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The original variable is $t=5x+20$.","difficultyBand":"hard","tier":null,"totalMarks":4,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"ddd97b3b-8856-4a45-9794-936381056a89","label":"a","content":"Find the mean of $t$.","metadata":null,"markscheme":"- Mean $=5(3.2)+20=36$ B1","marks":1,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]},{"id":"8e2e2987-1503-42a0-a7e1-232b0f46b81c","label":"b","content":"Find the variance of $t$.","metadata":null,"markscheme":"- Multiplies the variance by $5^2$ M1\n- Variance $=25(1.44)=36$ A1","marks":2,"order":1,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]},{"id":"05cb9312-af0b-4778-83d9-5a032a948818","label":"c","content":"Find the standard deviation of $t$.","metadata":null,"markscheme":"- Standard deviation $=\\sqrt{36}=6$ B1","marks":1,"order":2,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"c73b5846-e832-48d7-9122-25f0209e2e27","revisionId":"f9aea85f-4f89-4190-91e3-7da2e3c2119d","questionNumber":null,"questionType":"LA","specification":"A materials engineer is investigating the relationship between the cooling rate, \$R\$ (\$^{\\circ}\\text{C/min}\$), and the ultimate tensile strength, \$S\$ (\$\\text{MPa}\$), of a newly developed alloy. Data is collected from 8 different sample batches and is summarised in the table below.\n\n\\[\\begin{array}{|c|c|c|c|c|c|c|c|c|} \\hline R & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 \\\\ \\hline S & 582 & 545 & 535 & 502 & 495 & 458 & 435 & 410 \\\\ \\hline \\end{array}\\]\n\n[You may assume that \$\\sum R = 360\$, \$\\sum S = 3962\$, \$\\sum R^2 = 20400\$, \$\\sum RS = 168330\$ and \$S_{SS} = 23911.5\$]","difficultyBand":"hard","tier":null,"totalMarks":13,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"286b842d-248f-4236-8efc-c1c27a56b027","label":"a","content":"Calculate \$S_{RS}\$ and \$S_{RR}\$. Give your answers to 3 significant figures.","metadata":null,"markscheme":"- $S_{RS} = 168330 - \\frac{360 \\times 3962}{8} = 168330 - 178290 = -9960$ M1\n- $S_{RR} = 20400 - \\frac{360^2}{8} = 20400 - 16200 = 4200$ M1\n- Correct values to 3 s.f. ($S_{RS} = -9960$ and $S_{RR} = 4200$) A1","marks":3,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"formulaic","label":"Using Summary Statistics Formulas","steps":[{"type":"text","order":0,"title":"Identify given statistics","content":"To calculate the sum of products $S_{RS}$ and the sum of squares $S_{RR}$, we first extract the relevant summary statistics and sample size from the question:\n\n- Sample size $n = 8$ (since there are 8 batches)\n- $\\sum R = 360$\n- $\\sum S = 3962$\n- $\\sum R^2 = 20400$\n- $\\sum RS = 168330$","highlights":[{"text":"8 different sample batches","location":"specification"}]},{"type":"text","order":1,"title":"Calculate SRS","content":"The formula for the sum of products $S_{RS}$ is:\n\n$$S_{RS} = \\sum RS - \\frac{(\\sum R)(\\sum S)}{n}$$\n\nSubstituting our values into the formula:\n\n$$\\begin{aligned} S_{RS} &= 168330 - \\frac{360 \\times 3962}{8} \\\\ &= 168330 - 178290 \\\\ &= -9960 \\end{aligned}$$\n\nTo 3 significant figures, $S_{RS} = -9960$."},{"type":"text","order":2,"title":"Calculate SRR","content":"The formula for the sum of squares $S_{RR}$ is:\n\n$$S_{RR} = \\sum R^2 - \\frac{(\\sum R)^2}{n}$$\n\nSubstituting our values into the formula:\n\n$$\\begin{aligned} S_{RR} &= 20400 - \\frac{360^2}{8} \\\\ &= 20400 - 16200 \\\\ &= 4200 \\end{aligned}$$\n\nTo 3 significant figures, $S_{RR} = 4200$."},{"type":"text","order":3,"title":"State the final answers","content":"Our final values rounded to 3 significant figures are:\n\n- $S_{RS} = -9960$\n- $S_{RR} = 4200$"}]},{"id":"calculator","label":"Using Graphics Calculator","steps":[{"type":"text","order":0,"title":"Enter data into calculator","content":"To calculate the sum of products $S_{RS}$ and the sum of squares $S_{RR}$ using a graphing calculator, we first enter the raw data values of $R$ (independent variable) and $S$ (dependent variable) into our calculator lists.\n\nFrom the table, we have $n = 8$ pairs of data:\n\n$$\\begin{array}{|c|c|c|c|c|c|c|c|c|} \\hline R & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 \\\\ \\hline S & 582 & 545 & 535 & 502 & 495 & 458 & 435 & 410 \\\\ \\hline \\end{array}$$"},{"type":"text","order":1,"title":"Identify summary statistics","content":"Using the statistical calculation functions on your calculator, find the summary statistics. These calculations will provide:\n\n- $n = 8$\n- $\\sum R = 360$\n- $\\sum S = 3962$\n- $\\sum R^2 = 20400$\n- $\\sum RS = 168330$\n\nThese match the assumed values provided in the question."},{"type":"text","order":2,"title":"Calculate SRS","content":"Recall the formula for the sum of products of $R$ and $S$:\n\n$$S_{RS} = \\sum RS - \\frac{\\sum R \\sum S}{n}$$\n\nSubstituting our values into the formula:\n\n$$\\begin{aligned} S_{RS} &= 168330 - \\frac{360 \\times 3962}{8} \\\\ &= 168330 - 178290 \\\\ &= -9960 \\end{aligned}$$\n\nTo 3 significant figures, $S_{RS} = -9960$."},{"type":"text","order":3,"title":"Calculate SRR","content":"Next, recall the formula for the sum of squares of $R$:\n\n$$S_{RR} = \\sum R^2 - \\frac{\\left(\\sum R\\right)^2}{n}$$\n\nSubstituting our values into the formula:\n\n$$\\begin{aligned} S_{RR} &= 20400 - \\frac{360^2}{8} \\\\ &= 20400 - 16200 \\\\ &= 4200 \\end{aligned}$$\n\nTo 3 significant figures, $S_{RR} = 4200$."}]}],"generatedAt":"2026-06-09T08:46:50.810Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"5995551b-4d77-4cb9-9343-33ca3fc2a4dd","label":"b","content":"Calculate the product moment correlation coefficient for this data.","metadata":null,"markscheme":"- $r = \\frac{-9960}{\\sqrt{4200 \\times 23911.5}}$ M1\n- $r = \\frac{-9960}{10021.39...} = -0.99387... \\approx -0.994$ A1","marks":2,"order":1,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"analytical","label":"Analytical Formula","steps":[{"type":"text","order":0,"title":"Recall the PMCC formula","content":"To find the product moment correlation coefficient ($r$), we use the formula:\n\n$$r = \\frac{S_{RS}}{\\sqrt{S_{RR} \\times S_{SS}}}$$\n\nWhere $S_{RS}$ is the sum of products, $S_{RR}$ is the sum of squares for $R$, and $S_{SS}$ is the sum of squares for $S$ (with $n = 8$ batches of data)."},{"type":"text","order":1,"title":"Calculate sum of squares","content":"First, calculate the sum of squares for $R$, denoted as $S_{RR}$. The formula is:\n\n$$S_{RR} = \\sum R^2 - \\frac{\\left(\\sum R\\right)^2}{n}$$\n\nSubstituting the given values $\\sum R = 360$, $\\sum R^2 = 20400$, and $n = 8$:\n\n$$S_{RR} = 20400 - \\frac{360^2}{8} = 20400 - \\frac{129600}{8} = 20400 - 16200 = 4200$$"},{"type":"text","order":2,"title":"Calculate sum of products","content":"Next, calculate the sum of products, $S_{RS}$. The formula is:\n\n$$S_{RS} = \\sum RS - \\frac{\\left(\\sum R\\right)\\left(\\sum S\\right)}{n}$$\n\nSubstituting the given values $\\sum R = 360$, $\\sum S = 3962$, $\\sum RS = 168330$, and $n = 8$:\n\n$$S_{RS} = 168330 - \\frac{360 \\times 3962}{8} = 168330 - 178290 = -9960$$"},{"type":"text","order":3,"title":"Substitute into PMCC formula","content":"Now we substitute our calculated values ($S_{RS} = -9960$, $S_{RR} = 4200$) along with the given $S_{SS} = 23911.5$ into the PMCC formula:\n\n$$r = \\frac{-9960}{\\sqrt{4200 \\times 23911.5}}$$\n\nThis step earns the method mark (**M1**)."},{"type":"text","order":4,"title":"Find the final value","content":"Now calculate the final value of $r$:\n\n$$r = \\frac{-9960}{\\sqrt{100428300}} = \\frac{-9960}{10021.3921...} \\approx -0.994 \\text{ (to 3 s.f.)}$$\n\nThis final rounded answer earns the accuracy mark (**A1**).","calculator":[{"gdc":{"expression":"-9960 / \\sqrt{4200 * 23911.5}","instructions":"1. Open the run-matrix menu.\n2. Type `-9960 / sqrt(4200 * 23911.5)`.\n3. Press [EXE] to execute the calculation."},"ti84":{"expression":"-9960 / \\sqrt{4200 * 23911.5}","instructions":"Enter the expression on your home screen:\n1. Press [Alpha] [Y=] [Enter] to create a fraction.\n2. In the numerator, type: -9960\n3. In the denominator, press [2nd] [x^2] (for the square root symbol), then type: 4200 * 23911.5\n4. Press [Enter] to calculate."},"desmos":{"expression":"\\frac{-9960}{\\sqrt{4200 \\cdot 23911.5}}","instructions":"Type the expression into an empty cell to evaluate it immediately."},"description":"Evaluate the fraction to find the final correlation coefficient $r$."}]}]},{"id":"calculator","label":"Statistical Calculator","steps":[{"type":"text","order":0,"title":"Identify the goal","content":"The question asks us to calculate the **product moment correlation coefficient (PMCC)**, usually represented by $r$, for the cooling rate $R$ and the ultimate tensile strength $S$.\n\nInstead of calculating all the summary statistics by hand, we can input the raw data pairs directly into a statistical calculator to obtain the value of $r$ directly."},{"type":"text","order":1,"title":"Input data into calculator","content":"We will enter the raw data pairs from the table into the calculator's lists:\n- Let $R$ values be in List 1 ($x$-variable).\n- Let $S$ values be in List 2 ($y$-variable).\n\n$$\\begin{array}{|c|c|c|c|c|c|c|c|c|} \\hline R & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 \\\\ \\hline S & 582 & 545 & 535 & 502 & 495 & 458 & 435 & 410 \\\\ \\hline \\end{array}$$","calculator":[{"gdc":{"instructions":"1. Go to the Statistics/List menu.\n2. In List 1, enter the values: 10, 20, 30, 40, 50, 60, 70, 80.\n3. In List 2, enter the values: 582, 545, 535, 502, 495, 458, 435, 410.\n4. Select 'CALC', then 'REG' (Regression), and choose 'X' (Linear Regression of form $ax+b$ or $a+bx$).\n5. View the output value for $r$."},"ti84":{"instructions":"1. Press [STAT] and select 1:Edit...\n2. Enter the $R$ values: 10, 20, 30, 40, 50, 60, 70, 80 into L1.\n3. Enter the corresponding $S$ values: 582, 545, 535, 502, 495, 458, 435, 410 into L2.\n4. Ensure DiagnosticOn is enabled: press [2nd] [CATALOG] (above '0'), scroll to DiagnosticOn, press [ENTER] twice.\n5. Press [STAT], arrow over to CALC, and select 4:LinReg(ax+b).\n6. Set Xlist to L1, Ylist to L2, and press Calculate."},"desmos":{"expression":"y_1 \\sim m x_1 + c","instructions":"1. Click the '+' button and add a Table.\n2. In the $x_1$ column, enter: 10, 20, 30, 40, 50, 60, 70, 80.\n3. In the $y_1$ column, enter: 582, 545, 535, 502, 495, 458, 435, 410.\n4. In a new expression line, type the linear regression model: y1 ~ m x1 + c\n5. Look at the calculated parameters below the expression where $r$ is listed."},"description":"Enter the raw data into Lists and run a Linear Regression calculation to find the correlation coefficient $r$."}]},{"type":"text","order":2,"title":"Retrieve and round the result","content":"From our calculator's linear regression output, we get:\n\n$$r \\approx -0.99387...$$\n\nRounding to $3$ significant figures as standard:\n\n$$r \\approx -0.994$$"}]}],"generatedAt":"2026-06-09T08:48:21.086Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"4a516e6c-18a6-4394-bcdc-2140af0c6e7f","label":"c","content":"State whether or not your value supports the use of a linear regression equation to predict tensile strength based on cooling rates. Give a reason for your answer.","metadata":null,"markscheme":"- Yes, linear regression is appropriate. B1\n- Since $r = -0.994$ is very close to $-1$, indicating a very strong negative linear correlation. B1","marks":2,"order":2,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"correlation_interpretation","label":"Analyzing Correlation Strength","steps":[{"type":"text","order":0,"title":"Recall the value of r","content":"From the previous calculations, the product-moment correlation coefficient is:\n\n$$r = -0.994$$\n\nTo determine if a linear regression model is suitable, we must evaluate the strength of this correlation by looking at how close this value is to $-1$."},{"type":"text","order":1,"title":"Analyze the correlation strength","content":"A product-moment correlation coefficient ($r$) always lies in the range $[-1, 1]$:\n- An $r$ value close to $+1$ indicates a strong positive linear relationship.\n- An $r$ value close to $-1$ indicates a strong negative linear relationship.\n- An $r$ value close to $0$ indicates little to no linear relationship.\n\nSince our calculated value of $r = -0.994$ is **extremely close to $-1$**, it indicates a **very strong negative linear correlation** between the cooling rate ($R$) and the ultimate tensile strength ($S$)."},{"type":"text","order":2,"title":"Conclude and justify","content":"Because there is a very strong linear correlation, a linear relationship model is highly appropriate.\n\nTherefore, we conclude:\n- **Yes**, the value supports the use of a linear regression equation.\n- **Reason**: The correlation coefficient $r = -0.994$ is very close to $-1$, showing a very strong negative linear relationship between the cooling rate and ultimate tensile strength."}]}],"generatedAt":"2026-06-09T08:49:00.166Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"cb7215af-f0fd-4691-8593-71ba04f70176","label":"d","content":"Find the equation of the regression line of \$S\$ on \$R\$ giving your answer in the form \$S = a + bR\$.","metadata":null,"markscheme":"- $b = \\frac{S_{RS}}{S_{RR}} = \\frac{-9960}{4200} = -2.3714... \\approx -2.37$ M1\n- $a = \\bar{S} - b\\bar{R} = \\frac{3962}{8} - (-2.3714...)\\left(\\frac{360}{8}\\right) = 495.25 + 106.714... = 601.96...$ M1\n- $S = 602 - 2.37R$ A1","marks":3,"order":3,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"formulaic","label":"Formulaic Approach","steps":[{"type":"text","order":0,"title":"Identify summary statistics","content":"From the question, we are given the following summary statistics for $n = 8$ sample batches:\n\n* $n = 8$\n* $\\sum R = 360$\n* $\\sum S = 3962$\n* $\\sum R^2 = 20400$\n* $\\sum RS = 168330$\n\nWe need to find the regression line in the form:\n\n$$S = a + bR$$\n\nwhere $b = \\frac{S_{RS}}{S_{RR}}$ and $a = \\bar{S} - b\\bar{R}$."},{"type":"text","order":1,"title":"Calculate sum of squares","content":"First, we calculate the sum of products $S_{RS}$ and the sum of squares $S_{RR}$ using the standard statistical formulae:\n\n$$S_{RS} = \\sum RS - \\frac{(\\sum R)(\\sum S)}{n}$$\n\n$$S_{RR} = \\sum R^2 - \\frac{(\\sum R)^2}{n}$$\n\nSubstituting our values in:\n\n$$S_{RS} = 168330 - \\frac{360 \\times 3962}{8} = 168330 - 178290 = -9960$$\n\n$$S_{RR} = 20400 - \\frac{360^2}{8} = 20400 - 16200 = 4200$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$b = \\frac{\\htmlData{anchor=srs}{\\color{@red}{S_{RS}}}}{\\htmlData{anchor=srr}{\\color{@indigo}{S_{RR}}}}$"},{"latex":"$$b = \\frac{\\htmlData{anchor=srs-val}{\\color{@red}{-9960}}}{\\htmlData{anchor=srr-val}{\\color{@indigo}{4200}}} \\approx \\htmlData{anchor=b-val}{\\color{@purple}{-2.3714...}}$"}],"highlights":[{"color":"red","style":"text-color","anchor":"srs"},{"color":"indigo","style":"text-color","anchor":"srr"}],"connections":[{"to":"srs-val","from":"srs","color":"red","label":"substitute","style":"solid"},{"to":"srr-val","from":"srr","color":"indigo","label":"substitute","style":"solid"}]},"order":2,"title":"Calculate the gradient","description":"Now, we can compute the gradient $b$ of the regression line. This is worth the first method mark (M1)."},{"type":"text","order":3,"title":"Calculate the vertical intercept","content":"Next, we calculate the sample means $\\bar{R}$ and $\\bar{S}$:\n\n$$\\bar{R} = \\frac{\\sum R}{n} = \\frac{360}{8} = 45$$\n\n$$\\bar{S} = \\frac{\\sum S}{n} = \\frac{3962}{8} = 495.25$$\n\nNow, substitute these means and our calculated gradient $b \\approx -2.3714$ into the formula for $a$:\n\n$$a = \\bar{S} - b\\bar{R}$$\n\n$$a = 495.25 - (-2.3714) \\times 45$$\n\n$$a = 495.25 + 106.714...$$\n\n$$a \\approx 602$$"},{"type":"text","order":4,"title":"State the final equation","content":"Rounding our parameters to 3 significant figures:\n\n* $a \\approx 602$\n* $b \\approx -2.37$\n\nSubstituting these values back into our general form gives the final regression equation:\n\n$$S = 602 - 2.37R$$"}]},{"id":"calculator","label":"Using a Statistical Calculator","steps":[{"type":"text","order":0,"title":"Identify the variables","content":"We want to find the equation of the least squares regression line of $S$ on $R$ in the form:\n\n$$S = a + bR$$\n\nIn this relationship, $R$ (cooling rate) is the independent (explanatory) variable, and $S$ (ultimate tensile strength) is the dependent (response) variable."},{"type":"text","order":1,"title":"Input data into calculator","content":"Instead of calculating the sums and standard statistics by hand, we can input the paired data $(R, S)$ directly into a statistics calculator to find the coefficients $a$ and $b$.\n\nLet's enter our list of values:\n* **$R$ values (independent variable, $x$):** $10, 20, 30, 40, 50, 60, 70, 80$\n* **$S$ values (dependent variable, $y$):** $582, 545, 535, 502, 495, 458, 435, 410$","calculator":[{"gdc":{"instructions":"1. Open the Statistics menu/app.\n2. In the list editor, enter the $R$ values in the first column (e.g., `List 1`) and the $S$ values in the second column (e.g., `List 2`).\n3. Press the Calc/Regression option (usually `F2` or `F3` -> `REG` -> `X`).\n4. Select the linear model `a+bx` or `ax+b` (make sure your independent and dependent lists match your input)."},"ti84":{"instructions":"1. Press [STAT] and select `1:Edit...`\n2. Enter the $R$ values into list `L1` and the $S$ values into list `L2`.\n3. Press [STAT], arrow over to `CALC` and select `4:LinReg(ax+b)` (or `8:LinReg(a+bx)`).\n4. Set `Xlist` to `L1` and `Ylist` to `L2`.\n5. Select `Calculate` and press [ENTER]."},"desmos":{"expression":"y_1 ~ a + b*x_1","instructions":"1. Click the plus button (+) and choose `Table`.\n2. Enter the $R$ values into $x_1$ and $S$ values into $y_1$.\n3. In a new expression line, type: `y_1 ~ a + b*x_1` to run the regression."},"description":"How to input the bivariate data and calculate the linear regression parameters on different platforms."}]},{"type":"text","order":2,"title":"Read the regression coefficients","content":"From the calculator output screen, read the values for the y-intercept ($a$) and the slope ($b$):\n\n$$a \\approx 601.964...$$\n\n$$b \\approx -2.3714...$$\n\nRounding these to 3 significant figures gives:\n* $a \\approx 602$\n* $b \\approx -2.37$"},{"type":"text","order":3,"title":"Write the final equation","content":"Substitute the values of $a$ and $b$ back into the form $S = a + bR$:\n\n$$S = 602 - 2.37R$$\n\n* **Mark 1 (M1):** Finding the slope $b \\approx -2.37$ using the correct statistical calculations.\n* **Mark 2 (M1):** Finding the intercept $a \\approx 602$.\n* **Mark 3 (A1):** Writing down the final regression equation with the correct variables ($S$ and $R$)."}]}],"generatedAt":"2026-06-09T08:51:24.157Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"d5920614-177d-40b5-bfd7-8d65778650bb","label":"e","content":"Interpret the value of \$b\$.","metadata":null,"markscheme":"- For every 1 $^{\\circ}\\text{C/min}$ increase in the cooling rate, the tensile strength is estimated to decrease by 2.37 MPa. B1","marks":1,"order":4,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"direct_interpretation","label":"Direct Statistical Interpretation","steps":[{"type":"text","order":0,"title":"Recall the value of b","content":"From the equation of the least squares regression line of $S$ on $R$ in the form:\n\n$$S = a + bR$$\n\nwe found the gradient (slope) to be:\n\n$$b \\approx -2.37$$\n\nHere, $b$ represents how the dependent variable, ultimate tensile strength $S$ (in $\\text{MPa}$), changes in response to the independent variable, cooling rate $R$ (in $^{\\circ}\\text{C/min}$)."},{"type":"text","order":1,"title":"Define gradient statistically","content":"Statistically, the gradient $b$ represents the average change in the dependent variable for every single-unit increase in the independent variable.\n\nIn this context:\n- **Independent variable ($R$):** Cooling rate, measured in $^{\\circ}\\text{C/min}$\n- **Dependent variable ($S$):** Ultimate tensile strength, measured in $\\text{MPa}$"},{"type":"text","order":2,"title":"Interpret the negative sign","content":"Since $b = -2.37$ is negative, it indicates a negative relationship between the variables:\n\nAs the cooling rate $R$ **increases**, the ultimate tensile strength $S$ is expected to **decrease**."},{"type":"text","order":3,"title":"Write the final interpretation","content":"Combining these ideas, we state the final interpretation of $b$ as follows:\n\nFor every $1^{\\circ}\\text{C/min}$ increase in the cooling rate, the ultimate tensile strength is estimated to decrease by $2.37\\text{ MPa}$."}]}],"generatedAt":"2026-06-09T08:52:21.180Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"212acb07-2cf4-43bd-8be6-e5d9c481e8d4","label":"f","content":"Estimate the decrease in tensile strength when moving from a cooling rate of 25 \$^{\\circ}\\text{C/min}\$ to a cooling rate of 75 \$^{\\circ}\\text{C/min}\$.","metadata":null,"markscheme":"- Change in cooling rate $\\Delta R = 75 - 25 = 50$ M1\n- Estimated decrease $= 50 \\times 2.3714... = 118.57... \\approx 119$ MPa A1","marks":2,"order":5,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"gradient_method","label":"Using Regression Slope","steps":[{"type":"text","order":0,"title":"Identify the goal","content":"We want to estimate the decrease in tensile strength, $S$, when the cooling rate, $R$, increases from $25^\\circ\\text{C/min}$ to $75^\\circ\\text{C/min}$.\n\nFirst, let's find the change in the cooling rate, which we can call $\\Delta R$:\n\n$$\\Delta R = 75 - 25 = 50^\\circ\\text{C/min}$$","highlights":[{"text":"moving from a cooling rate of 25 \$^{\\circ}\\text{C/min}\$ to a cooling rate of 75 \$^{\\circ}\\text{C/min}\$","location":"specification"}]},{"type":"text","order":1,"title":"Calculate summary statistics","content":"To find the slope $b$ of the regression line $S = a + bR$, we first calculate the sum of products $S_{RS}$ and sum of squares $S_{RR}$.\n\nThe formulas are:\n\n$$S_{RR} = \\sum R^2 - \\frac{(\\sum R)^2}{n}$$\n\n$$S_{RS} = \\sum RS - \\frac{(\\sum R)(\\sum S)}{n}$$\n\nWith $n = 8$ and our given summary values:\n- $\\sum R = 360$\n- $\\sum S = 3962$\n- $\\sum R^2 = 20400$\n- $\\sum RS = 168330$\n\nNow we calculate $S_{RR}$:\n\n$$S_{RR} = 20400 - \\frac{360^2}{8} = 20400 - 16200 = 4200$$\n\nAnd we calculate $S_{RS}$:\n\n$$S_{RS} = 168330 - \\frac{360 \\times 3962}{8} = 168330 - 178290 = -9960$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$b = \\frac{\\htmlData{anchor=srs-val}{\\color{@sky}{S_{RS}}}}{\\htmlData{anchor=srr-val}{\\color{@amber}{S_{RR}}}}$"},{"latex":"$$b = \\frac{\\htmlData{anchor=srs-num}{\\color{@sky}{-9960}}}{\\htmlData{anchor=srr-num}{\\color{@amber}{4200}}} \\approx \\htmlData{anchor=slope-val}{\\color{@purple}{-2.3714}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"srs-val"},{"color":"amber","style":"text-color","anchor":"srr-val"},{"color":"purple","style":"box","anchor":"slope-val"}],"connections":[{"to":"srs-num","from":"srs-val","color":"sky","label":"substitute","style":"solid"},{"to":"srr-num","from":"srr-val","color":"amber","label":"substitute","style":"solid"}]},"order":2,"title":"Calculate the regression slope","description":"The slope represents the rate of change of tensile strength per unit increase in cooling rate. A value of $-2.3714$ means that for every $1^\\circ\\text{C/min}$ increase in cooling rate, the tensile strength is estimated to decrease by $2.3714\\text{ MPa}$."},{"type":"text","order":3,"title":"Estimate the decrease","content":"To find the estimated decrease, we multiply the rate of decrease (the magnitude of the slope $b$) by our change in cooling rate, $\\Delta R = 50$:\n\n$$\\text{Estimated decrease} = 50 \\times 2.3714 \\approx 118.57\\text{ MPa}$$\n\nRounding to 3 significant figures gives:\n\n$$\\text{Estimated decrease} \\approx 119\\text{ MPa}$$"}]},{"id":"full_equation_method","label":"Evaluating the Regression Equation","steps":[{"type":"text","order":0,"title":"Calculate regression slope","content":"We first find the gradient $b$ of the least-squares regression line $S = a + bR$ using the formula:\n\n$$b = \\frac{S_{RS}}{S_{RR}}$$\n\nUsing the summary values provided:\n- $S_{RS} = 168330 - \\frac{360 \\times 3962}{8} = -9960$\n- $S_{RR} = 20400 - \\frac{360^2}{8} = 4200$\n\nSubstituting these values gives:\n\n$$b = \\frac{-9960}{4200} \\approx -2.3714$$"},{"type":"text","order":1,"title":"Calculate intercept","content":"Now we compute the intercept $a$ using the means of $S$ and $R$:\n\n$$\\bar{R} = \\frac{360}{8} = 45$$\n\n$$\\bar{S} = \\frac{3962}{8} = 495.25$$\n\nSubstitute these into the equation $a = \\bar{S} - b\\bar{R}$:\n\n$$a = 495.25 - (-2.3714 \\times 45) \\approx 601.964$$\n\nThis gives us our full regression line equation:\n\n$$S = 601.964 - 2.3714R$$"},{"type":"text","order":2,"title":"Predict strength at 25","content":"Using our regression equation, let's predict the tensile strength when $R = 25^\\circ\\text{C/min}$:\n\n$$S_{25} = 601.964 - 2.3714 \\times 25 \\approx 542.68\\text{ MPa}$$"},{"type":"text","order":3,"title":"Predict strength at 75","content":"Similarly, we find the predicted tensile strength when the cooling rate is $R = 75^\\circ\\text{C/min}$:\n\n$$S_{75} = 601.964 - 2.3714 \\times 75 \\approx 424.11\\text{ MPa}$$"},{"type":"text","order":4,"title":"Find the difference","content":"Now we calculate the estimated decrease by finding the difference between our two predictions:\n\n$$\\text{Decrease} = S_{25} - S_{75}$$\n\n$$\\text{Decrease} = 542.68 - 424.11 \\approx 118.57\\text{ MPa}$$\n\nRounding to 3 significant figures, we get an estimated decrease of **$119\\text{ MPa}$**."}]}],"generatedAt":"2026-06-09T08:54:30.070Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"286b842d-248f-4236-8efc-c1c27a56b027:formulaic","label":"(a) Using Summary Statistics Formulas","steps":[{"type":"text","order":0,"title":"Identify given statistics","content":"To calculate the sum of products $S_{RS}$ and the sum of squares $S_{RR}$, we first extract the relevant summary statistics and sample size from the question:\n\n- Sample size $n = 8$ (since there are 8 batches)\n- $\\sum R = 360$\n- $\\sum S = 3962$\n- $\\sum R^2 = 20400$\n- $\\sum RS = 168330$","highlights":[{"text":"8 different sample batches","location":"specification"}]},{"type":"text","order":1,"title":"Calculate SRS","content":"The formula for the sum of products $S_{RS}$ is:\n\n$$S_{RS} = \\sum RS - \\frac{(\\sum R)(\\sum S)}{n}$$\n\nSubstituting our values into the formula:\n\n$$\\begin{aligned} S_{RS} &= 168330 - \\frac{360 \\times 3962}{8} \\\\ &= 168330 - 178290 \\\\ &= -9960 \\end{aligned}$$\n\nTo 3 significant figures, $S_{RS} = -9960$."},{"type":"text","order":2,"title":"Calculate SRR","content":"The formula for the sum of squares $S_{RR}$ is:\n\n$$S_{RR} = \\sum R^2 - \\frac{(\\sum R)^2}{n}$$\n\nSubstituting our values into the formula:\n\n$$\\begin{aligned} S_{RR} &= 20400 - \\frac{360^2}{8} \\\\ &= 20400 - 16200 \\\\ &= 4200 \\end{aligned}$$\n\nTo 3 significant figures, $S_{RR} = 4200$."},{"type":"text","order":3,"title":"State the final answers","content":"Our final values rounded to 3 significant figures are:\n\n- $S_{RS} = -9960$\n- $S_{RR} = 4200$"}]},{"id":"286b842d-248f-4236-8efc-c1c27a56b027:calculator","label":"(a) Using Graphics Calculator","steps":[{"type":"text","order":0,"title":"Enter data into calculator","content":"To calculate the sum of products $S_{RS}$ and the sum of squares $S_{RR}$ using a graphing calculator, we first enter the raw data values of $R$ (independent variable) and $S$ (dependent variable) into our calculator lists.\n\nFrom the table, we have $n = 8$ pairs of data:\n\n$$\\begin{array}{|c|c|c|c|c|c|c|c|c|} \\hline R & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 \\\\ \\hline S & 582 & 545 & 535 & 502 & 495 & 458 & 435 & 410 \\\\ \\hline \\end{array}$$"},{"type":"text","order":1,"title":"Identify summary statistics","content":"Using the statistical calculation functions on your calculator, find the summary statistics. These calculations will provide:\n\n- $n = 8$\n- $\\sum R = 360$\n- $\\sum S = 3962$\n- $\\sum R^2 = 20400$\n- $\\sum RS = 168330$\n\nThese match the assumed values provided in the question."},{"type":"text","order":2,"title":"Calculate SRS","content":"Recall the formula for the sum of products of $R$ and $S$:\n\n$$S_{RS} = \\sum RS - \\frac{\\sum R \\sum S}{n}$$\n\nSubstituting our values into the formula:\n\n$$\\begin{aligned} S_{RS} &= 168330 - \\frac{360 \\times 3962}{8} \\\\ &= 168330 - 178290 \\\\ &= -9960 \\end{aligned}$$\n\nTo 3 significant figures, $S_{RS} = -9960$."},{"type":"text","order":3,"title":"Calculate SRR","content":"Next, recall the formula for the sum of squares of $R$:\n\n$$S_{RR} = \\sum R^2 - \\frac{\\left(\\sum R\\right)^2}{n}$$\n\nSubstituting our values into the formula:\n\n$$\\begin{aligned} S_{RR} &= 20400 - \\frac{360^2}{8} \\\\ &= 20400 - 16200 \\\\ &= 4200 \\end{aligned}$$\n\nTo 3 significant figures, $S_{RR} = 4200$."}]},{"id":"5995551b-4d77-4cb9-9343-33ca3fc2a4dd:analytical","label":"(b) Analytical Formula","steps":[{"type":"text","order":0,"title":"Recall the PMCC formula","content":"To find the product moment correlation coefficient ($r$), we use the formula:\n\n$$r = \\frac{S_{RS}}{\\sqrt{S_{RR} \\times S_{SS}}}$$\n\nWhere $S_{RS}$ is the sum of products, $S_{RR}$ is the sum of squares for $R$, and $S_{SS}$ is the sum of squares for $S$ (with $n = 8$ batches of data)."},{"type":"text","order":1,"title":"Calculate sum of squares","content":"First, calculate the sum of squares for $R$, denoted as $S_{RR}$. The formula is:\n\n$$S_{RR} = \\sum R^2 - \\frac{\\left(\\sum R\\right)^2}{n}$$\n\nSubstituting the given values $\\sum R = 360$, $\\sum R^2 = 20400$, and $n = 8$:\n\n$$S_{RR} = 20400 - \\frac{360^2}{8} = 20400 - \\frac{129600}{8} = 20400 - 16200 = 4200$$"},{"type":"text","order":2,"title":"Calculate sum of products","content":"Next, calculate the sum of products, $S_{RS}$. The formula is:\n\n$$S_{RS} = \\sum RS - \\frac{\\left(\\sum R\\right)\\left(\\sum S\\right)}{n}$$\n\nSubstituting the given values $\\sum R = 360$, $\\sum S = 3962$, $\\sum RS = 168330$, and $n = 8$:\n\n$$S_{RS} = 168330 - \\frac{360 \\times 3962}{8} = 168330 - 178290 = -9960$$"},{"type":"text","order":3,"title":"Substitute into PMCC formula","content":"Now we substitute our calculated values ($S_{RS} = -9960$, $S_{RR} = 4200$) along with the given $S_{SS} = 23911.5$ into the PMCC formula:\n\n$$r = \\frac{-9960}{\\sqrt{4200 \\times 23911.5}}$$\n\nThis step earns the method mark (**M1**)."},{"type":"text","order":4,"title":"Find the final value","content":"Now calculate the final value of $r$:\n\n$$r = \\frac{-9960}{\\sqrt{100428300}} = \\frac{-9960}{10021.3921...} \\approx -0.994 \\text{ (to 3 s.f.)}$$\n\nThis final rounded answer earns the accuracy mark (**A1**).","calculator":[{"gdc":{"expression":"-9960 / \\sqrt{4200 * 23911.5}","instructions":"1. Open the run-matrix menu.\n2. Type `-9960 / sqrt(4200 * 23911.5)`.\n3. Press [EXE] to execute the calculation."},"ti84":{"expression":"-9960 / \\sqrt{4200 * 23911.5}","instructions":"Enter the expression on your home screen:\n1. Press [Alpha] [Y=] [Enter] to create a fraction.\n2. In the numerator, type: -9960\n3. In the denominator, press [2nd] [x^2] (for the square root symbol), then type: 4200 * 23911.5\n4. Press [Enter] to calculate."},"desmos":{"expression":"\\frac{-9960}{\\sqrt{4200 \\cdot 23911.5}}","instructions":"Type the expression into an empty cell to evaluate it immediately."},"description":"Evaluate the fraction to find the final correlation coefficient $r$."}]}]},{"id":"5995551b-4d77-4cb9-9343-33ca3fc2a4dd:calculator","label":"(b) Statistical Calculator","steps":[{"type":"text","order":0,"title":"Identify the goal","content":"The question asks us to calculate the **product moment correlation coefficient (PMCC)**, usually represented by $r$, for the cooling rate $R$ and the ultimate tensile strength $S$.\n\nInstead of calculating all the summary statistics by hand, we can input the raw data pairs directly into a statistical calculator to obtain the value of $r$ directly."},{"type":"text","order":1,"title":"Input data into calculator","content":"We will enter the raw data pairs from the table into the calculator's lists:\n- Let $R$ values be in List 1 ($x$-variable).\n- Let $S$ values be in List 2 ($y$-variable).\n\n$$\\begin{array}{|c|c|c|c|c|c|c|c|c|} \\hline R & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 \\\\ \\hline S & 582 & 545 & 535 & 502 & 495 & 458 & 435 & 410 \\\\ \\hline \\end{array}$$","calculator":[{"gdc":{"instructions":"1. Go to the Statistics/List menu.\n2. In List 1, enter the values: 10, 20, 30, 40, 50, 60, 70, 80.\n3. In List 2, enter the values: 582, 545, 535, 502, 495, 458, 435, 410.\n4. Select 'CALC', then 'REG' (Regression), and choose 'X' (Linear Regression of form $ax+b$ or $a+bx$).\n5. View the output value for $r$."},"ti84":{"instructions":"1. Press [STAT] and select 1:Edit...\n2. Enter the $R$ values: 10, 20, 30, 40, 50, 60, 70, 80 into L1.\n3. Enter the corresponding $S$ values: 582, 545, 535, 502, 495, 458, 435, 410 into L2.\n4. Ensure DiagnosticOn is enabled: press [2nd] [CATALOG] (above '0'), scroll to DiagnosticOn, press [ENTER] twice.\n5. Press [STAT], arrow over to CALC, and select 4:LinReg(ax+b).\n6. Set Xlist to L1, Ylist to L2, and press Calculate."},"desmos":{"expression":"y_1 \\sim m x_1 + c","instructions":"1. Click the '+' button and add a Table.\n2. In the $x_1$ column, enter: 10, 20, 30, 40, 50, 60, 70, 80.\n3. In the $y_1$ column, enter: 582, 545, 535, 502, 495, 458, 435, 410.\n4. In a new expression line, type the linear regression model: y1 ~ m x1 + c\n5. Look at the calculated parameters below the expression where $r$ is listed."},"description":"Enter the raw data into Lists and run a Linear Regression calculation to find the correlation coefficient $r$."}]},{"type":"text","order":2,"title":"Retrieve and round the result","content":"From our calculator's linear regression output, we get:\n\n$$r \\approx -0.99387...$$\n\nRounding to $3$ significant figures as standard:\n\n$$r \\approx -0.994$$"}]},{"id":"4a516e6c-18a6-4394-bcdc-2140af0c6e7f:correlation_interpretation","label":"(c) Analyzing Correlation Strength","steps":[{"type":"text","order":0,"title":"Recall the value of r","content":"From the previous calculations, the product-moment correlation coefficient is:\n\n$$r = -0.994$$\n\nTo determine if a linear regression model is suitable, we must evaluate the strength of this correlation by looking at how close this value is to $-1$."},{"type":"text","order":1,"title":"Analyze the correlation strength","content":"A product-moment correlation coefficient ($r$) always lies in the range $[-1, 1]$:\n- An $r$ value close to $+1$ indicates a strong positive linear relationship.\n- An $r$ value close to $-1$ indicates a strong negative linear relationship.\n- An $r$ value close to $0$ indicates little to no linear relationship.\n\nSince our calculated value of $r = -0.994$ is **extremely close to $-1$**, it indicates a **very strong negative linear correlation** between the cooling rate ($R$) and the ultimate tensile strength ($S$)."},{"type":"text","order":2,"title":"Conclude and justify","content":"Because there is a very strong linear correlation, a linear relationship model is highly appropriate.\n\nTherefore, we conclude:\n- **Yes**, the value supports the use of a linear regression equation.\n- **Reason**: The correlation coefficient $r = -0.994$ is very close to $-1$, showing a very strong negative linear relationship between the cooling rate and ultimate tensile strength."}]},{"id":"cb7215af-f0fd-4691-8593-71ba04f70176:formulaic","label":"(d) Formulaic Approach","steps":[{"type":"text","order":0,"title":"Identify summary statistics","content":"From the question, we are given the following summary statistics for $n = 8$ sample batches:\n\n* $n = 8$\n* $\\sum R = 360$\n* $\\sum S = 3962$\n* $\\sum R^2 = 20400$\n* $\\sum RS = 168330$\n\nWe need to find the regression line in the form:\n\n$$S = a + bR$$\n\nwhere $b = \\frac{S_{RS}}{S_{RR}}$ and $a = \\bar{S} - b\\bar{R}$."},{"type":"text","order":1,"title":"Calculate sum of squares","content":"First, we calculate the sum of products $S_{RS}$ and the sum of squares $S_{RR}$ using the standard statistical formulae:\n\n$$S_{RS} = \\sum RS - \\frac{(\\sum R)(\\sum S)}{n}$$\n\n$$S_{RR} = \\sum R^2 - \\frac{(\\sum R)^2}{n}$$\n\nSubstituting our values in:\n\n$$S_{RS} = 168330 - \\frac{360 \\times 3962}{8} = 168330 - 178290 = -9960$$\n\n$$S_{RR} = 20400 - \\frac{360^2}{8} = 20400 - 16200 = 4200$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$b = \\frac{\\htmlData{anchor=srs}{\\color{@red}{S_{RS}}}}{\\htmlData{anchor=srr}{\\color{@indigo}{S_{RR}}}}$"},{"latex":"$$b = \\frac{\\htmlData{anchor=srs-val}{\\color{@red}{-9960}}}{\\htmlData{anchor=srr-val}{\\color{@indigo}{4200}}} \\approx \\htmlData{anchor=b-val}{\\color{@purple}{-2.3714...}}$"}],"highlights":[{"color":"red","style":"text-color","anchor":"srs"},{"color":"indigo","style":"text-color","anchor":"srr"}],"connections":[{"to":"srs-val","from":"srs","color":"red","label":"substitute","style":"solid"},{"to":"srr-val","from":"srr","color":"indigo","label":"substitute","style":"solid"}]},"order":2,"title":"Calculate the gradient","description":"Now, we can compute the gradient $b$ of the regression line. This is worth the first method mark (M1)."},{"type":"text","order":3,"title":"Calculate the vertical intercept","content":"Next, we calculate the sample means $\\bar{R}$ and $\\bar{S}$:\n\n$$\\bar{R} = \\frac{\\sum R}{n} = \\frac{360}{8} = 45$$\n\n$$\\bar{S} = \\frac{\\sum S}{n} = \\frac{3962}{8} = 495.25$$\n\nNow, substitute these means and our calculated gradient $b \\approx -2.3714$ into the formula for $a$:\n\n$$a = \\bar{S} - b\\bar{R}$$\n\n$$a = 495.25 - (-2.3714) \\times 45$$\n\n$$a = 495.25 + 106.714...$$\n\n$$a \\approx 602$$"},{"type":"text","order":4,"title":"State the final equation","content":"Rounding our parameters to 3 significant figures:\n\n* $a \\approx 602$\n* $b \\approx -2.37$\n\nSubstituting these values back into our general form gives the final regression equation:\n\n$$S = 602 - 2.37R$$"}]},{"id":"cb7215af-f0fd-4691-8593-71ba04f70176:calculator","label":"(d) Using a Statistical Calculator","steps":[{"type":"text","order":0,"title":"Identify the variables","content":"We want to find the equation of the least squares regression line of $S$ on $R$ in the form:\n\n$$S = a + bR$$\n\nIn this relationship, $R$ (cooling rate) is the independent (explanatory) variable, and $S$ (ultimate tensile strength) is the dependent (response) variable."},{"type":"text","order":1,"title":"Input data into calculator","content":"Instead of calculating the sums and standard statistics by hand, we can input the paired data $(R, S)$ directly into a statistics calculator to find the coefficients $a$ and $b$.\n\nLet's enter our list of values:\n* **$R$ values (independent variable, $x$):** $10, 20, 30, 40, 50, 60, 70, 80$\n* **$S$ values (dependent variable, $y$):** $582, 545, 535, 502, 495, 458, 435, 410$","calculator":[{"gdc":{"instructions":"1. Open the Statistics menu/app.\n2. In the list editor, enter the $R$ values in the first column (e.g., `List 1`) and the $S$ values in the second column (e.g., `List 2`).\n3. Press the Calc/Regression option (usually `F2` or `F3` -> `REG` -> `X`).\n4. Select the linear model `a+bx` or `ax+b` (make sure your independent and dependent lists match your input)."},"ti84":{"instructions":"1. Press [STAT] and select `1:Edit...`\n2. Enter the $R$ values into list `L1` and the $S$ values into list `L2`.\n3. Press [STAT], arrow over to `CALC` and select `4:LinReg(ax+b)` (or `8:LinReg(a+bx)`).\n4. Set `Xlist` to `L1` and `Ylist` to `L2`.\n5. Select `Calculate` and press [ENTER]."},"desmos":{"expression":"y_1 ~ a + b*x_1","instructions":"1. Click the plus button (+) and choose `Table`.\n2. Enter the $R$ values into $x_1$ and $S$ values into $y_1$.\n3. In a new expression line, type: `y_1 ~ a + b*x_1` to run the regression."},"description":"How to input the bivariate data and calculate the linear regression parameters on different platforms."}]},{"type":"text","order":2,"title":"Read the regression coefficients","content":"From the calculator output screen, read the values for the y-intercept ($a$) and the slope ($b$):\n\n$$a \\approx 601.964...$$\n\n$$b \\approx -2.3714...$$\n\nRounding these to 3 significant figures gives:\n* $a \\approx 602$\n* $b \\approx -2.37$"},{"type":"text","order":3,"title":"Write the final equation","content":"Substitute the values of $a$ and $b$ back into the form $S = a + bR$:\n\n$$S = 602 - 2.37R$$\n\n* **Mark 1 (M1):** Finding the slope $b \\approx -2.37$ using the correct statistical calculations.\n* **Mark 2 (M1):** Finding the intercept $a \\approx 602$.\n* **Mark 3 (A1):** Writing down the final regression equation with the correct variables ($S$ and $R$)."}]},{"id":"d5920614-177d-40b5-bfd7-8d65778650bb:direct_interpretation","label":"(e) Direct Statistical Interpretation","steps":[{"type":"text","order":0,"title":"Recall the value of b","content":"From the equation of the least squares regression line of $S$ on $R$ in the form:\n\n$$S = a + bR$$\n\nwe found the gradient (slope) to be:\n\n$$b \\approx -2.37$$\n\nHere, $b$ represents how the dependent variable, ultimate tensile strength $S$ (in $\\text{MPa}$), changes in response to the independent variable, cooling rate $R$ (in $^{\\circ}\\text{C/min}$)."},{"type":"text","order":1,"title":"Define gradient statistically","content":"Statistically, the gradient $b$ represents the average change in the dependent variable for every single-unit increase in the independent variable.\n\nIn this context:\n- **Independent variable ($R$):** Cooling rate, measured in $^{\\circ}\\text{C/min}$\n- **Dependent variable ($S$):** Ultimate tensile strength, measured in $\\text{MPa}$"},{"type":"text","order":2,"title":"Interpret the negative sign","content":"Since $b = -2.37$ is negative, it indicates a negative relationship between the variables:\n\nAs the cooling rate $R$ **increases**, the ultimate tensile strength $S$ is expected to **decrease**."},{"type":"text","order":3,"title":"Write the final interpretation","content":"Combining these ideas, we state the final interpretation of $b$ as follows:\n\nFor every $1^{\\circ}\\text{C/min}$ increase in the cooling rate, the ultimate tensile strength is estimated to decrease by $2.37\\text{ MPa}$."}]},{"id":"212acb07-2cf4-43bd-8be6-e5d9c481e8d4:gradient_method","label":"(f) Using Regression Slope","steps":[{"type":"text","order":0,"title":"Identify the goal","content":"We want to estimate the decrease in tensile strength, $S$, when the cooling rate, $R$, increases from $25^\\circ\\text{C/min}$ to $75^\\circ\\text{C/min}$.\n\nFirst, let's find the change in the cooling rate, which we can call $\\Delta R$:\n\n$$\\Delta R = 75 - 25 = 50^\\circ\\text{C/min}$$","highlights":[{"text":"moving from a cooling rate of 25 \$^{\\circ}\\text{C/min}\$ to a cooling rate of 75 \$^{\\circ}\\text{C/min}\$","location":"specification"}]},{"type":"text","order":1,"title":"Calculate summary statistics","content":"To find the slope $b$ of the regression line $S = a + bR$, we first calculate the sum of products $S_{RS}$ and sum of squares $S_{RR}$.\n\nThe formulas are:\n\n$$S_{RR} = \\sum R^2 - \\frac{(\\sum R)^2}{n}$$\n\n$$S_{RS} = \\sum RS - \\frac{(\\sum R)(\\sum S)}{n}$$\n\nWith $n = 8$ and our given summary values:\n- $\\sum R = 360$\n- $\\sum S = 3962$\n- $\\sum R^2 = 20400$\n- $\\sum RS = 168330$\n\nNow we calculate $S_{RR}$:\n\n$$S_{RR} = 20400 - \\frac{360^2}{8} = 20400 - 16200 = 4200$$\n\nAnd we calculate $S_{RS}$:\n\n$$S_{RS} = 168330 - \\frac{360 \\times 3962}{8} = 168330 - 178290 = -9960$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$b = \\frac{\\htmlData{anchor=srs-val}{\\color{@sky}{S_{RS}}}}{\\htmlData{anchor=srr-val}{\\color{@amber}{S_{RR}}}}$"},{"latex":"$$b = \\frac{\\htmlData{anchor=srs-num}{\\color{@sky}{-9960}}}{\\htmlData{anchor=srr-num}{\\color{@amber}{4200}}} \\approx \\htmlData{anchor=slope-val}{\\color{@purple}{-2.3714}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"srs-val"},{"color":"amber","style":"text-color","anchor":"srr-val"},{"color":"purple","style":"box","anchor":"slope-val"}],"connections":[{"to":"srs-num","from":"srs-val","color":"sky","label":"substitute","style":"solid"},{"to":"srr-num","from":"srr-val","color":"amber","label":"substitute","style":"solid"}]},"order":2,"title":"Calculate the regression slope","description":"The slope represents the rate of change of tensile strength per unit increase in cooling rate. A value of $-2.3714$ means that for every $1^\\circ\\text{C/min}$ increase in cooling rate, the tensile strength is estimated to decrease by $2.3714\\text{ MPa}$."},{"type":"text","order":3,"title":"Estimate the decrease","content":"To find the estimated decrease, we multiply the rate of decrease (the magnitude of the slope $b$) by our change in cooling rate, $\\Delta R = 50$:\n\n$$\\text{Estimated decrease} = 50 \\times 2.3714 \\approx 118.57\\text{ MPa}$$\n\nRounding to 3 significant figures gives:\n\n$$\\text{Estimated decrease} \\approx 119\\text{ MPa}$$"}]},{"id":"212acb07-2cf4-43bd-8be6-e5d9c481e8d4:full_equation_method","label":"(f) Evaluating the Regression Equation","steps":[{"type":"text","order":0,"title":"Calculate regression slope","content":"We first find the gradient $b$ of the least-squares regression line $S = a + bR$ using the formula:\n\n$$b = \\frac{S_{RS}}{S_{RR}}$$\n\nUsing the summary values provided:\n- $S_{RS} = 168330 - \\frac{360 \\times 3962}{8} = -9960$\n- $S_{RR} = 20400 - \\frac{360^2}{8} = 4200$\n\nSubstituting these values gives:\n\n$$b = \\frac{-9960}{4200} \\approx -2.3714$$"},{"type":"text","order":1,"title":"Calculate intercept","content":"Now we compute the intercept $a$ using the means of $S$ and $R$:\n\n$$\\bar{R} = \\frac{360}{8} = 45$$\n\n$$\\bar{S} = \\frac{3962}{8} = 495.25$$\n\nSubstitute these into the equation $a = \\bar{S} - b\\bar{R}$:\n\n$$a = 495.25 - (-2.3714 \\times 45) \\approx 601.964$$\n\nThis gives us our full regression line equation:\n\n$$S = 601.964 - 2.3714R$$"},{"type":"text","order":2,"title":"Predict strength at 25","content":"Using our regression equation, let's predict the tensile strength when $R = 25^\\circ\\text{C/min}$:\n\n$$S_{25} = 601.964 - 2.3714 \\times 25 \\approx 542.68\\text{ MPa}$$"},{"type":"text","order":3,"title":"Predict strength at 75","content":"Similarly, we find the predicted tensile strength when the cooling rate is $R = 75^\\circ\\text{C/min}$:\n\n$$S_{75} = 601.964 - 2.3714 \\times 75 \\approx 424.11\\text{ MPa}$$"},{"type":"text","order":4,"title":"Find the difference","content":"Now we calculate the estimated decrease by finding the difference between our two predictions:\n\n$$\\text{Decrease} = S_{25} - S_{75}$$\n\n$$\\text{Decrease} = 542.68 - 424.11 \\approx 118.57\\text{ MPa}$$\n\nRounding to 3 significant figures, we get an estimated decrease of **$119\\text{ 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2.5 Data presentation and interpretation

Question 52

2.5 Data presentation and interpretation Questions

2.5 Data presentation and interpretation

Question 52

2.5 Data presentation and interpretation Questions