Overall AI SL mastery
0 / 582 skills mastered
0 %
1
Number and Algebra
0 / 90
Easy — Foundations
1 Writing numbers in standard form Express a number as \(a\times10^{k}\) with \(1\le a<10\). Application A galaxy \(9{,}500{,}000{,}000{,}000{,}000{,}000\) km away is written \(9.5\times10^{18}\) km, and a \(0.000000275\) m wavelength becomes \(2.75\times10^{-7}\) m. The shorthand keeps huge and tiny figures readable.Easy 2 Index laws: zero, negative, fractional Use \(a^{0}=1,\ a^{-n}=\tfrac1{a^{n}},\ a^{1/n}=\sqrt[n]a\). Application Writing \(\sqrt{x}\) as \(x^{1/2}\) lets you later differentiate it as \(\tfrac12x^{-1/2}\). A bank’s \(1.03^{12}\) annual growth factor relies on the same rules.Easy 3 Rounding (d.p., s.f., integer) Round to decimal places, significant figures or integers. Application A shop totals three \(\$4.99\) items as \(\$14.97\) to the cent, while a population of \(8{,}012{,}433\) rounds to \(8.0\) million for a headline. Matching precision to the context avoids false accuracy.Easy 4 Arithmetic in standard form Combine \(a\times10^{k}\) values. Application Multiplying light speed \(3\times10^{8}\) m/s by a \(4.6\times10^{2}\) s travel time gives \(1.38\times10^{11}\) m. Physicists combine such figures routinely.Easy 5 Logarithm basics (log, ln) Use \(\log\) (base 10) and \(\ln\). Application A magnitude-6 earthquake on the \(\log\) scale releases ten times the energy of magnitude 5. The natural log appears in continuous growth, like \(\ln2\) for a doubling time.Easy 6 Percentage error Use \(\varepsilon=\left|\tfrac{v_A-v_E}{v_E}\right|\times100\%\). Application Estimating a \(2.0\) m wall as \(2.1\) m is a 5% error, flagged before you trust it. Labs quote this whenever a measured value is compared with the true one.Easy 7 Upper and lower bounds Give the interval a rounded value lies in. Application A length stated as 5 cm to the nearest cm could be anywhere from \(4.5\) to \(5.5\) cm. Builders use these bounds so parts still fit at the extremes.Easy 8 Estimation to 1 s.f. Round each value to one significant figure. Application Checking \(312\times19\approx300\times20=6000\) catches a calculator slip in seconds. Shoppers use the same trick to sanity-check a trolley total.Easy Hard — Structured IB Questions
18 Trig formula, round, percentage error Evaluate a trig formula and find the error. Application Computing a roof brace as \(8\sin52^{\circ}=6.3\) m, then rounding to 6 m, gives a 4.8% error. Builders judge whether that error is acceptable.Hard 19 Reverse index equations Work back to an unknown index. Application If a doubling experiment gives \(2^{n}=512\), working back finds \(n=9\) generations. Biologists count cell-division rounds this way.Hard 20 Standard form with bounds Combine standard form and bounds. Application A star’s \(4.0\times10^{16}\) m distance, to 2 s.f., lies between \(3.95\) and \(4.05\times10^{16}\) m. Astronomers state both ends to be honest about precision.Hard 21 Exponential model Use \(y=ab^{x}\). Application A \(\$2000\) investment under \(y=2000(1.06)^{x}\) reaches about \(\$2676\) after 5 years. The same curve models bacteria or new followers.Hard 22 Reverse percentage error Work back from an error. Application If a reading is within 2% of 50 kg, the true mass lies between 49 and 51 kg. Quality control works backwards from the allowed error.Hard 23 Bounds of a quotient Propagate bounds through division. Application A 100 km trip to the nearest km in \(2.0\) h to the nearest \(0.1\) h gives a speed between \(48.5\) and \(51.5\) km/h. Speed estimates allow for this spread.Hard 24 Linear systems on the GDC Solve simultaneous equations. Application Solving \(2x+3y=12,\ x-y=1\) finds where two pricing plans cost the same. Businesses compare suppliers this way in seconds.Hard 25 Volume, s.f., percentage error Combine volume, rounding and error. Application A 5 mm ball bearing has volume \(\approx524\) mm\(^3\); rounding the radius to 1 s.f. shifts it noticeably. Manufacturers track that error against tolerance.Hard Very Hard — Integrated AI SL Challenge
Hard — Structured IB Questions
Very Hard — Integrated AI SL Challenge
Easy — Foundations
1 \(n\)th term of an arithmetic sequence Use \(u_n=u_1+(n-1)d\). Application A theatre with 20 seats in row 1 and 2 more each row has \(20+2(n-1)\) seats in row \(n\). This jumps straight to row 15 without listing every row.Easy 2 \(n\)th term of a geometric sequence Use \(u_n=u_1r^{\,n-1}\). Application A \(\$1000\) investment growing 5% a year follows \(1000(1.05)^{\,n-1}\). The same formula models a population multiplying each generation.Easy 3 Arithmetic vs geometric Distinguish a constant difference from a constant ratio. Application Saving \(\$50\) more each month is arithmetic; earning 5% more each year is geometric. Spotting which fits picks the right model.Easy 4 Sum of an arithmetic series Use \(S_n=\tfrac n2(2u_1+(n-1)d)\). Application Stacking logs in rows of 1, 2, 3, … up to 20 totals 210 logs. Totalling steady increases is instant with the formula.Easy 5 Sum of a geometric series Use \(S_n=\tfrac{u_1(r^{n}-1)}{r-1}\). Application Saving \(\$100\) and growing the deposit 10% each month totals quickly over a year. Loan balances use the same sum.Easy 6 Evaluating sigma notation Compute \(\sum u_r\). Application \(\sum_{r=1}^{10}(3r+1)\) compactly adds ten terms to 175. Spreadsheets and exams both use sigma to compress sums.Easy 7 Reverse: \(d\) from two terms Solve for \(d\). Application If a runner’s week-3 distance is 12 km and week-7 is 20 km, the weekly increase is 2 km. Coaches recover the training step this way.Easy 8 Arithmetic word problem Model a real situation with an AP. Application A taxi charging \(\$3\) plus \(\$2\)/km gives fares 5, 7, 9 for 1, 2, 3 km. Modelling it as an AP predicts any fare.Easy Hard — Structured IB Questions
Very Hard — Integrated AI SL Challenge
25 Algebraic terms of a GP Solve a geometric sequence algebraically. Application Terms \(x,\ x+4,\ 2x+2\) being geometric forces a quadratic in \(x\). Solving it pins down the unknown.Very Hard 26 Algebraic terms of an AP Solve an arithmetic sequence algebraically. Application Terms \(2x,\ 3x+1,\ 5x-1\) being arithmetic give an equation for \(x\). This tests reverse reasoning with unknowns.Very Hard 27 Geometric revenue model Model revenue with a geometric sequence. Application A service adding 8% more revenue monthly follows a geometric series. Forecasters sum it to project the year.Very Hard 28 Comparing strategies (sigma, inequalities) Compare using sums. Application Two saving schemes written in sigma form are compared by inequality to find the better one. Decisions rest on the totals.Very Hard 29 Salary system: "show that", simultaneous Prove and solve. Application Proving two salary offers meet in year 8 needs simultaneous equations from the sequences. Such "show that" steps appear in exams.Very Hard 30 Theatre seating model Sum the rows; solve. Application A fan-shaped theatre with 18 seats in row 1 and 3 more each row seats hundreds; the sum finds capacity. Designers solve for the row count too.Very Hard 31 Shared terms of an AP and GP Find common terms. Application A number in both a "+4" sequence and a "×2" sequence is found by matching terms. These overlap problems are a known challenge.Very Hard 32 Streaming growth, sigma, capacity Model growth with limits. Application A platform’s subscriber growth, summed with sigma, is checked against a server-capacity ceiling. Engineers model both growth and the limit.Very Hard
2
Functions
0 / 86
Easy — Foundations
1 Linear model: read, evaluate, solve Use \(y=mx+c\). Application A phone plan costing \(\$20\) plus \(\$0.10\)/min is \(y=20+0.1x\), so 200 minutes costs \(\$40\). Reading \(m\) and \(c\) gives the rate and fixed fee at a glance.Easy 2 Direct variation basics Use \(y=kx\). Application If 3 kg of apples costs \(\$6\), then \(y=2x\) prices any weight. Recipes and currency conversion scale the same proportional way.Easy 3 Quadratic model: evaluate, symmetry Use \(ax^{2}+bx+c\); axis \(x=-\tfrac b{2a}\). Application A ball’s height \(h=-5t^{2}+20t\) peaks at \(t=2\) s by symmetry. Engineers use the axis to find a parabola’s maximum quickly.Easy 4 Exponential model: initial value, factor Use \(y=ab^{x}\). Application A \(\$5000\) car under \(y=5000(0.85)^{x}\) keeps 85% of its value each year. The factor 0.85 captures the yearly drop.Easy 5 Inverse variation Use \(y=\tfrac kx\). Application At a fixed distance, doubling a journey’s speed halves its time via \(t=\tfrac dv\). Gas pressure and volume relate the same inverse way.Easy 6 Piecewise model: evaluate, solve Pick the right piece. Application A tax system charging 0% under \(\$10{,}000\) and 20% above needs the right piece for each income. Parking and postage costs switch rules similarly.Easy 7 Sinusoidal model: amplitude, midline Read \(a\) and \(d\). Application A tide swinging between 1 m and 5 m has midline 3 m and amplitude 2 m. The same reading describes daily temperature or daylight curves.Easy 8 Choosing a model from a table Identify the model family. Application Constant differences signal a linear fit; a constant ratio signals exponential. Picking correctly stops you forcing the wrong curve onto data.Easy Hard — Structured IB Questions
17 Comparing two linear models Compare and decide. Application Two gyms, one \(\$40\) joining plus \(\$20\)/month and one \(\$10\)/month, cross after some months. Members pick the cheaper plan for their usage.Hard 18 Quadratic revenue optimisation Maximise revenue. Application Revenue \(R=p(100-2p)\) for a price \(p\) peaks at \(p=\$25\). Shops set the price that maximises takings.Hard 19 Exponential from two points, percentage error Fit and assess. Application Fitting \(y=ab^{x}\) through two population counts, then checking a prediction, reveals the model’s error. Demographers report how trustworthy it is.Hard 20 Sinusoidal from max and min Build from the extremes. Application A tide reaching 5.2 m high and 0.8 m low fixes the amplitude and midline. Harbour masters build the model from two readings.Hard 21 Cubic regression and optimisation Fit and optimise. Application Fitting a cubic to yield-versus-fertiliser data finds the dose giving the best yield. Farmers optimise input from the curve.Hard 22 Direct variation with a cube Use \(y=kx^{3}\). Application A sphere’s mass varies with the cube of its radius, so doubling the radius gives eight times the mass. Foundries scale castings this way.Hard 23 Piecewise tax model, reverse Solve a tax-band model. Application Given a \(\$4300\) tax bill across two bands, you back-solve for the income. Accountants recover earnings from tax paid.Hard 24 Comparing models, judging fit Compare via \(R^{2}\). Application Choosing between a linear and an exponential fit, the higher \(R^{2}\) wins. Data scientists pick the model explaining more variation.Hard Very Hard — Integrated AI SL Challenge
Easy — Foundations
1 Gradient and intercepts of \(y=mx+c\) Read \(m\) and \(c\). Application A taxi graph \(y=2x+3\) shows a \(\$3\) flag fall and \(\$2\)/km rate. Reading \(m\) and \(c\) gives the fare structure at a glance.Easy 2 Gradient from two points Use \(m=\tfrac{\Delta y}{\Delta x}\). Application A road rising 50 m over 1000 m has gradient 0.05, a 5% slope. Civil engineers grade ramps and roads this way.Easy 3 Midpoint of a segment Use the midpoint formula. Application The midpoint of homes at \((2,3)\) and \((8,7)\) is \((5,5)\), a fair meeting spot. Planners place a shared facility at the midpoint.Easy 4 Distance between two points Use \(d=\sqrt{\Delta x^{2}+\Delta y^{2}}\). Application Two towns at \((1,2)\) and \((4,6)\) are 5 units apart on a scaled map. GPS uses the same formula for straight-line distance.Easy 5 Equation from gradient and point Use \(y-y_1=m(x-x_1)\). Application A trend rising \(\$2\)/day through \((0,50)\) gives \(y=2x+50\). Forecasters build a line from one point and a known rate.Easy 6 Converting between line forms Rearrange \(ax+by+d=0\). Application Rewriting \(2x+4y-8=0\) as \(y=-\tfrac12x+2\) reveals the slope. Switching forms makes a graph or comparison easier.Easy 7 Parallel/perpendicular gradients Use \(m_1=m_2,\ m_1m_2=-1\). Application A path parallel to \(y=3x\) keeps slope 3; a perpendicular has slope \(-\tfrac13\). Designers lay out parallel streets and right-angle junctions.Easy 8 Reading a line from a graph Read \(m,c\) from a graph. Application Reading a phone-bill graph gives the monthly fee and per-text cost. Consumers extract the deal directly from the plot.Easy Hard — Structured IB Questions
Very Hard — Integrated AI SL Challenge
25 Linear growth model, suitability Model and critique. Application A straight-line population fit works short term but overpredicts long term. Demographers judge when a linear model stops being suitable.Very Hard 26 Triangle from intersecting lines Find the vertices and area. Application Three roads meeting pairwise enclose a triangular block whose area you compute. Planners size the parcel from the line equations.Very Hard 27 Perpendicular bisector, point at distance Combine the two. Application Locating a transmitter equidistant from two towns, a set distance up a bisector, blends both skills. Engineers site towers this way.Very Hard 28 Lattice points on a line Find the integer points. Application Counting whole-number points on \(y=\tfrac23x+1\) finds valid grid positions. Games and tiling use these integer solutions.Very Hard 29 Distance to find a point, parallel line Combine distance and a parallel line. Application Placing a point a fixed distance away, then drawing a parallel guide line, sets up a design. Drafters use both steps together.Very Hard 30 Carpenter model with tax Model costs with tax. Application A quote of \(\$50\) call-out plus \(\$40\)/hour, then 10% tax, is a line scaled by 1.1. Tradespeople build taxed quotes this way.Very Hard 31 Exact area as a fraction Find an exact fractional area. Application A grid triangle may have area exactly \(\tfrac{15}{2}\) units, not a rounded decimal. Exam answers keep the exact fraction.Very Hard 32 Quadrilateral: parallel lines, area Analyse a composite shape. Application A trapezium plot bounded by two parallel roads has an area found from its lines. Surveyors compute composite parcels this way.Very Hard More Functions & Their Graphs Hard — Structured IB Questions
Very Hard — Integrated AI SL Challenge
3
Geometry and Trigonometry
0 / 122
Easy — Foundations
1 Distance between two points Use \(d=\sqrt{\Delta x^{2}+\Delta y^{2}}\). Application Two trees at \((1,2)\) and \((5,5)\) on a garden plan are 5 m apart. Landscapers measure straight-line spacing this way.Easy 2 Midpoint of a segment Use the midpoint formula. Application The midpoint of a 12 m fence from \((0,0)\) to \((12,0)\) is \((6,0)\), where a gate goes. Builders centre features with it.Easy 3 Gradient of a line Use \(m=\tfrac{\Delta y}{\Delta x}\). Application A wheelchair ramp rising 0.5 m over 6 m has gradient \(\tfrac1{12}\), the legal limit. Architects check accessibility this way.Easy 4 Equation from gradient and point Use \(y-y_1=m(x-x_1)\). Application A path of slope 2 through \((1,3)\) is \(y=2x+1\). Designers draw a guide line from one point and a direction.Easy 5 Arc length in degrees Use \(\ell=\tfrac\theta{360}2\pi r\). Application A 90\(^\circ\) slice of a 10 cm-radius pizza has a 15.7 cm crust edge. Bakers and designers measure curved edges this way.Easy 6 Sector area in degrees Use \(A=\tfrac\theta{360}\pi r^{2}\). Application A 60\(^\circ\) sector of a 12 m sprinkler covers about 75 m\(^2\). Gardeners size watering zones with it.Easy 7 Perpendicular gradient and line Use \(m_1m_2=-1\). Application A path crossing a road \(y=2x\) at a right angle has slope \(-\tfrac12\). Planners set perpendicular junctions this way.Easy 8 Equation through two points Find the line. Application A pipe through \((0,1)\) and \((4,9)\) follows \(y=2x+1\). Engineers route a straight line through two fixed points.Easy Hard — Structured IB Questions
Very Hard — Integrated AI SL Challenge
Three-Dimensional Geometry Easy — Foundations
1 Distance in 3D Use \(\sqrt{\Delta x^{2}+\Delta y^{2}+\Delta z^{2}}\). Application A drone at \((0,0,0)\) and a target at \((3,4,12)\) are 13 m apart. Pilots compute straight-line range in space this way.Easy 2 Midpoint in 3D Average the coordinates. Application The midpoint of a beam from \((0,0,0)\) to \((4,2,6)\) is \((2,1,3)\), where a support goes. Engineers centre 3D members.Easy 3 Cuboid volume and surface area Use \(V=abc\). Application A 2 m by 1 m by 0.5 m crate holds 1 m\(^3\) and needs 7 m\(^2\) of card. Packers size boxes from both.Easy 4 Volume of a sphere Use \(V=\tfrac43\pi r^{3}\). Application A football of radius 11 cm holds about 5575 cm\(^3\) of air. Manufacturers spec capacity from the radius.Easy 5 Surface area of a cylinder Use \(A=2\pi rh+2\pi r^{2}\). Application A can 12 cm tall with a 3 cm radius needs about 283 cm\(^2\) of metal. Drinks firms cost the material this way.Easy 6 Volume of a cone Use \(V=\tfrac13\pi r^{2}h\). Application An ice-cream cone 4 cm wide and 10 cm deep holds about 42 cm\(^3\). Makers size servings from it.Easy 7 Volume of a pyramid Use \(V=\tfrac13(\text{base})h\). Application A square pyramid roof of base 6 m and height 4 m encloses 48 m\(^3\). Builders compute attic space.Easy 8 Reverse: sphere surface area Solve \(4\pi r^{2}=A\) for \(r\). Application A ball needing 200 cm\(^2\) of leather has radius about 4 cm. Designers work back from a material target.Easy Hard — Structured IB Questions
Very Hard — Integrated AI SL Challenge
25 Similar spheres: mass, area ratio Use cube and square ratios. Application A ball twice the radius has 4 times the surface and 8 times the mass. Manufacturers scale products by these ratios.Very Hard 26 Frustum: volume, surface, diagonal Carry out a full frustum analysis. Application A lampshade frustum is specified by its volume, surface and slant diagonal. Makers cut and cost it.Very Hard 27 Cuboid with surds: diagonal, angle Carry out exact 3D work. Application A cuboid with sides \(\sqrt2,\ \sqrt3,\ \sqrt6\) m has an exact diagonal and angle. Exact work avoids rounding drift.Very Hard 28 Cone fill: melting sphere Equate the volumes. Application A melted ice sphere poured into a cone fills it to a height you can find. Volume conservation gives the level.Very Hard 29 Packing: spheres vs cylinders Compare the packing. Application Comparing how balls and tins pack a crate shows which wastes less space. Logistics firms choose the better packing.Very Hard 30 Pool prism: capacity, draining Combine volume and rate. Application A sloping-floor pool has a capacity and a draining time from its volume and flow rate. Operators schedule maintenance.Very Hard 31 Spherical cap (given formula) Apply a cap formula. Application The volume of liquid in a tilted tank uses the spherical-cap formula. Engineers gauge partial fills.Very Hard 32 Dome + cuboid: paint, cost, point Combine area, cost and geometry. Application A domed building’s paintable area, cost and a key 3D coordinate combine several skills. Contractors quote the job.Very Hard Hard — Structured IB Questions
17 Layered elevation across a gap Combine sight lines. Application Sighting a peak across a valley from two stations finds its height. Surveyors handle inaccessible bases.Hard 18 Two-leg bearings journey Combine bearings. Application A plane flying 200 km then 150 km on new bearings ends a computable distance from base. Pilots find the direct return.Hard 19 Reverse area, percentage error Combine area and accuracy. Application Recovering an angle from a measured area, then assessing rounding, gives the error. Surveyors report reliability.Hard 20 Ratio point, cosine rule, areas Combine the skills. Application A point dividing a side in a ratio, with the cosine rule, splits a triangle into known areas. Designers partition shapes.Hard 21 Area to a quadratic in a side Form a quadratic. Application Setting a triangle’s area to a target produces a quadratic in the side length. Solving it sizes the shape.Hard 22 3D prism: length, angle, area Find 3D measures. Application A wedge ramp’s longest edge, slope angle and face area come from 3D trig. Builders set it out.Hard 23 Structural model: angle, length Apply trig in a structure. Application A bracing strut’s length and angle in a frame are found by trigonometry. Engineers size the member.Hard 24 Mast and bearings combined Combine a mast and bearings. Application A mast viewed on a bearing, with an elevation, gives its height and position. Surveyors fix both.Hard Very Hard — Integrated AI SL Challenge
Hard — Structured IB Questions
Very Hard — Integrated AI SL Challenge
4
Probability and Statistics
0 / 220
Easy — Foundations
1 Identifying sampling methods Classify random, systematic, stratified and other methods. Application Polling every 10th shopper is systematic, while drawing names from a hat is random. Choosing the right method keeps a survey representative.Easy 2 Mean, median, mode of a list Compute the three averages. Application For salaries 30, 32, 35, 35, 90 (in thousands), the median 35 describes the typical worker better than the mean 44. Each average suits a different question.Easy 3 Range; discrete vs continuous Find the range; classify the data. Application Shoe sizes are discrete while heights are continuous, and a 12 cm height range shows spread. Recognising the type picks the right chart.Easy 4 Mean from a frequency table Use \(\bar x=\tfrac{\sum fx}{\sum f}\). Application If 10 families have 1 child, 8 have 2 and 2 have 3, the mean is 1.6 children. Census data is summarised this way.Easy 5 Reading a box plot Read the five-number summary. Application A box plot of exam marks shows the median, quartiles and any outliers at a glance. Teachers compare classes from it.Easy 6 Drawing a box plot Build from a summary. Application From a five-number summary of race times, you draw a box plot to show the spread. Coaches present results clearly.Easy 7 Cumulative frequency: median Read the median. Application Reading the 50% point on a cumulative-frequency curve gives the median wage. Economists locate it directly.Easy 8 Modal class and mid-interval Find the modal class. Application The most common height band, 160–170 cm, is the modal class with a 165 cm midpoint. Tailors stock for it.Easy Hard — Structured IB Questions
17 CF curve to box plot, outliers, percentage Combine the skills. Application Turning a cumulative-frequency curve into a box plot, then flagging outliers, summarises exam data fully. Examiners present the whole picture.Hard 18 Table from CF, then mean, sd, percentage error Combine the skills. Application Recovering class frequencies from a CF table feeds a mean, sd and a rounding error. Analysts chain the steps.Hard 19 Combined and weighted means Use \(\sum x=n\bar x\). Application Merging two classes of 20 and 30 students weights their means correctly. Schools report a fair combined average.Hard 20 Reverse linear transformation Work back from a transformation. Application Given a transformed mean and sd, you recover the original scale. Analysts undo a rescaling.Hard 21 Outlier from a frequency table Find an outlier. Application A frequency table can hide an outlier the IQR test reveals. Quality teams catch faulty readings.Hard 22 Histogram interpretation, median Read a histogram. Application Reading a skewed histogram locates the median by equal area. Statisticians describe the shape.Hard 23 Comparing two box plots Contrast distributions. Application Two box plots of rival products’ lifetimes show which is more reliable. Buyers compare spread and centre.Hard 24 Missing frequency, best measure Find a frequency; choose a measure. Application With a gap in the data, you find the frequency and pick the fairest average. Analysts justify their choice.Hard Very Hard — Integrated AI SL Challenge
25 Unknown from grouped mean Solve for an unknown. Application A grouped-mean condition solves for a hidden class frequency. Census analysts fill data gaps.Very Hard 26 Transformation and standardisation Standardise data. Application Standardising heights to z-scores compares people across different groups. Researchers rank fairly.Very Hard 27 Portfolio percentage change and volatility Combine percentage change and sd. Application A stock’s mean return and its standard deviation measure reward and risk together. Investors weigh both.Very Hard 28 Mean puzzle and range Solve a mean-and-range puzzle. Application Given a mean and range, you reconstruct possible data values. Puzzles and audits use this reasoning.Very Hard 29 Histogram to statistics, percentage error Combine the skills. Application Estimating the mean from a histogram, then comparing to the true value, gives the error. Analysts report accuracy.Very Hard 30 Box plot algebra, outlier bound Solve algebraically. Application Algebra on quartiles finds the exact outlier threshold. Quality control sets the cut-off.Very Hard 31 Comparing consistency (full stats) Compare via full statistics. Application Comparing two machines’ full statistics shows which produces more uniform parts. Engineers pick the steadier one.Very Hard 32 Grouped data \(p=kq\), mean, sd, percentage error Solve with a constraint. Application A constraint linking two frequencies, with mean, sd and error, is a full data challenge. Exam capstones test the lot.Very Hard Hard — Structured IB Questions
16 Scatter, line, prediction, reliability Combine the skills. Application A full analysis plots data, fits a line, predicts and judges reliability. Researchers present the whole case.Hard 17 Comparing \(r\) and \(r_s\) Contrast the measures. Application Comparing Pearson and Spearman shows whether a link is linear or just monotonic. Analysts pick the right measure.Hard 18 Outlier effect on \(r\) and line Analyse an outlier. Application Removing an outlier can change both \(r\) and the regression line. Analysts report the impact.Hard 19 Regression and percentage error Assess accuracy. Application A prediction compared to the actual value gives the percentage error. Forecasters report accuracy.Hard 20 Revenue model, profit, break-even Model business data. Application A revenue regression, minus costs, finds the break-even sales. Businesses plan from it.Hard 21 Robustness of Spearman’s \(r_s\) Discuss robustness. Application Spearman’s \(r_s\) resists outliers better than Pearson’s \(r\). Analysts prefer it for skewed data.Hard 22 Mean point on the line Show \((\bar x,\bar y)\) lies on it. Application The regression line always passes through the mean point. Analysts use this as a check.Hard 23 BMI model and formula link Link a model to a formula. Application A regression of weight on height links to the BMI formula. Health studies connect data to theory.Hard Very Hard — Integrated AI SL Challenge
24 Two regression lines: intersection Find where they meet. Application The \(y\)-on-\(x\) and \(x\)-on-\(y\) lines cross at the mean point. Statisticians interpret the crossing.Very Hard 25 \(r\) vs \(r_s\) with full reasoning Contrast and justify. Application A full comparison of \(r\) and \(r_s\) justifies which describes the data. Analysts argue the choice.Very Hard 26 Tip model, percentage error, back-calculation Combine modelling and accuracy. Application A tip-versus-bill regression predicts tips, then back-checks against data. Restaurants model gratuities.Very Hard 27 Petrol model, scientific notation, percentage error Combine the skills. Application A consumption model in scientific notation predicts fuel use with a stated error. Engineers report both.Very Hard 28 Profit target, least clients Solve for a threshold. Application A revenue regression finds the fewest clients to hit a profit target. Firms set sales goals.Very Hard 29 Energy model, extrapolation, percentage error Combine and critique. Application An energy-use model extrapolated, with its error, warns of an unreliable forecast. Planners judge the risk.Very Hard 30 Capstone: \(r,r_s\), line, judgement Synthesise everything. Application A capstone combines \(r\), \(r_s\), a line and a reasoned judgement. It mirrors a full IA analysis.Very Hard Hard — Structured IB Questions
Very Hard — Integrated AI SL Challenge
Probability Distributions Hard — Structured IB Questions
17 Algebraic die, \(E(X)\), two-roll sum Combine the skills. Application A die with algebraic face values has an expected value and a two-roll sum distribution. Designers tune it.Hard 18 Raffle, arithmetic prize, fair game Use expectation. Application A raffle with prizes in arithmetic progression is made fair by its expected value. Organisers set the ticket price.Hard 19 Target, integer constraint, fair game Solve with a constraint. Application A prize that must be a whole number constrains a fair-game solution. Designers round sensibly.Hard 20 Repeated scores, \(E(X)\), two-round Combine the skills. Application A two-round game’s expected total comes from combining single-round expectations. Designers balance it.Hard 21 Quadratic in \(k\), reject a root Solve and select. Application A distribution condition gives a quadratic in \(k\), one root invalid. Modellers select the valid value.Hard 22 Sum of two unlike dice Build the distribution. Application Two differently-numbered dice give a sum distribution from a grid. Game designers explore it.Hard 23 Real data, ratio, estimate Build and estimate. Application Observed ratios build a distribution used to estimate a population figure. Analysts scale up.Hard Very Hard — Integrated AI SL Challenge
Easy — Foundations
1 Conditions for a binomial Fixed \(n\), constant \(p\), independent trials. Application Flipping a fair coin 20 times fits a binomial: fixed trials, constant chance, independent. Modellers check this first.Easy 2 \(n,p\) and \(E(X)=np,\ \operatorname{Var}=np(1-p)\) Identify and summarise. Application For 50 free throws at 80%, you expect 40 successes with a known variance. Coaches predict performance.Easy 3 \(E(X)=np\), \(\sigma=\sqrt{np(1-p)}\) Apply in context. Application A factory making 200 items with 3% defects expects 6 faulty, plus a spread. Managers plan inspections.Easy 4 \(P(X=x)\) by formula Use \(\binom nx p^{x}(1-p)^{n-x}\). Application The chance of exactly 3 sixes in 10 rolls uses the binomial formula. Players compute precise odds.Easy 5 \(P(X=x)\) for two values, \(P(X=0)\) Compute several. Application The chance of zero defects in a batch is \((0.97)^{n}\). Quality teams quote it.Easy 6 Cumulative \(P(X\le x)\) Add probabilities. Application The chance of at most 2 failures in 20 tests sums three terms. Engineers assess reliability.Easy 7 Complement \(P(X\ge1)=1-P(X=0)\) Use the complement. Application The chance of at least one win in 10 tries is \(1-P(X=0)\). Players gauge their odds.Easy 8 \(E(X)\), variance, sd together Compute all three. Application A binomial’s mean, variance and sd summarise an exam-pass model. Schools predict outcomes.Easy Medium — Applications
9 \(E(X)\) with exact/cumulative probabilities Combine the skills. Application Combining the mean with exact and cumulative chances analyses a sales target. Managers plan.Medium 10 Interval \(P(a\le X\le b)\), \(P(X>a)\) Compute intervals. Application The chance of between 4 and 6 successes in 10 trials is an interval probability. Analysts use it.Medium 11 Selecting pdf, cdf, complement Choose the method. Application Choosing the right tool for "at least", "exactly" or "at most" speeds a solution. Students pick efficiently.Medium 12 Distribution and \(P(X<a)\) State and compute. Application Stating \(X\sim B(15,0.4)\) and finding \(P(X<5)\) models a survey. Researchers compute it.Medium 13 \(p\) from a sub-experiment Find \(p\), then apply. Application A spinner defines a success chance \(p\) feeding a binomial of repeated spins. Designers chain them.Medium 14 \(E(X)\) and a small cumulative Combine the skills. Application The mean plus a tail probability flags a rare event. Risk teams watch for it.Medium 15 Mean, sd, interval probability Combine the skills. Application Combining mean, sd and an interval probability profiles a process. Engineers monitor it.Medium 16 Exact, cumulative, most-likely Find the mode. Application Finding the most likely number of successes uses the peak of the distribution. Planners target it.Medium Hard — Structured IB Questions
17 \(E(X)\) and cumulative intervals Combine the skills. Application Mean plus cumulative intervals analyses a quality-control sample. Managers set limits.Hard 18 Conditional \(P(A\mid B)\) with cumulative Condition within a binomial. Application A conditional binomial probability refines a reliability estimate. Engineers use it.Hard 19 Forming \(p\), then expectation Find \(p\); compute. Application Deriving \(p\) from a described event, then finding the mean, models a campaign. Marketers plan.Hard 20 Mean, sd, interval probability Combine the skills. Application A full profile of mean, sd and interval probability describes a production run. Engineers monitor it.Hard 21 Selecting pdf, complement, interval Choose the methods. Application One question may need exact, complement and interval methods together. Students switch fluently.Hard 22 Least \(k\) with \(P(X\le k)\) target Solve a threshold. Application Finding the smallest \(k\) so \(P(X\le k)\ge0.9\) sets a stock level. Retailers plan inventory.Hard 23 Binomial count to expected value Link to money. Application Linking the number of sales to profit per sale gives expected revenue. Firms forecast.Hard 24 Independent binomials, add expectations Combine models. Application Two independent binomials’ expectations add to a combined total. Analysts merge models.Hard Very Hard — Integrated AI SL Challenge
Hard — Structured IB Questions
17 Sketch, probability, expected count Combine the skills. Application A sketch, a probability and an expected count together model defects. Engineers present the case.Hard 18 IQR of one, proportion of another Compare distributions. Application Comparing one model’s IQR with another’s tail contrasts two processes. Managers choose.Hard 19 Inverse normal both ways, interval Combine the tails. Application Finding symmetric cut-offs builds a central band of values. Engineers set tolerances.Hard 20 Predicting a count, top-percentage range Combine count and cut-off. Application Predicting how many score in the top 10% sets a scholarship quota. Schools plan.Hard 21 \(\sigma\) and a binomial model Combine a normal and a binomial. Application A normal tail probability feeds a binomial count of failures. Engineers chain the models.Hard 22 Justify a probability, extreme count Combine and justify. Application Justifying a rare-event probability flags an unusual result. Quality teams investigate.Hard 23 Boundary, sub-range count, classification Combine the skills. Application Cut-offs classify products into grades by count. Manufacturers sort output.Hard 24 \(\sigma\) from the rule, sample size Recover \(\sigma\) and \(n\). Application The empirical rule recovers the sd and then a sample size. Researchers plan studies.Hard Very Hard — Integrated AI SL Challenge
25 Graph, inverse normal, sample size Read and recover. Application Reading a graphed boundary, inverting and recovering the sample size combines several skills. Analysts synthesise.Very Hard 26 Probability, IQR, percentile, binomial Combine many skills. Application A capstone links a probability, IQR, percentile and a binomial count. Examiners test the lot.Very Hard 27 Inverse normal, conditional, binomial Combine the models. Application A cut-off, a conditional probability and a binomial model a screening programme. Health teams use it.Very Hard 28 Comparing two models, warranty binomial Carry out real reliability modelling. Application Comparing two normal models, then a warranty-failure binomial, guides a guarantee. Firms set terms.Very Hard 29 Probability, inverse normal, binomial proportion Combine the skills. Application A normal probability defines a binomial proportion of customers. Marketers model uptake.Very Hard 30 Recover \(\sigma\), class size, binomial Recover and compute. Application Recovering the sd, a class size and a binomial count chains the skills. Analysts work through it.Very Hard 31 Variance from the rule, binomial Recover variance; combine. Application The empirical rule gives the variance feeding a binomial model. Engineers combine them.Very Hard 32 Two groups, Bayesian-style finale Combine and reason. Application Two normal groups with conditional reasoning end in a Bayesian-style result. Analysts conclude.Very Hard Hard — Structured IB Questions
Very Hard — Integrated AI SL Challenge
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Hard — Structured IB Questions
Very Hard — Integrated AI SL Challenge
Medium — Applications
9 Gradient, tangent, curve Combine these. Application Integrating a known gradient function recovers the original curve. Designers rebuild a shape.Medium 10 Gradient, tangent, cubic curve Combine these. Application A cubic’s gradient function, integrated, gives the curve up to a constant. Modellers reconstruct it.Medium 11 Trapezoidal vs exact, percentage error Compare the estimates. Application Comparing the trapezoidal estimate to the exact area gives its percentage error. Analysts judge accuracy.Medium 12 Area between a curve and the \(x\)-axis Integrate. Application The area between a profit curve and the axis is total profit. Firms read it off.Medium 13 Rate model, function, optimisation Integrate and optimise. Application Integrating a rate gives a total that can then be optimised. Engineers chain the steps.Medium 14 Area of a region (GDC) Compute numerically. Application A calculator finds the area of a region bounded by a real-world curve. Analysts get it quickly.Medium 15 Quadratic-arch modelling, area Model and find the area. Application A parabolic bridge arch’s enclosed area comes from integration. Architects compute material.Medium 16 Trapezoidal (\(n=6\)), over/under Judge the estimate. Application With six strips, a concave curve makes the trapezoidal rule overestimate. Analysts state the bias.Medium Hard — Structured IB Questions
Very Hard — Integrated AI SL Challenge
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