more than absolute values for nutritional balance; 10:20:70 simplifies to 1:2:7 which aligns with recommended proportions ✓

2. Both wrong - need more information

3. Student A correct - lowest number means unhealthy

4. Neither - ratios don't apply to nutrition

1. Both wrong - ratios don't matter in business

2. Neither addresses the real issue

3. CEO correct - more sales is always better

4. CFO correct - new ratio 5:3:5 makes sales equal to research, changing the relative priorities from original strategy where research was dominant (5 vs 2) ✓

1. The belief that gambling always loses money

2. The belief that probability doesn't apply to gambling

3. The mistaken belief that past independent events affect future independent events ✓

4. The belief that all games are fair

1. You win immediately

2. The host opens a door with a goat and offers you a chance to switch ✓

3. You must open your door

4. The game ends

1. P(person has condition | positive test)

2. P(positive test | person has condition) - ability to detect true cases ✓

3. The emotional impact of the test

4. How sensitive the test equipment is

1. The test's precision

2. How specific the disease is

3. P(positive test | has condition)

4. P(negative test | person doesn't have condition) - ability to correctly identify negatives ✓

1. Test misses disease that's present

2. Test gives unclear result

3. Test indicates disease when person doesn't have it ✓

4. Test correctly identifies disease

1. 23 people ✓

2. 183 people

3. 365 people

4. 50 people

1. The average outcome weighted by probabilities over many trials ✓

2. The most common outcome

3. The highest possible value

4. The value you expect to get every time

1. Because the host helps you

2. Because your initial choice had 1/3 chance of being right, so the other door has 2/3 chance after host reveals goat ✓

3. It's random - 50-50

4. Because there are two doors left

1. It can't - accuracy determines diagnostic value

2. Because the test is actually not accurate

3. Because when disease is rare (low base rate), false positives from large healthy population can outnumber true positives from small diseased population ✓

4. Because doctors make mistakes

1. They're the same thing

2. Relative risk is always higher

3. Absolute risk is for individuals, relative for groups

4. Relative risk compares risks as a ratio (e.g., 'doubled'), while absolute risk shows actual probability difference (e.g., '+0.01%') ✓

1. Only for probability calculations

2. To calculate defect rates, expected costs, and optimize inspection strategies based on probabilities ✓

3. It's not important

4. To make products randomly

1. It doesn't - business is about certainty

2. Only used for gambling businesses

3. Probability quantifies likelihood of outcomes, allowing expected value calculations and informed risk-benefit decisions ✓

4. Business ignores probability

1. Inheritance probabilities depend on parents' genotypes (conditions), making P(child trait | parent genotypes) essential for predictions ✓

2. They're not - genetics uses different math

3. Only for plant genetics

4. Genetics is always 50-50

1. Only for temperature

2. They don't use probability

3. They calculate conditional probabilities based on current conditions and historical patterns: P(rain tomorrow | current conditions) ✓

4. They guess randomly

1. 11% ✓

2. 2%

3. 85%

4. 90%

1. $14,000

2. $70,000

3. $26,000 ✓

4. -$10,000

1. 93.75% ✓

2. 75%

3. 50%

4. 56.25%

1. 3 defects ✓

2. 10 defects

3. 30 defects

4. 1 defect

1. 57.8% ✓

2. 25%

3. 75%

4. 42.2%

1. 65%

2. 60%

3. 100%

4. 75% ✓

1. Should say quadruples

2. Technically accurate (3× increase) but misleading emphasis on relative risk (tripled) obscures tiny absolute risk increase of only 0.04%; sensationalist framing ✓

3. Completely false

4. Accurate and fair presentation

1. Depends on: disease prevalence, test accuracy, costs, risk tolerance. If disease rare in general population but common in high-risk group, Strategy B may optimize cost-effectiveness while maintaining acceptable detection rate ✓

2. Strategy A always better

3. Strategy B always better

4. They're equivalent

1. Results support theoretical prediction; experimental probability close to theoretical with reasonable sample size (small difference due to random variation) ✓

2. Disproves theory

3. Need exactly 10 and 20 to match theory

4. Proves switching doesn't help - only 60% not 67%

1. Cannot determine from given information - need P(succeed | hire) which requires base rates and may use Bayes' theorem; the two given conditionals are not what's needed ✓

2. 90% - that's P(positive | succeed)

3. 80% - that's P(hire | positive)

4. Average: 85%

1. Critic clearly correct - duty overrides probability

2. Question is invalid

3. Utilitarian clearly correct - lower expected value

4. Probability can inform (expected value analysis) but cannot resolve moral dilemma; highlights tension between minimizing expected harm vs. duties/rights; different ethical frameworks interpret same probabilities differently ✓

1. Calculate the answer immediately

2. Check your answer

3. Understand the problem - read carefully and identify what's being asked ✓

4. Draw a diagram

1. For 'OR' questions

2. For 'AND' questions ✓

3. Never multiply probabilities

4. For mutually exclusive events

1. For dependent events

2. For 'OR' questions ✓

3. For conditional probability

4. For 'AND' questions

1. Conditional probability

2. 'AND' problems

3. 'Exactly one' problems

4. 'At least one' or 'one or more' problems ✓

1. Pie chart

2. Two-way table

3. Tree diagram ✓

4. Venn diagram

1. Two-way table ✓

2. Tree diagram

3. Organized list

4. Venn diagram

1. Must be between 0 and 1 inclusive ✓

2. Must be a whole number

3. Must be greater than 1

4. Must be exactly 0.5

1. It always gives the same answer

2. 'At least one' includes many cases (1, 2, 3...), but 'none' is just one case, making complement much simpler ✓

3. It's not easier - just different

4. Because 1 is easier to remember

1. You never subtract for OR problems

2. You always subtract - it's a rule

3. If events overlap (can occur together), P(A) + P(B) counts the overlap twice, so must subtract P(A and B) to correct ✓

4. To make the calculation harder

1. To make the problem harder

2. You don't need to update

3. Only if drawing the same color

4. Because removing an item changes both the number of favorable outcomes and the total, altering the probability for subsequent draws ✓

1. Independent events can't be solved

2. Determines whether to multiply probabilities directly (independent) or use conditional probabilities (dependent) ✓

3. It's not important

4. Only matters for coin flips

1. You should use the grand total

2. It doesn't matter which total you use

3. Because B is the condition - we're restricting to cases where B occurred, making B's total the new sample space ✓

4. Because B comes second alphabetically

1. Complex problems often require intermediate calculations that can be labeled and combined using probability rules ✓

2. To make the problem longer

3. You shouldn't break down problems

4. Sub-problems are easier to guess

1. Only for hard problems

2. It's not necessary if math is correct

3. Helps catch errors - if answer seems illogical (e.g., P(both) > P(one)), likely made mistake ✓

4. Intuition is unreliable

1. 3/8

2. 5/8

3. 7/8

4. 1/2 ✓

1. 3/45

2. 9/100

3. 6/90 = 1/15 ✓

4. 9/10

1. 5 ✓

2. 50

3. 95

4. 48

1. 0.70

2. 0.49

3. 1.40

4. 0.91 ✓

1. 15/40 = 37.5% ✓

2. 15/55

3. 15/100 = 15%

4. 40/100

1. 1/3

2. 5/6

3. 1/2

4. 2/3 ✓

1. Incorrect - should be 2%

2. Incorrect - used P(A) instead of conditional; correct calculation: P(A and def) = 0.60×0.02 = 0.012, P(B and def) = 0.40×0.05 = 0.020, P(def total) = 0.032. P(A|def) = 0.012/0.032 = 37.5% ✓

3. Correct - A makes 60% of items

4. Correct - conditional probability not needed

1. Student A's approach is wrong

2. Student A's complement approach is MUCH simpler - one calculation vs. many complex terms with different orderings ✓

3. Both equally difficult

4. Student B's approach is simpler

1. Both correct - different valid approaches to same problem! ✓

2. Only Student Y

3. Only Student X

4. Both wrong

1. Incorrect - recognized impossible answer but wrong fix; events overlap (can win both), so must use general addition rule: P(1 or 2) = 0.6 + 0.7 - (0.6×0.7) = 1.3 - 0.42 = 0.88 = 88% ✓

2. Incorrect - should multiply instead

3. Correct - winning at least one game is certain

4. Correct reasoning and answer

1. Both interpretations valid

2. Neither correct - need more information

3. Teacher correct - student confused P(positive | disease) with P(disease | positive); these are different conditionals affected by base rate; need full calculation with false positives ✓

4. Student correct - 90% accuracy means 90% probability

1. Probability of A or B

2. Probability of A and B

3. Probability of A given that B has occurred ✓

4. Probability of A divided by B

1. P(A|B) = P(A) + P(B)

2. P(A|B) = P(A and B) / P(B) ✓

3. P(A|B) = P(A) × P(B)

4. P(A|B) = P(B) / P(A)

1. Pie chart

2. Contingency table ✓

3. Line graph

4. Scatter plot

1. Probability of both variables together

2. Probability written in the margins

3. Probability with a condition

4. Probability from the table's edges/margins (row or column totals) ✓

1. Probability with a condition

2. Probability from row totals

3. Probability of both variables together (from a specific cell) ✓

4. Probability from column totals

1. Test is positive when person doesn't have condition ✓

2. Test is negative when person has condition

3. Test gives wrong percentage

4. Person lies about symptoms

1. They always occur together

2. They are mutually exclusive

3. They are independent ✓

4. They are dependent

1. To make calculations easier

2. Because once we know B occurred, we only consider outcomes where B is true ✓

3. To reduce the number of outcomes

4. It doesn't - uses full sample space

1. Yes, but only for dependent events

2. Yes, always equal

3. No, generally different - they answer different questions ✓

4. Yes, but only for independent events

1. No - it's actually higher

2. Yes, they're the same

3. Yes, but only if disease is common

4. No - test accuracy often refers to P(positive | disease), not P(disease | positive); the latter depends on disease prevalence and false positive rate ✓

1. Sum of A and B

2. Total for B (the condition) ✓

3. Grand total

4. Total for A

1. P(B)

2. 0

3. P(A) ✓

4. P(A and B)

1. Because false positives from large healthy population can outnumber true positives from small diseased population ✓

2. Because doctors make mistakes

3. They don't - it's always high

4. Because the formula is wrong

1. A and B are mutually exclusive

2. A and B are independent

3. B makes A more likely - positive association ✓

4. B makes A less likely

1. 1/6

2. 1/2

3. 2/3

4. 1/3 ✓

1. 4/52

2. 4/51

3. 3/51 = 1/17 ✓

4. 3/52

1. 45/60 = 75% ✓

2. 60/100

3. 45/100 = 45%

4. 45/65

1. 60/100 = 60%

2. 45/100 = 45%

3. 45/60 = 75%

4. 45/65 ≈ 69.2% ✓

1. 45/145 ≈ 31% ✓

2. 100/145

3. 45/1000 = 4.5%

4. 45/50 = 90%

1. 0.48

2. 0.52

3. 0.12

4. 0.3 ✓

1. Incorrect - should add probabilities

2. Incorrect - used wrong total; should be P(smartphone | tablet) = 50/80 = 62.5% using tablet total as denominator ✓

3. Correct method, wrong numbers

4. Correct calculation

1. Cannot determine

2. Need to calculate: A produces 0.70×0.03=0.021 defects, B produces 0.30×0.07=0.021 defects. Exactly equal! P(A|defect) = P(B|defect) = 50% ✓

3. Factory A - makes more products

4. Factory B - higher defect rate

1. Doctor correct: 100 have disease (99 test +), 99,900 don't (999 false +). So 999 false positives >> 99 true positives! P(disease | +) = 99/1098 ≈ 9% only ✓

2. Doctor wrong - 99% accurate means reliable

3. Need more information

4. Doctor wrong - 99% means 99% chance

1. Incorrect reasoning - P(A|B) and P(B|A) answer different questions and shouldn't be averaged; they're related by P(A|B)×P(B) = P(A and B) = P(B|A)×P(A), but need individual P(A) and P(B) to use them ✓

2. Average should be weighted

3. Correct approach

4. Correct but need to multiply

1. Both wrong - insurance should not use statistics

2. Critic correct - should use same probability for all

3. Critic correct - conditional probability invalid here

4. Company correct from probability standpoint - conditional probabilities reflect real risk differences; whether this justifies differential treatment is ethical/legal question beyond pure mathematics ✓

1. Events that always happen together

2. Events that have equal probability

3. Events that cannot happen at the same time ✓

4. Events that are independent

1. P(A or B) = P(A) - P(B)

2. P(A or B) = P(A) + P(B) ✓

3. P(A or B) = P(A) + P(B) - P(A and B)

4. P(A or B) = P(A) × P(B)

1. P(A or B) = P(A) + P(B)

2. P(A or B) = P(A) + P(B) - P(A and B) ✓

3. P(A or B) = P(A) - P(A and B)

4. P(A or B) = P(A) × P(B)

1. P(A) + P(B)

2. P(A) × P(B)

3. 1 (one)

4. 0 (zero) ✓

1. Events that cannot occur together

2. Events that are independent

3. Events that can occur at the same time ✓

4. Events that always occur together

1. P(A and B) - the intersection ✓

2. P(A) + P(B)

3. P(neither A nor B)

4. P(A or B)

1. Disjoint events ✓

2. Dependent events

3. Conditional events

4. Independent events

1. Because it's a mathematical rule

2. To correct for double-counting the overlap when adding P(A) and P(B) ✓

3. To make the answer smaller

4. Only for mutually exclusive events

1. It depends on the sample space

2. Yes, always

3. No - mutually exclusive events are dependent because if A occurs, P(B) becomes 0 ✓

4. Yes, but only for coin flips

1. Always

2. Only when P(A) = P(B)

3. Only when events are independent

4. Only when events are mutually exclusive ✓

1. 0.9

2. 0.7 ✓

3. 0.3

4. 1.1

1. 0.35

2. 0.15

3. 0.8 ✓

4. 0.2

1. No, because 6 is in both sets (they overlap) ✓

2. Yes, because one die can only show one number

3. Cannot determine

4. Yes, because they're different sets

1. Because of the multiplication rule

2. Only true for independent events

3. Because we subtract the overlap (or 0 for mutually exclusive), so result ≤ sum ✓

4. It's not - can be greater

1. 16/52

2. 1/2

3. 4/52

4. 8/52 = 2/13 ✓

1. 5/6

2. 1/2

3. 2/3 ✓

4. 1/3

1. 32/52 = 8/13 ✓

2. 12/52

3. 38/52

4. 26/52

1. 18/30

2. 33/30

3. 15/30

4. 25/30 = 5/6 ✓

1. 8/36 = 2/9 ✓

2. 7/36

3. 6/36 = 1/6

4. 9/36 = 1/4

1. 8/10

2. 7/10

3. 3/10

4. 5/10 ✓

1. Correct - everyone likes at least one

2. Incorrect - forgot to subtract overlap; correct is 60/100 + 40/100 - 25/100 = 75/100 = 75% ✓

3. Incorrect - should multiply

4. Correct calculation

1. Neither - cannot calculate this

2. A is correct - even and odd are mutually exclusive (no overlap), so simple addition gives 100%, which makes sense (all outcomes are even or odd) ✓

3. B is correct - must always subtract

4. Both wrong - answer should be 50%

1. Partially correct - minimum is 0.6 (when B⊆A), but maximum is 1.0 (probability can't exceed 1); also note: if mutually exclusive, P(A or B)=1.1 is impossible, so max overlap must be 0.1 to keep sum ≤1 ✓

2. Correct range

3. Wrong - must be exactly 1.1

4. Wrong - should be 0 to 1.1

1. Questionable assumption - while rare, a car could be in accident AND then stolen (or stolen then crashed), so not perfectly mutually exclusive; however, for practical insurance purposes, treating as mutually exclusive (0.07 or 7%) is reasonable approximation since overlap is negligible ✓

2. Incorrect - should multiply probabilities

3. Correct reasoning and calculation

4. Completely wrong - probabilities don't add

1. Direct count must be wrong - formula is always correct; student likely miscounted outcomes

2. Both methods wrong

3. Formula doesn't apply to this situation

4. Formula is wrong - should not subtract ✓

1. Events that happen at the same time

2. Events that are unrelated to probability

3. Events where one does NOT affect the probability of the other ✓

4. Events that cancel each other out

1. Events that always happen together

2. Events where one DOES affect the probability of the other ✓

3. Events that are impossible

4. Events with low probability

1. P(A AND B) = P(A) × P(B) ✓

2. P(A AND B) = P(A) + P(B)

3. P(A AND B) = P(A) ÷ P(B)

4. P(A AND B) = P(A) - P(B)

1. Replacing broken items

2. Selecting a different item

3. Item is thrown away after selection

4. Item is returned to the sample after selection ✓

1. Item is returned after selection

2. Item is replaced with a different one

3. Item is NOT returned after selection ✓

4. No items are selected

1. Probability of B given that A has occurred ✓

2. Probability of B or A

3. Probability of B divided by A

4. Probability of B and A together

1. P(A AND B) = P(A) + P(B|A)

2. P(A AND B) = P(A) × P(B|A) ✓

3. P(A AND B) = P(A) × P(B)

4. P(A AND B) = P(A|B) × P(B|A)

1. Because you're replacing old items with new ones

2. Because returning the item maintains the same sample composition and probabilities for each selection ✓

3. It doesn't - they're still dependent

4. Because replacement makes events more likely

1. Because it's more complicated

2. It doesn't - they're still independent

3. Because removing an item changes the sample composition, altering probabilities for subsequent selections ✓

4. Because you run out of items

1. Because three is a small number

2. They're not independent

3. Because you use the same coin

4. Because coins don't have memory - each flip has the same 50/50 probability regardless of previous results ✓

1. Scenario A (with replacement) ✓

2. Scenario B (without replacement)

3. Both equal

4. Cannot determine

1. 0.7

2. 0.1

3. 0.12 ✓

4. 0.6

1. Without replacement: fewer cards and fewer favorable outcomes for second draw reduce probability ✓

2. It's not lower - they're equal

3. With replacement is always wrong

4. It's actually higher without replacement

1. Add: 0.9 + 0.9 + 0.9 = 2.7

2. Average: 2.7 ÷ 3 = 0.9

3. Multiply: 0.9 × 0.9 × 0.9 = 0.729 ✓

4. Use complement: 1 - 0.9 = 0.1

1. 1/4 ✓

2. 3/4

3. 1/8

4. 1/2

1. 15/64

2. 5/28

3. 15/56 ✓

4. 1/4

1. 1/6

2. 1/12

3. 1/36 ✓

4. 2/6

1. 1/13 × 1/13

2. 4/52 × 3/51 = 1/221 ✓

3. 4/52 + 3/51

4. 4/52 × 4/52 = 1/169

1. 94.1% ✓

2. 98%

3. 29%

4. 294%

1. 1/20

2. 1/4

3. 5/4

4. 1/1024 ✓

1. Correct - outcomes must balance

2. Incorrect - Gambler's Fallacy; coin flips are independent, so next flip still has 50% chance for heads and 50% for tails regardless of previous results ✓

3. Correct - the law of averages applies immediately

4. Incorrect - next flip is more likely to be heads

1. Scenario A (larger sample)

2. Scenario A is very slightly higher because the ratio changes less with large samples ✓

3. Both equal - same ratio

4. Scenario B (smaller sample)

1. 99.9% - P(at least 1 works) = 1 - P(all fail) = 1 - (0.1)³ = 1 - 0.001 = 0.999 ✓

2. 90% - same as one alarm

3. 270% - add all three

4. 99% - multiply 90% three times

1. Incorrect - treated as independent (with replacement) when should be dependent; correct is (3/5) × (2/4) = 6/20 = 3/10 ✓

2. Correct calculation

3. Incorrect - should add probabilities

4. Correct method, wrong numbers

1. Meteorologist wrong - weather is always independent

2. Correlation doesn't matter

3. Independence always overestimates

4. Meteorologist correct - if rainy days are positively correlated (dependent), P(both rain) > 0.30×0.40 = 12% because P(Tue rain | Mon rain) > 40%; independence assumption underestimates when events are positively dependent ✓

1. An event that is very complicated

2. An event that happens twice

3. An event consisting of two or more simple events ✓

4. An event with multiple outcomes

1. A way to add all possible outcomes

2. If one event has m ways and another has n ways, together they have m × n ways ✓

3. The principle that you must count systematically

4. A formula for calculating probability

1. 2 outcomes

2. 4 outcomes ✓

3. 3 outcomes

4. 6 outcomes

1. 12 outcomes

2. 18 outcomes

3. 72 outcomes

4. 36 outcomes ✓

1. A table or grid

2. A pie chart

3. A tree diagram ✓

4. A bar graph

1. You do one or the other

2. You do both actions ✓

3. The outcomes must be the same

4. You have a choice

1. 8 outcomes ✓

2. 6 outcomes

3. 3 outcomes

4. 9 outcomes

1. It looks more organized

2. To avoid missing outcomes or counting duplicates, ensuring accuracy ✓

3. It's required by mathematical rules

4. Random listing works just as well

1. They're not different - it's an error

2. To make more outcomes

3. Because order matters - HT means first coin H, second T; TH means first T, second H ✓

4. Only when the coins are different colors

1. Adding is wrong because it's too simple

2. It doesn't matter which you use

3. Addition only works for simple events

4. Multiplication counts all combinations; each appetizer can be paired with each main course and each dessert ✓

1. When you want it to look pretty

2. When there are exactly two events and you want to see all combinations visually ✓

3. Always - tables are always better

4. Never - trees are always better

1. Listing is actually easier

2. FCP gives 1,024 outcomes instantly; listing all 1,024 would take hours and be error-prone

3. You get the same answer faster ✓

4. You don't need to know all outcomes

1. OR includes more favorable outcomes (at least one condition), while AND requires all conditions (fewer outcomes) ✓

2. OR is mathematically stronger

3. They have the same probability

4. AND is actually higher

1. It's not easier - just different

2. Complement always gives wrong answer

3. Instead of counting many outcomes with ≥1 head, just find P(all tails) and subtract from 1 ✓

4. Only works for coins

1. 1/6

2. 1/2

3. 1/3

4. 1/4 ✓

1. 720 codes

2. 100 codes

3. 30 codes

4. 1000 codes ✓

1. 1/6 ✓

2. 1/9

3. 1/12

4. 1/3

1. 1/8

2. 1/4

3. 1/2

4. 3/8 ✓

1. 24 outfits ✓

2. 14 outfits

3. 9 outfits

4. 48 outfits

1. 1/4

2. 1/2

3. 5/6

4. 3/4 ✓

1. No - only 26,000 passwords possible (26+10+10+10)

2. Yes - 26 × 10 × 10 × 10 = 26,000, which exceeds 50,000... wait, no: only 26,000 < 50,000, so NO ✓

3. Yes - 26,000 passwords possible (26×10×10×10)

4. No - only 2,600 passwords possible

1. Wrong - should be 6/12

2. Correct - there are exactly 6 doubles (1-1, 2-2, 3-3, 4-4, 5-5, 6-6) out of 36 outcomes ✓

3. Wrong - there are more ways to win

4. Correct but simplified wrong - should be 1/12

1. Scenario A: P = 7/8 = 87.5% is higher than Scenario B: P = 11/36 ≈ 30.6% ✓

2. Scenario B is more likely

3. Both have equal probability

4. Cannot compare different experiments

1. Incorrect - those 4 outcomes are NOT equally likely; actual sample space has 8 equally-likely outcomes (HHH, HHT, etc.), where 1H occurs 3 ways and 2H occurs 3 ways ✓

2. Correct - there are 4 outcomes

3. Correct - probabilities always divide evenly

4. Incorrect - there are only 3 outcomes

1. Correct calculation and reasoning

2. Wrong - should only count grand prize

3. Wrong - should multiply the probabilities

4. Correct reasoning (add probabilities for mutually exclusive events) and correct calculation: P = 111/1,000,000 ≈ 0.011% ✓

1. Probability that cannot be calculated

2. Probability calculated from mathematical theory

3. Probability based on actual experiments and observations ✓

4. Probability of conducting an experiment

1. Number of favorable outcomes / Total possible outcomes

2. Times event occurred / Total number of trials ✓

3. Total trials / Times event occurred

4. Sample space / Event outcomes

1. Larger numbers have higher probability

2. As trials increase, experimental probability approaches theoretical probability ✓

3. You need large numbers to calculate probability

4. Large samples always give exact theoretical probability

1. The final result of an experiment

2. A court proceeding

3. A test to see if equipment is fair

4. One repetition of an experiment ✓

1. How often you conduct experiments

2. The theoretical probability

3. How many times a specific event occurred ✓

4. The speed of conducting trials

1. Another term for experimental probability ✓

2. The speed relative to time

3. The difference between frequencies

4. Frequency of similar events

1. Yes, it updates with new data ✓

2. No, it stays constant like theoretical probability

3. Only for the first 100 trials

4. Only if the equipment is unfair

1. They should never be different

2. Because of random variation, especially with small samples ✓

3. Because theoretical probability is always wrong

4. Because experiments are always done incorrectly

1. The experiment was done wrong

2. The coin is definitely unfair

3. This is normal random variation for a small sample ✓

4. Heads has higher probability than tails

1. Larger samples always give exact theoretical results

2. It takes more time so seems more professional

3. You can stop early if you get the right answer

4. Random variations cancel out, giving results closer to theoretical probability ✓

1. 50% ✓

2. 52%

3. 48%

4. 100%

1. Only when you don't know the formula

2. Never - theoretical is always better

3. When outcomes are not equally likely or are unknown/complex ✓

4. Always - theoretical probability is outdated

1. The die appears to be fair ✓

2. The die is definitely unfair

3. More trials are needed

4. The results are suspicious because they're too perfect

1. The equipment must be perfectly fair

2. The experiment was perfect

3. It's either coincidence (with reasonable trials) or suspicious (if too few trials) ✓

4. This always happens

1. 17/100 = 17%

2. 25/17 = 1.47 = 147%

3. 8/25 = 0.32 = 32%

4. 17/25 = 0.68 = 68% ✓

1. 30/50 = 60%

2. 1/4 = 25%

3. 30/80 = 0.375 = 37.5% ✓

4. 30/100 = 30%

1. 6 items

2. Cannot determine

3. 200 items ✓

4. 150 items

1. (8+9+10)/60 = 45%

2. 3/6 = 50%

3. 33/100 = 33%

4. (11+12+10)/60 = 33/60 = 55% ✓

1. 14 times ✓

2. Cannot determine

3. 35 times

4. 0.35 times

1. 11/20 = 55%

2. (12+9+11)/3 = 32/3 ≈ 10.67

3. 12/20 = 60%

4. (12+9+11)/(20+20+20) = 32/60 ≈ 0.533 = 53.3% ✓

1. First - it matches theoretical perfectly

2. Second - larger sample is more reliable, and natural variation is realistic; first is suspiciously perfect ✓

3. Neither - both seem wrong

4. Both equally trustworthy

1. The experiment was done incorrectly

2. Likely indicates an unfair coin or systematic bias; 8% deviation over 1000 trials is substantial ✓

3. The coin is definitely weighted

4. Normal random variation - 58% is close enough to 50%

1. 85% is correct (170/200), so claim is valid

2. Need to consider: control group comparison, sample representativeness, placebo effect, natural recovery rate, and long-term effects ✓

3. Just need more patients to be sure

4. 170 is a large number, so claim is automatically valid

1. Incorrect reasoning - probability doesn't guarantee outcomes in finite trials; even with 1% chance, it's possible (though unlikely) to lose 100 times; probability is about long-term patterns over many sets of 100 tickets, not guarantees ✓

2. Need to buy exactly 100 more tickets

3. Correct - the company is lying

4. Correct - if 1 in 100 win, then 100 tickets should guarantee a win

1. Partially correct - only scientists need experimental probability

2. Correct - theoretical is always better

3. Correct - experiments always have errors

4. Incorrect - experimental probability is essential when theoretical is impossible/impractical (complex real-world systems, unknown probabilities, testing equipment fairness), validates theories, and is foundation of scientific method; both have important roles ✓

1. The actual outcome of an event

2. The number of favorable outcomes

3. A measure of how likely an event is to occur ✓

4. A type of statistics

1. From -1 to +1

2. From 0 to 1 ✓

3. From 0 to 10

4. From 0 to 100

1. The event is certain to happen

2. The event is impossible ✓

3. The event is equally likely

4. The event is very likely

1. The event is impossible

2. The event might happen

3. The event is unlikely

4. The event is certain to happen ✓

1. The space where experiments are conducted

2. The favorable outcomes

3. The set of all possible outcomes of an experiment ✓

4. The most likely outcome

1. P(E) = Favorable outcomes / Total outcomes ✓

2. P(E) = Total outcomes / Favorable outcomes

3. P(E) = Favorable outcomes × Total outcomes

4. P(E) = Total outcomes - Favorable outcomes

1. The event 'NOT E' (E does not occur) ✓

2. The same as event E

3. The opposite outcome

4. The favorable outcomes

1. It's just a mathematical rule with no meaning

2. Because one of the outcomes must occur (certainty), and 1 represents certainty ✓

3. To make calculations easier

4. Because there's always one favorable outcome

1. 0.3

2. 1.3

3. 0.7 ✓

4. Cannot determine

1. All outcomes have different probabilities

2. The most favorable outcome

3. Only two outcomes are possible

4. All outcomes have the same probability ✓

1. Gambling is always wrong

2. Past outcomes of independent events don't affect future outcomes ✓

3. Probabilities change after each event

4. The fallacy is actually correct

1. The coin looks nice

2. The coin always lands on heads

3. Both outcomes (heads and tails) are equally likely ✓

4. The coin is expensive

1. Probability describes long-term patterns, not individual outcomes ✓

2. Probability can be greater than 1 if very likely

3. Probability predicts exactly what will happen next

4. Probability changes based on past results

1. Certain

2. Unlikely (1 in 4 chance)

3. Impossible ✓

4. Equally likely

1. 1/4

2. 1/2

3. 4/6

4. 1/6 ✓

1. 5/8

2. 1/2

3. 5/10 ✓

4. 3/10

1. 1/4 ✓

2. 1/13

3. 4/52

4. 1/52

1. 1/3

2. 3/6

3. 2/3

4. 1/2 ✓

1. 0.65 ✓

2. 0.35

3. 1.35

4. 0.15

1. 3%

2. 25%

3. 50%

4. 37.5% ✓

1. Friend is correct - should be exactly 5 and 5

2. Friend is wrong - with small samples, variation from expected 50-50 is normal; doesn't prove unfairness ✓

3. The coin is definitely unfair

4. Need exactly 100 flips to determine fairness

1. Game A (coin) with 50% chance

2. Both are exactly equal - both give 50% chance to win ✓

3. Game B (die) with 50% chance

4. Cannot determine without playing

1. P(blue) = 3/5 (assuming only red and blue exist) ✓

2. P(blue) could be anything

3. P(blue) must be 2/5

4. P(blue) = 1/5

1. Friend confuses individual probability (very low: 1 in 10M) with certainty that someone among millions will win; both statements can be true ✓

2. Friend is correct - someone winning proves high probability

3. The lottery is lying about odds

4. Probability doesn't apply to lotteries

1. Could be true - luck is real

2. 1/3 is a reasonable increase

3. Need to test with experiments

4. Impossible - dice outcomes are determined by physics (roll force, angle), not charms; claim violates probability principles for fair dice ✓

1. Read, Write, Calculate, Graph, Conclude, Present

2. Question, Answer, Check, Revise, Report, Finish

3. Formulate, Plan, Collect, Analyze, Conclude, Communicate ✓

4. Observe, Measure, Record, Calculate, Graph, Done

1. Simple, Measurable, Accurate, Relevant, Testable

2. Specific, Measurable, Achievable, Relevant, Time-bound ✓

3. Scientific, Meaningful, Applicable, Rigorous, Transparent

4. Statistical, Mathematical, Analytical, Reasonable, Timely

1. The variable you measure as the outcome

2. The variable you manipulate or compare (predictor) ✓

3. The variable that stays constant

4. The variable that depends on other variables

1. 100

2. 1000

3. 10

4. 30 ✓

1. Systematic sampling

2. Convenience sampling

3. Simple random sampling ✓

4. Stratified sampling

1. A clear, specific description of how a variable will be measured or identified ✓

2. The definition found in a dictionary

3. The theoretical meaning of a concept

4. The general understanding of a term

1. Participants understand the purpose and procedures and voluntarily agree to participate ✓

2. Researchers inform participants after the study ends

3. Participants agree to anything the researcher wants

4. Only parents need to know about the study

1. It's grammatically incorrect

2. It's vague (what is 'happy'?), not measurable, no defined population or timeframe ✓

3. It's too short

4. Happiness cannot be studied

1. Observational studies are always worse

2. Experiments always have more participants

3. In experiments, researchers manipulate variables; in observational studies, they observe existing conditions ✓

4. Observational studies can prove causation

1. It's too expensive

2. There's nothing wrong with it

3. It's too easy

4. Available people often share characteristics, creating bias and unrepresentative samples ✓

1. Only professional researchers need to pilot test

2. To identify problems (confusing questions, timing issues, equipment failures) and fix them before collecting all data ✓

3. Pilot testing is unnecessary

4. To waste time before the real study

1. To make your work look bad

2. It's not important - hide weaknesses

3. To be honest and help readers interpret findings appropriately; shows critical thinking and scientific integrity ✓

4. Only failed studies have limitations

1. Memory errors, social desirability bias (reporting what sounds good), and lack of awareness can distort self-reports ✓

2. Self-reports are always accurate

3. Only surveys have this problem

4. People are always dishonest

1. Technical jargon impresses all audiences

2. One presentation style works for everyone

3. Different audiences have different knowledge levels and interests - adapting ensures understanding and relevance ✓

4. To manipulate people

1. Do phones affect all students everywhere?

2. What do students think about phones?

3. Are phones good or bad?

4. Is there a relationship between daily smartphone usage (hours) and self-reported stress levels (1-10 scale) among Year 8 students? ✓

1. Survey only your friends

2. Only measure tall students

3. Use the entire class (census) since it's small and accessible ✓

4. Randomly select 5 students

1. Experiment: randomly assign participants to music/silence conditions and measure puzzle completion time ✓

2. Review existing research only

3. Survey asking if people think music helps

4. Observe people naturally and record if music is present

1. Skip analysis and jump to conclusions

2. Only calculate the mean

3. Just draw a graph

4. Calculate mean, median, SD for both variables; create scatter plot; identify correlation pattern; look for outliers ✓

1. The 1-percentage-point difference is minimal and could be random variation; need larger sample or more tests to determine if meaningful ✓

2. Music has no effect whatsoever

3. Everyone should study with music

4. Music dramatically improves scores!

1. Make it as short as possible

2. Only report findings that support your hypothesis

3. Use as much jargon as possible

4. Answer research question, provide evidence, acknowledge limitations, discuss implications ✓

1. The sample size is the only problem

2. Invalid: tiny sample (n=5), biased (friends only), overgeneralization (5 people → all teenagers), correlation ≠ causation ✓

3. The conclusion is valid - correlation was found

4. Everything is fine except the wording

1. Both are equally good

2. Design B is better: random assignment allows causal inference, controls confounds, directly tests effect; A only shows correlation ✓

3. Design A is better because surveys are easier

4. Neither can answer the question

1. Low credibility: tiny sample, correlation not causation, conflict of interest (industry funding), probable data dredging (testing many foods), needs replication ✓

2. Funding source doesn't matter

3. Sample size is the only issue

4. Highly credible - they found significance

1. Good: clear SMART question, adequate random sample, objective measurements, appropriate analysis and visualization, appropriate conclusion (correlation not causation), limitations acknowledged ✓

2. Poor - too simple a question

3. Mediocre - needs experimental design

4. Bad - should have claimed causation

1. Claim is valid - they measured improvement

2. Perfect study design

3. Only needs larger sample

4. Weak: tiny sample (n=10), very short duration (1 week), no control group (improvement might be practice effect/maturation), needs: larger sample, control group, longer duration, independent testing ✓

1. Read, Write, Calculate, Draw

2. Mean, Median, Mode, Range

3. Describe, Compare Central Tendency, Compare Spread, Conclude ✓

4. Collect, Organize, Analyze, Present

1. Only the mean

2. Mean, median, and mode ✓

3. Only the median

4. None - central tendency isn't important

1. Dataset A has a larger mean

2. Dataset A is more consistent/less variable ✓

3. Dataset A has more data points

4. Dataset A has higher values

1. Bimodal

2. Heterogeneous

3. Skewed

4. Homogeneous ✓

1. Larger sample size

2. Higher median

3. Greater variability in the middle 50% of data ✓

4. More outliers

1. They must be measured using the same method and units ✓

2. They must have the same number of data points

3. They must have the same mean

4. They must be collected on the same day

1. One or both datasets are skewed or have different distributions ✓

2. The datasets cannot be compared

3. The datasets are identical

4. There's a calculation error

1. There's no reason - means are sufficient

2. Means don't show variability - datasets can have same mean but very different spreads and distributions ✓

3. Means are always inaccurate

4. Means are too difficult to calculate

1. Dataset X is better

2. Dataset Y has more data points

3. Both have same average but Dataset X is more consistent (lower variability), Dataset Y more variable ✓

4. Both datasets are identical since means are equal

1. Sample size doesn't affect comparisons

2. Small samples are always wrong

3. Large samples are always biased

4. The small sample (n=10) has much higher variability and less reliability, making comparison less stable ✓

1. The test was too easy

2. There are two distinct groups (e.g., well-prepared vs. unprepared students) vs. one homogeneous group ✓

3. The data is incorrect

4. All students performed equally

1. Independent samples comparison

2. Unrelated comparison

3. Paired comparison (dependent/matched samples) ✓

4. Cross-sectional comparison

1. Dataset A is both higher on average AND more consistent - clearly superior on both measures ✓

2. Dataset A is worse

3. The datasets are equal

4. Dataset B is more reliable

1. Temperature can't be averaged

2. Averages are always accurate for temperature

3. Two cities could have same average but very different climates - one stable year-round, one with extreme seasons ✓

4. Cities are too different to compare

1. Both classes performed identically

2. Class B is more consistent

3. Class A has lower typical performance

4. Class A: left-skewed (or symmetric), more consistent. Class B: right-skewed, more variable ✓

1. Athlete 2 because they're faster on average

2. Both are equally reliable

3. Athlete 1 because times are more consistent (lower variability) ✓

4. Cannot determine from this data