89:"UserProgressProvider" 8a:I[651451,["$1","$2","$3","$4","$5","$6","$7","$8","$9","$a","$b","$c","$d","$e","$f","$10","$11","$12","$13","$19","$1a","$1b","$1c","$1d","$1e","$1f","$20","$21","$22","$23","$24","$25","$26","$27","$28","$51","$52","$53","$54","$55","$56","$57","$58","$59","$5a","$5b","$5c","$5d","$5e","$5f","$60","$61","$62","$63","$64","$65","$66","$67","$68","$69","$6a","$6b","$6c","$6d","$6e"],"$89"] 8b:I[196793,["$1","$2","$3","$4","$5","$6","$7","$8","$9","$a","$b","$c","$d","$e","$f","$10","$11","$12","$13","$19","$1a","$1b","$1c","$1d","$1e","$1f","$20","$21","$22","$23","$24","$25","$26","$27","$28","$51","$52","$53","$54","$55","$56","$57","$58","$59","$5a","$5b","$5c","$5d","$5e","$5f","$60","$61","$62","$63","$64","$65","$66","$67","$68","$69","$6a","$6b","$6c","$6d","$6e"],"Gingerbread"] 8d:"TopicRecentlyViewed" 8e:I[716297,["$1","$2","$3","$4","$5","$6","$7","$8","$9","$a","$b","$c","$d","$e","$f","$10","$11","$12","$13","$19","$1a","$1b","$1c","$1d","$1e","$1f","$20","$21","$22","$23","$24","$25","$26","$27","$28","$51","$52","$53","$54","$55","$56","$57","$58","$59","$5a","$5b","$5c","$5d","$5e","$5f","$60","$61","$62","$63","$64","$65","$66","$67","$68","$69","$6a","$6b","$6c","$6d","$6e"],"$8d"] 8f:"RecordResourceResume" 90:I[647935,["$1","$2","$3","$4","$5","$6","$7","$8","$9","$a","$b","$c","$d","$e","$f","$10","$11","$12","$13","$19","$1a","$1b","$1c","$1d","$1e","$1f","$20","$21","$22","$23","$24","$25","$26","$27","$28","$51","$52","$53","$54","$55","$56","$57","$58","$59","$5a","$5b","$5c","$5d","$5e","$5f","$60","$61","$62","$63","$64","$65","$66","$67","$68","$69","$6a","$6b","$6c","$6d","$6e"],"$8f"] 91:"TopicQuestionBank" 92:I[426249,["$1","$2","$3","$4","$5","$6","$7","$8","$9","$a","$b","$c","$d","$e","$f","$10","$11","$12","$13","$19","$1a","$1b","$1c","$1d","$1e","$1f","$20","$21","$22","$23","$24","$25","$26","$27","$28","$51","$52","$53","$54","$55","$56","$57","$58","$59","$5a","$5b","$5c","$5d","$5e","$5f","$60","$61","$62","$63","$64","$65","$66","$67","$68","$69","$6a","$6b","$6c","$6d","$6e"],"$91"] 93:T43f,To calculate Spearman's rank correlation coefficient, we first need to assign ranks to the data values for both variables: Distance ($D$) and Price ($P$). We can rank them either in ascending or descending order, as long as we are consistent for both variables. Let's rank both in **ascending order** (from smallest to largest): * **For Distance ($D$):** The values are already ordered from smallest ($1.5$) to largest ($9.5$). | Office Block | A | B | C | D | E | F | G | H | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | **Distance $D$ (km)** | 1.5 | 2.4 | 3.6 | 4.2 | 5.9 | 6.8 | 8.1 | 9.5 | | **Rank of $D$** | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | * **For Price ($P$):** The smallest price is $48$ (Rank 1), and the largest is $94$ (Rank 8). | Office Block | A | B | C | D | E | F | G | H | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | **Price $P$ (£100s)** | 94 | 82 | 88 | 70 | 75 | 60 | 52 | 48 | | **Rank of $P$** | 8 | 6 | 7 | 4 | 5 | 3 | 2 | 1 | *Award 1 mark ($M1$) for correctly ranking at least one variable.*94:T409,**Conclusion (Part 2):** There is sufficient evidence at the 5% level of significance to support the claim that there is a negative correlation between the distance from the city center and office rental price. *Award 1 mark ($A1$) for a clear, contextualized conclusion.* --- **Comment on the developer's claim (Part 3):** The developer claims that moving an office building closer to the city center will *automatically* increase its value. * **Correlation does not imply causation:** A strong correlation between distance and price does not mean that distance is the direct cause of the price difference. * **Other factors:** Other variables such as building age, size, local amenities, transport links, and overall condition of the property also play a major role in determining rental prices. Simply relocating a building does not guarantee an increase in its market value. *Award 1 mark ($C1$) for explaining that correlation does not equal causation.* *Award 1 mark ($C1$) for suggesting other relevant contextual factors.*95:T4ff,Let's rank both datasets. We can rank them either in ascending order (smallest value gets rank 1) or descending order, as long as we are consistent. Let's use **ascending order** (rank 1 for the smallest value, rank 8 for the largest): ### 1. Ranking Distance $D$ (km) The values are already ordered from smallest to largest: * $1.5 \rightarrow \mathbf{1}$ * $2.4 \rightarrow \mathbf{2}$ * $3.6 \rightarrow \mathbf{3}$ * $4.2 \rightarrow \mathbf{4}$ * $5.9 \rightarrow \mathbf{5}$ * $6.8 \rightarrow \mathbf{6}$ * $8.1 \rightarrow \mathbf{7}$ * $9.5 \rightarrow \mathbf{8}$ ### 2. Ranking Price $P$ (£100s) Sorting from smallest to largest: * $48$ is 1st $\rightarrow \mathbf{1}$ * $52$ is 2nd $\rightarrow \mathbf{2}$ * $60$ is 3rd $\rightarrow \mathbf{3}$ * $70$ is 4th $\rightarrow \mathbf{4}$ * $75$ is 5th $\rightarrow \mathbf{5}$ * $82$ is 6th $\rightarrow \mathbf{6}$ * $88$ is 7th $\rightarrow \mathbf{7}$ * $94$ is 8th $\rightarrow \mathbf{8}$ Now, map these ranks back to the original order of the office blocks (A to H): | Office Block | A | B | C | D | E | F | G | H | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | **Rank of $D$ ($R_D$)** | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | **Rank of $P$ ($R_P$)** | 8 | 6 | 7 | 4 | 5 | 3 | 2 | 1 |96:T43f,To calculate Spearman's rank correlation coefficient, we first need to assign ranks to the data values for both variables: Distance ($D$) and Price ($P$). We can rank them either in ascending or descending order, as long as we are consistent for both variables. Let's rank both in **ascending order** (from smallest to largest): * **For Distance ($D$):** The values are already ordered from smallest ($1.5$) to largest ($9.5$). | Office Block | A | B | C | D | E | F | G | H | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | **Distance $D$ (km)** | 1.5 | 2.4 | 3.6 | 4.2 | 5.9 | 6.8 | 8.1 | 9.5 | | **Rank of $D$** | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | * **For Price ($P$):** The smallest price is $48$ (Rank 1), and the largest is $94$ (Rank 8). | Office Block | A | B | C | D | E | F | G | H | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | **Price $P$ (£100s)** | 94 | 82 | 88 | 70 | 75 | 60 | 52 | 48 | | **Rank of $P$** | 8 | 6 | 7 | 4 | 5 | 3 | 2 | 1 | *Award 1 mark ($M1$) for correctly ranking at least one variable.*97:T409,**Conclusion (Part 2):** There is sufficient evidence at the 5% level of significance to support the claim that there is a negative correlation between the distance from the city center and office rental price. *Award 1 mark ($A1$) for a clear, contextualized conclusion.* --- **Comment on the developer's claim (Part 3):** The developer claims that moving an office building closer to the city center will *automatically* increase its value. * **Correlation does not imply causation:** A strong correlation between distance and price does not mean that distance is the direct cause of the price difference. * **Other factors:** Other variables such as building age, size, local amenities, transport links, and overall condition of the property also play a major role in determining rental prices. Simply relocating a building does not guarantee an increase in its market value. *Award 1 mark ($C1$) for explaining that correlation does not equal causation.* *Award 1 mark ($C1$) for suggesting other relevant contextual factors.*98:T4ff,Let's rank both datasets. We can rank them either in ascending order (smallest value gets rank 1) or descending order, as long as we are consistent. Let's use **ascending order** (rank 1 for the smallest value, rank 8 for the largest): ### 1. Ranking Distance $D$ (km) The values are already ordered from smallest to largest: * $1.5 \rightarrow \mathbf{1}$ * $2.4 \rightarrow \mathbf{2}$ * $3.6 \rightarrow \mathbf{3}$ * $4.2 \rightarrow \mathbf{4}$ * $5.9 \rightarrow \mathbf{5}$ * $6.8 \rightarrow \mathbf{6}$ * $8.1 \rightarrow \mathbf{7}$ * $9.5 \rightarrow \mathbf{8}$ ### 2. Ranking Price $P$ (£100s) Sorting from smallest to largest: * $48$ is 1st $\rightarrow \mathbf{1}$ * $52$ is 2nd $\rightarrow \mathbf{2}$ * $60$ is 3rd $\rightarrow \mathbf{3}$ * $70$ is 4th $\rightarrow \mathbf{4}$ * $75$ is 5th $\rightarrow \mathbf{5}$ * $82$ is 6th $\rightarrow \mathbf{6}$ * $88$ is 7th $\rightarrow \mathbf{7}$ * $94$ is 8th $\rightarrow \mathbf{8}$ Now, map these ranks back to the original order of the office blocks (A to H): | Office Block | A | B | C | D | E | F | G | H | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | **Rank of $D$ ($R_D$)** | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | **Rank of $P$ ($R_P$)** | 8 | 6 | 7 | 4 | 5 | 3 | 2 | 1 |72:["$","$L8a",null,{"bundle":{"topicIds":["ab5d3401-bbd2-443e-8b86-dfca29fff489","724d8cf5-191f-4af5-adf8-943269cbae70","2e78a1f2-b374-4de1-bd84-82bb5e4fdfc2","7a582e41-7d4b-40a4-b97e-8af1bc73957e","cfb2f92a-baf1-4f2c-8812-3993aec2c0b0","61531822-c179-4da2-8077-7b4458ea8d90"],"paperIds":[],"stageIds":[],"stageTopicIds":[],"stageAssessmentIds":[],"topicNoteIds":[]},"children":[["$","$L8b",null,{"feature":"question-bank"}],"$L8c",["$","$L8e",null,{"subjectCode":"a_level_maths__ocr_mei","subjectName":"Maths","topicSlug":"statistics","topicTitle":"2 Statistics","sectionTitle":null,"activity":"worksheet","resourceTitle":"2 Statistics Questions","resourceHref":"/a-level/maths/ocr/mei/statistics/questions/q1","saveTrail":false}],["$","$L90",null,{"href":"/a-level/maths/ocr/mei/statistics/questions/q1","label":"2 Statistics","subjectCode":"a_level_maths__ocr_mei","surface":"questions"}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"LearningResource\",\"name\":\"Question 308 – 2 Statistics - A Level Maths Questions\",\"description\":\"Practise 2 Statistics A Level Maths exam-style questions with worked solutions on MathsGenie.\",\"learningResourceType\":\"Practice questions\",\"educationalLevel\":\"A Level\",\"about\":{\"@type\":\"Thing\",\"name\":\"2 Statistics\"},\"url\":\"https://mathsgenie.co.uk/a-level/maths/ocr/mei/statistics/questions/q308\",\"inLanguage\":\"en-GB\",\"provider\":{\"@id\":\"https://mathsgenie.co.uk/#organization\"}}"}}],["$","div",null,{"className":"min-h-0 w-full flex-1 overflow-hidden bg-background","children":["$","$L92",null,{"backHref":"/a-level/maths/ocr/mei/questions","basePath":"/a-level/maths/ocr/mei/statistics/questions","formulaUrl":null,"initialIndex":307,"initialQuestionWindow":[{"id":"1173f323-e9e4-4464-9352-babe67c3c2eb","revisionId":"87e568d2-7060-4aa8-9a5e-414767250e60","questionNumber":null,"questionType":"LA","specification":"Events $A$ and $B$ satisfy $P(A)=\\frac3{20}$, $P(A\\cap B)=\\frac1{20}$ and $P(A\\mid B)=\\frac5{14}$.","difficultyBand":"hard","tier":null,"totalMarks":9,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"1b1ad072-78cf-4558-b44b-72eac1787480","label":"a","content":"Find $P(B)$.","metadata":null,"markscheme":"- $P(B)=\\dfrac{P(A\\cap B)}{P(A\\mid B)}$. M1\n- $P(B)=\\frac1{20}\\div\\frac5{14}$. M1\n- $P(B)=\\frac7{50}$. A1","marks":3,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]},{"id":"140cd82e-fbc9-4b3e-aa0e-0c4853877d15","label":"b","content":"Find $P(A\\cup B)$.","metadata":null,"markscheme":"- $P(A\\cup B)=\\frac3{20}+\\frac7{50}-\\frac1{20}$. M1\n- $P(A\\cup B)=\\frac{24}{100}$. M1\n- $P(A\\cup B)=\\frac6{25}$. A1","marks":3,"order":1,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]},{"id":"83aaac33-8edb-4195-a280-7e2af85e7d92","label":"c","content":"Find $P(B\\mid A')$.","metadata":null,"markscheme":"- $P(B\\cap A')=\\frac7{50}-\\frac1{20}=\\frac9{100}$. M1\n- $P(A')=1-\\frac3{20}=\\frac{17}{20}$. M1\n- $P(B\\mid A')=\\dfrac{9/100}{17/20}=\\frac9{85}$. A1","marks":3,"order":2,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"11fd8de4-cf73-4540-97bc-cfe5983eb30c","revisionId":"47a4ac37-ab19-477f-b639-a97ff132e86c","questionNumber":null,"questionType":"LA","specification":"Osman employs 150 people and wants a systematic sample of 30 to investigate views about the staff canteen.","difficultyBand":"hard","tier":null,"totalMarks":3,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"6f6dcc6a-210a-4cc4-8866-8af63717d6a8","label":null,"content":"Describe how Osman should select the sample.","metadata":null,"markscheme":"- Calculate the interval $150/30=5$ B1\n- Choose a random starting position from 1 to 5 M1\n- Select that employee and every fifth employee thereafter from a complete list A1","marks":3,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"130278fc-351d-4522-9bdf-f0d8540321af","revisionId":"53da78b0-7f83-486e-99cf-ab3d73e23106","questionNumber":null,"questionType":"LA","specification":"An urban planner is analyzing the market value of office rentals in a growing metropolitan area. He suspects that as the distance from the central business district increases, the average rental price per square meter decreases. He collects data from 8 randomly selected office blocks. The results for distance ($D$ km) and rental price ($P$ in £100s per m$^2$) are shown in the table below:\n\n| Office Block | A | B | C | D | E | F | G | H |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| Distance $D$ (km) | 1.5 | 2.4 | 3.6 | 4.2 | 5.9 | 6.8 | 8.1 | 9.5 |\n| Price $P$ (£100s) | 94 | 82 | 88 | 70 | 75 | 60 | 52 | 48 |","difficultyBand":"hard","tier":null,"totalMarks":10,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"64506f36-d24a-4a48-90a9-409a1065dbb9","label":"a","content":"Calculate Spearman's rank correlation coefficient for these data.","metadata":null,"markscheme":"- Assigns ranks correctly for at least one variable. For $D$ (ascending): 1, 2, 3, 4, 5, 6, 7, 8. For $P$ (ascending): 8, 6, 7, 4, 5, 3, 2, 1 M1\n- Calculates the differences $d$ and their squares $d^2$: $49, 16, 16, 0, 0, 9, 25, 49$ M1\n- Calculates $\\sum d^2 = 164$ M1\n- Uses $r_s = 1 - \\frac{6 \\times 164}{8(8^2 - 1)}$ to find $r_s \\approx -0.952$ A1","marks":4,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"manual_formula","label":"Manual Calculation with Formula","steps":[{"type":"text","order":0,"title":"Rank the data","content":"$93","highlights":[{"text":"Distance D (km) | 1.5 | 2.4 | 3.6 | 4.2 | 5.9 | 6.8 | 8.1 | 9.5","location":"specification"},{"text":"Price P (£100s) | 94 | 82 | 88 | 70 | 75 | 60 | 52 | 48","location":"specification"}]},{"type":"text","order":1,"title":"Calculate differences and squares","content":"Next, we calculate the difference $d$ between the ranks of $D$ and $P$ for each office block ($d = \\text{Rank of } D - \\text{Rank of } P$) and then square each difference ($d^2$):\n\n| Office Block | A | B | C | D | E | F | G | H |\n| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |\n| **Rank of $D$** | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |\n| **Rank of $P$** | 8 | 6 | 7 | 4 | 5 | 3 | 2 | 1 |\n| **Difference $d$** | -7 | -4 | -4 | 0 | 0 | 3 | 5 | 7 |\n| **Difference $d^2$** | 49 | 16 | 16 | 0 | 0 | 9 | 25 | 49 |\n\n*Award 1 mark ($M1$) for calculating the values of $d$ and $d^2$ correctly.*"},{"type":"text","order":2,"title":"Sum the squared differences","content":"Now, we find the sum of all the squared differences, denoted as $\\sum d^2$:\n\n$$\\sum d^2 = 49 + 16 + 16 + 0 + 0 + 9 + 25 + 49 = 164$$\n\n*Award 1 mark ($M1$) for finding the correct sum of $164$.*"},{"type":"text","order":3,"title":"Apply Spearman's formula","content":"We use the Spearman's rank correlation coefficient formula:\n\n$$r_s = 1 - \\frac{6 \\sum d^2}{n(n^2 - 1)}$$\n\nHere, the number of data pairs is $n = 8$, and $\\sum d^2 = 164$. Substitute these values into the formula:\n\n$$\\begin{aligned} r_s &= 1 - \\frac{6 \\times 164}{8(8^2 - 1)} \\\\ &= 1 - \\frac{984}{8(64 - 1)} \\\\ &= 1 - \\frac{984}{8(63)} \\\\ &= 1 - \\frac{984}{504} \\\\ &= 1 - 1.95238... \\\\ &\\approx -0.952 \\text{ (to 3 s.f.)} \\end{aligned}$$\n\n*Award 1 mark ($A1$) for calculating the final value of $r_s \\approx -0.952$.*"},{"type":"text","order":4,"title":"State hypotheses","content":"For Part 2, we need to test whether there is evidence of a negative correlation between distance and rental price.\n\nLet $\\rho$ represent the population correlation coefficient. We state our null and alternative hypotheses:\n* Null Hypothesis, $H_0: \\rho = 0$\n* Alternative Hypothesis, $H_1: \\rho < 0$\n\n*Award 1 mark ($B1$) for stating the hypotheses correctly.*"},{"type":"text","order":5,"title":"Critical value and test","content":"Next, we find the critical value and perform the test at the 5% level of significance:\n* For a sample size of $n = 8$ and a one-tailed test at the $5\\%$ level of significance, the critical value for Spearman's rank correlation coefficient is $-0.6429$ (since we are testing for a negative correlation).\n* Our critical region is $r_s \\le -0.6429$.\n\nNow, we compare our calculated test statistic to this critical value:\n$$-0.952 < -0.6429$$\n\nSince our test statistic falls within the critical region, the result is significant. We reject the null hypothesis $H_0$.\n\n*Award 1 mark ($B1$) for stating the critical value of $-0.6429$.*\n*Award 1 mark ($M1$) for comparing the test statistic to the critical value.*"},{"type":"text","order":6,"title":"Conclusions and evaluation","content":"$94"}]},{"id":"graphing_calculator","label":"Using a Graphing Calculator","steps":[{"type":"text","order":0,"title":"Understand the method","content":"To calculate Spearman's rank correlation coefficient ($r_s$) using a graphing calculator, we follow two main phases:\n\n1. **Rank the data manually**: Assign ranks to both variables ($D$ and $P$).\n2. **Use the calculator**: Enter these ranks into two separate lists in our Graphic Display Calculator (GDC) and calculate the standard Pearson product-moment correlation coefficient ($r$) on the ranks. Since Spearman's coefficient is mathematically equivalent to the Pearson correlation coefficient of the ranked data, this directly yields $r_s$."},{"type":"text","order":1,"title":"Rank the data","content":"$95"},{"type":"text","order":2,"title":"Input ranks into GDC","content":"Now we will enter the rank pairs into our calculator. Let list 1 contain the ranks of $D$ ($R_D$):\n$$\\{1, 2, 3, 4, 5, 6, 7, 8\\}$$\n\nAnd let list 2 contain the corresponding ranks of $P$ ($R_P$):\n$$\\{8, 6, 7, 4, 5, 3, 2, 1\\}$$","calculator":[{"gdc":{"instructions":"1. Open the Statistics menu (or Spreadsheet app).\n2. Input the ranks of D into List 1 and the ranks of P into List 2.\n3. Navigate to Calc -> Regression -> Linear Reg (ax+b) (often represented as X).\n4. Select List 1 as X-List and List 2 as Y-List, then calculate."},"ti84":{"instructions":"1. Press [STAT] and select 1:Edit...\n2. Input the ranks of D in L1: {1, 2, 3, 4, 5, 6, 7, 8}.\n3. Input the ranks of P in L2: {8, 6, 7, 4, 5, 3, 2, 1}.\n4. Press [STAT], arrow over to CALC, and select 4:LinReg(ax+b).\n5. Ensure Xlist is L1, Ylist is L2, and select Calculate. (If r value does not show, press [2nd][CATALOG], scroll to 'DiagnosticOn', and press [ENTER] twice)."},"desmos":{"expression":"y_1 \\sim m x_1 + c","instructions":"1. Click the '+' button in the top left and select 'Table'.\n2. In the x1 column, enter the ranks: 1, 2, 3, 4, 5, 6, 7, 8.\n3. In the y1 column, enter the ranks: 8, 6, 7, 4, 5, 3, 2, 1.\n4. In an empty cell below, type the regression model: y1 ~ m*x1 + c. Look for the 'r' value under Statistics."},"description":"Enter the lists and calculate 2-Variable Statistics or Linear Regression."}]},{"type":"text","order":3,"title":"Obtain the final value","content":"From the calculator's regression output, we obtain the correlation coefficient $r$, which represents Spearman's rank correlation coefficient $r_s$ for this ranked data:\n\n$$r_s \\approx -0.952 \\quad (3\\text{ sig. figs.})$$\n\nThis strong negative value confirms that as distance from the central business district increases, the rental price decreases."},{"type":"text","order":4,"title":"Hypothesis testing","content":"With $r_s \\approx -0.952$ calculated, we now conduct the hypothesis test:\n\n1. **State the hypotheses:**\n * Null hypothesis: $H_0: \\rho = 0$\n * Alternative hypothesis: $H_1: \\rho < 0$ (where $\\rho$ is the population correlation coefficient)\n\n2. **Identify the critical value:**\n For $n = 8$ at the $5\\%$ level of significance (one-tailed), the critical value is $-0.6429$.\n\n3. **Compare and conclude:**\n Since our test statistic $-0.952$ is less than the critical value $-0.6429$, we reject $H_0$. There is significant evidence at the 5% level of significance to suggest a negative correlation between distance from the city center and office rental price."},{"type":"text","order":5,"title":"Evaluate the claim","content":"For Part 3, we evaluate the developer's claim that moving an office building 2 km closer to the city center will automatically increase its value:\n\n* **Correlation is not causation:** Although there is a strong negative correlation, it does not mean that distance itself directly causes the change in price.\n* **Confounding variables:** Other factors such as the age of the building, size, facilities, and proximity to transport links are also critical in determining rental value. Relocating a building does not guarantee an automatic increase in value if these other factors are unfavorable."}]}],"generatedAt":"2026-06-09T03:24:33.508Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"da0cbe91-6f7b-4efc-b1db-3d8f7f136a5a","label":"b","content":"Stating your hypotheses clearly, test at the 5% level of significance whether there is evidence of a negative correlation between distance from the city center and office rental price. State the critical value used.","metadata":null,"markscheme":"- States null hypothesis $H_0: \\rho = 0$ and alternative hypothesis $H_1: \\rho < 0$ (where $\\rho$ is the population correlation coefficient) B1\n- Identifies the critical value for $n=8$ at the 5% level of significance (one-tailed) as $-0.6429$ B1\n- Compares the test statistic to the critical value: $-0.952 < -0.6429$ M1\n- Concludes that the result is significant, rejects $H_0$, and states there is evidence of a negative correlation between distance and price A1","marks":4,"order":1,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"critical_value","label":"Critical Value Comparison","steps":[{"type":"text","order":0,"title":"State the hypotheses","content":"To test for negative correlation between the distance from the city center ($D$) and the office rental price ($P$), we state our hypotheses in terms of the population correlation coefficient, $\\rho$.\n\n- The null hypothesis ($H_0$) represents no correlation.\n- The alternative hypothesis ($H_1$) represents the suspected negative correlation.\n\n$$\\begin{aligned} H_0 &: \\rho = 0 \\\\ H_1 &: \\rho < 0 \\end{aligned}$$\n\nWhere $\\rho$ is the product moment correlation coefficient of the population.","highlights":[{"text":"Stating your hypotheses clearly, test at the 5% level of significance whether there is evidence of a negative correlation","location":"specification"}]},{"type":"text","order":1,"title":"Calculate the sample correlation","content":"Using a scientific or graphing calculator, enter the bivariate data for $D$ and $P$ to calculate the product moment correlation coefficient, $r$. \n\nInputting the 8 pairs of data gives our test statistic:\n$$r \\approx -0.9523$$\n\nTo three decimal places, $r = -0.952$.","calculator":[{"gdc":{"instructions":"1. Go to 'Statistics' or 'Lists & Spreadsheet'.\n2. Input $D$ into list 1 (or column A) and $P$ into list 2 (or column B).\n3. Perform a 'Linear Regression' or '2-Variable Statistics' calculation to find the value of $r$."},"ti84":{"instructions":"1. Press [STAT] and select '1:Edit...'.\n2. Enter the $D$ values into L1 and the $P$ values into L2.\n3. Press [STAT], arrow over to 'CALC', and select '4:LinReg(ax+b)'.\n4. Ensure Xlist is L1, Ylist is L2, then select 'Calculate'. (If $r$ does not show, press [2nd][CATALOG], scroll to 'DiagnosticOn', and press [ENTER] twice)."},"desmos":{"instructions":"1. Add a table and input the $D$ values into $x_1$ and $P$ values into $y_1$.\n2. In a new expression line, type: `y_1 ~ m*x_1 + c`.\n3. Desmos will display the correlation coefficient $r$ under the statistics section."},"description":"Enter the bivariate data into lists to find the correlation coefficient $r$."}]},{"type":"text","order":2,"title":"Determine the critical value","content":"Since we are testing for a negative correlation ($H_1: \\rho < 0$), we perform a **one-tailed test** at the **5% significance level** with a sample size of $n = 8$.\n\nLooking up the critical value for the product moment correlation coefficient in the statistical tables for $n = 8$ at the $5\\%$ level (one-tailed) gives $0.6429$.\n\nSince we are testing for a *negative* correlation, the critical value is negative:\n$$\\text{Critical Value} = -0.6429$$","highlights":[{"text":"test at the 5% level of significance","location":"specification"},{"text":"State the critical value used.","location":"specification"}]},{"type":"text","order":3,"title":"Compare and conclude","content":"We now compare our sample correlation coefficient $r = -0.952$ to the critical value $-0.6429$.\n\nSince $r$ is more extreme than the critical value:\n$$-0.952 < -0.6429$$\n\nOur test statistic lies inside the critical region. Therefore, we **reject $H_0$**.\n\nThere is **sufficient evidence** at the 5% significance level to conclude that there is a significant negative correlation between the distance from the city center and the office rental price."}]},{"id":"p_value","label":"P-value Comparison","steps":[{"type":"text","order":0,"title":"State the Hypotheses","content":"We begin by stating our hypotheses. We want to test if there is a negative correlation between the distance from the city center ($D$) and the rental price ($P$). \n\nLet $\\rho$ represent the population correlation coefficient between distance and office rental price.\n\n$$\\begin{aligned} H_0 &: \\rho = 0 \\\\ H_1 &: \\rho < 0 \\end{aligned}$$\n\nWhere:\n- $H_0$ is the null hypothesis (there is no correlation).\n- $H_1$ is the alternative hypothesis (there is a negative correlation). This is a **one-tailed test**.","highlights":[{"text":"Stating your hypotheses clearly","location":"specification"},{"text":"negative correlation","location":"specification"}]},{"type":"text","order":1,"title":"Calculate the P-value","content":"Using a graphing calculator, we can enter the data for $D$ into list $L_1$ and $P$ into list $L_2$, then run a **Linear Regression $t$-Test** (or finding the Pearson correlation coefficient $r$ and its associated $p$-value) to test the hypothesis $\\beta < 0$ and $\\rho < 0$.\n\nFrom the calculator, we find:\n- The product-moment correlation coefficient: $r \\approx -0.952$\n- The test $p$-value: $P \\approx 0.000133$\n\n$$\\text{P-value} \\approx 0.00013$$\n\n*(Note: Your calculator might also show $t \\approx -7.64$ with $6$ degrees of freedom.)*","calculator":[{"gdc":{"instructions":"1. Go to the Statistics application.\n2. Enter the Distance values in List 1 and Price values in List 2.\n3. Choose Calc / Test / Reg / Linear Reg t-test.\n4. Set the alternative hypothesis to < 0.\n5. Execute the calculation to find the p-value and correlation coefficient r."},"ti84":{"instructions":"1. Press [STAT] and select 1:Edit...\n2. Enter the Distance values in L1: {1.5, 2.4, 3.6, 4.2, 5.9, 6.8, 8.1, 9.5}\n3. Enter the Price values in L2: {94, 82, 88, 70, 75, 60, 52, 48}\n4. Press [STAT], arrow over to [TESTS], and select F:LinRegTTest...\n5. Set Xlist: L1, Ylist: L2, Freq: 1\n6. For $\\beta$ & $\\rho$:, select <0 (since we are testing for a negative correlation)\n7. Highlight Calculate and press [ENTER].\n8. Read the value for p and r."},"desmos":{"instructions":"1. Create a table with columns x1 (Distance) and y1 (Price).\n2. Enter the data points into the table.\n3. In a new cell, type the regression model: y1 ~ m*x1 + c\n4. Desmos will output the correlation coefficient r. To find the p-value for the t-test, use the formula for the t-statistic: t = r * sqrt((n-2)/(1-r^2)) and then calculate the CDF of a t-distribution with df = 6."},"description":"Perform a Linear Regression $t$-test using list data."}]},{"type":"text","order":2,"title":"Find the Critical Value","content":"For a sample size of $n = 8$, at a $5\\%$ level of significance for a one-tailed test of negative correlation, we look up the critical value for the Pearson product-moment correlation coefficient ($r$) in the formula booklet:\n\n$$\\text{Critical Value} = -0.6429$$\n\nAny correlation coefficient $r$ more extreme (i.e., less than $-0.6429$) lies in the critical region.","highlights":[{"text":"State the critical value used.","location":"specification"},{"text":"5% level of significance","location":"specification"}]},{"type":"text","order":3,"title":"Compare and Conclude","content":"Now we compare our calculated $p$-value to the significance level $\\alpha = 0.05$:\n\n$$0.00013 < 0.05$$\n\nSince the $p$-value is significantly less than $0.05$, we have extremely strong evidence to reject the null hypothesis $H_0$.\n\nAlternatively, comparing our calculated test statistic $r = -0.952$ to the critical value $-0.6429$:\n\n$$-0.952 < -0.6429$$\n\nSince $r$ is in the critical region, we reject $H_0$.\n\n**Conclusion:**\nThere is sufficient evidence at the $5\\%$ level of significance to suggest a significant negative correlation between the distance from the city center and the office rental price."}]}],"generatedAt":"2026-06-09T03:26:02.004Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"838e855f-d4e1-499c-9def-874d3b3c88e3","label":"c","content":"A developer claims that moving an existing office building 2 km closer to the city center will automatically increase its market value. Comment on this claim.","metadata":null,"markscheme":"- Explains that correlation does not imply causation C1\n- Suggests other factors in context (e.g., building age, local amenities, transport links) that might influence the price beyond just distance C1","marks":2,"order":2,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"statistical_interpretation","label":"Statistical Interpretation","steps":[{"type":"text","order":0,"title":"Analyze the claim","content":"The developer claims that moving an existing office building 2 km closer to the city center will **automatically** increase its market value.\n\nTo evaluate this, we need to look at the statistical relationship between distance $D$ and rental price $P$. Although we observe a negative correlation (as distance from the center increases, price generally decreases), we must be careful about how we interpret this relationship.","highlights":[{"text":"moving an existing office building 2 km closer to the city center will automatically increase its market value.","location":"specification"}]},{"type":"text","order":1,"title":"Correlation vs. causation","content":"A fundamental rule of statistics is that **correlation does not imply causation**.\n\nJust because there is a strong correlation between being closer to the city center and having a higher rental price, it does not mean that changing the location of a building will *cause* its value to increase. \n\nThis key insight earns the first mark C1."},{"type":"text","order":2,"title":"Consider confounding factors","content":"In reality, the market value of an office building is influenced by many other confounding factors. Even if a building is physically moved closer to the city center, its price might not increase if these other factors are unfavourable.\n\nSome of these other context-specific factors include:\n- **Age and condition** of the building\n- **Local amenities** and nearby transport links\n- **Size and facilities** of the office space\n\nSuggesting other realistic factors in context earns the second mark C1."}]}],"generatedAt":"2026-06-09T03:36:25.244Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"64506f36-d24a-4a48-90a9-409a1065dbb9:manual_formula","label":"(a) Manual Calculation with Formula","steps":[{"type":"text","order":0,"title":"Rank the data","content":"$96","highlights":[{"text":"Distance D (km) | 1.5 | 2.4 | 3.6 | 4.2 | 5.9 | 6.8 | 8.1 | 9.5","location":"specification"},{"text":"Price P (£100s) | 94 | 82 | 88 | 70 | 75 | 60 | 52 | 48","location":"specification"}]},{"type":"text","order":1,"title":"Calculate differences and squares","content":"Next, we calculate the difference $d$ between the ranks of $D$ and $P$ for each office block ($d = \\text{Rank of } D - \\text{Rank of } P$) and then square each difference ($d^2$):\n\n| Office Block | A | B | C | D | E | F | G | H |\n| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |\n| **Rank of $D$** | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |\n| **Rank of $P$** | 8 | 6 | 7 | 4 | 5 | 3 | 2 | 1 |\n| **Difference $d$** | -7 | -4 | -4 | 0 | 0 | 3 | 5 | 7 |\n| **Difference $d^2$** | 49 | 16 | 16 | 0 | 0 | 9 | 25 | 49 |\n\n*Award 1 mark ($M1$) for calculating the values of $d$ and $d^2$ correctly.*"},{"type":"text","order":2,"title":"Sum the squared differences","content":"Now, we find the sum of all the squared differences, denoted as $\\sum d^2$:\n\n$$\\sum d^2 = 49 + 16 + 16 + 0 + 0 + 9 + 25 + 49 = 164$$\n\n*Award 1 mark ($M1$) for finding the correct sum of $164$.*"},{"type":"text","order":3,"title":"Apply Spearman's formula","content":"We use the Spearman's rank correlation coefficient formula:\n\n$$r_s = 1 - \\frac{6 \\sum d^2}{n(n^2 - 1)}$$\n\nHere, the number of data pairs is $n = 8$, and $\\sum d^2 = 164$. Substitute these values into the formula:\n\n$$\\begin{aligned} r_s &= 1 - \\frac{6 \\times 164}{8(8^2 - 1)} \\\\ &= 1 - \\frac{984}{8(64 - 1)} \\\\ &= 1 - \\frac{984}{8(63)} \\\\ &= 1 - \\frac{984}{504} \\\\ &= 1 - 1.95238... \\\\ &\\approx -0.952 \\text{ (to 3 s.f.)} \\end{aligned}$$\n\n*Award 1 mark ($A1$) for calculating the final value of $r_s \\approx -0.952$.*"},{"type":"text","order":4,"title":"State hypotheses","content":"For Part 2, we need to test whether there is evidence of a negative correlation between distance and rental price.\n\nLet $\\rho$ represent the population correlation coefficient. We state our null and alternative hypotheses:\n* Null Hypothesis, $H_0: \\rho = 0$\n* Alternative Hypothesis, $H_1: \\rho < 0$\n\n*Award 1 mark ($B1$) for stating the hypotheses correctly.*"},{"type":"text","order":5,"title":"Critical value and test","content":"Next, we find the critical value and perform the test at the 5% level of significance:\n* For a sample size of $n = 8$ and a one-tailed test at the $5\\%$ level of significance, the critical value for Spearman's rank correlation coefficient is $-0.6429$ (since we are testing for a negative correlation).\n* Our critical region is $r_s \\le -0.6429$.\n\nNow, we compare our calculated test statistic to this critical value:\n$$-0.952 < -0.6429$$\n\nSince our test statistic falls within the critical region, the result is significant. We reject the null hypothesis $H_0$.\n\n*Award 1 mark ($B1$) for stating the critical value of $-0.6429$.*\n*Award 1 mark ($M1$) for comparing the test statistic to the critical value.*"},{"type":"text","order":6,"title":"Conclusions and evaluation","content":"$97"}]},{"id":"64506f36-d24a-4a48-90a9-409a1065dbb9:graphing_calculator","label":"(a) Using a Graphing Calculator","steps":[{"type":"text","order":0,"title":"Understand the method","content":"To calculate Spearman's rank correlation coefficient ($r_s$) using a graphing calculator, we follow two main phases:\n\n1. **Rank the data manually**: Assign ranks to both variables ($D$ and $P$).\n2. **Use the calculator**: Enter these ranks into two separate lists in our Graphic Display Calculator (GDC) and calculate the standard Pearson product-moment correlation coefficient ($r$) on the ranks. Since Spearman's coefficient is mathematically equivalent to the Pearson correlation coefficient of the ranked data, this directly yields $r_s$."},{"type":"text","order":1,"title":"Rank the data","content":"$98"},{"type":"text","order":2,"title":"Input ranks into GDC","content":"Now we will enter the rank pairs into our calculator. Let list 1 contain the ranks of $D$ ($R_D$):\n$$\\{1, 2, 3, 4, 5, 6, 7, 8\\}$$\n\nAnd let list 2 contain the corresponding ranks of $P$ ($R_P$):\n$$\\{8, 6, 7, 4, 5, 3, 2, 1\\}$$","calculator":[{"gdc":{"instructions":"1. Open the Statistics menu (or Spreadsheet app).\n2. Input the ranks of D into List 1 and the ranks of P into List 2.\n3. Navigate to Calc -> Regression -> Linear Reg (ax+b) (often represented as X).\n4. Select List 1 as X-List and List 2 as Y-List, then calculate."},"ti84":{"instructions":"1. Press [STAT] and select 1:Edit...\n2. Input the ranks of D in L1: {1, 2, 3, 4, 5, 6, 7, 8}.\n3. Input the ranks of P in L2: {8, 6, 7, 4, 5, 3, 2, 1}.\n4. Press [STAT], arrow over to CALC, and select 4:LinReg(ax+b).\n5. Ensure Xlist is L1, Ylist is L2, and select Calculate. (If r value does not show, press [2nd][CATALOG], scroll to 'DiagnosticOn', and press [ENTER] twice)."},"desmos":{"expression":"y_1 \\sim m x_1 + c","instructions":"1. Click the '+' button in the top left and select 'Table'.\n2. In the x1 column, enter the ranks: 1, 2, 3, 4, 5, 6, 7, 8.\n3. In the y1 column, enter the ranks: 8, 6, 7, 4, 5, 3, 2, 1.\n4. In an empty cell below, type the regression model: y1 ~ m*x1 + c. Look for the 'r' value under Statistics."},"description":"Enter the lists and calculate 2-Variable Statistics or Linear Regression."}]},{"type":"text","order":3,"title":"Obtain the final value","content":"From the calculator's regression output, we obtain the correlation coefficient $r$, which represents Spearman's rank correlation coefficient $r_s$ for this ranked data:\n\n$$r_s \\approx -0.952 \\quad (3\\text{ sig. figs.})$$\n\nThis strong negative value confirms that as distance from the central business district increases, the rental price decreases."},{"type":"text","order":4,"title":"Hypothesis testing","content":"With $r_s \\approx -0.952$ calculated, we now conduct the hypothesis test:\n\n1. **State the hypotheses:**\n * Null hypothesis: $H_0: \\rho = 0$\n * Alternative hypothesis: $H_1: \\rho < 0$ (where $\\rho$ is the population correlation coefficient)\n\n2. **Identify the critical value:**\n For $n = 8$ at the $5\\%$ level of significance (one-tailed), the critical value is $-0.6429$.\n\n3. **Compare and conclude:**\n Since our test statistic $-0.952$ is less than the critical value $-0.6429$, we reject $H_0$. There is significant evidence at the 5% level of significance to suggest a negative correlation between distance from the city center and office rental price."},{"type":"text","order":5,"title":"Evaluate the claim","content":"For Part 3, we evaluate the developer's claim that moving an office building 2 km closer to the city center will automatically increase its value:\n\n* **Correlation is not causation:** Although there is a strong negative correlation, it does not mean that distance itself directly causes the change in price.\n* **Confounding variables:** Other factors such as the age of the building, size, facilities, and proximity to transport links are also critical in determining rental value. Relocating a building does not guarantee an automatic increase in value if these other factors are unfavorable."}]},{"id":"da0cbe91-6f7b-4efc-b1db-3d8f7f136a5a:critical_value","label":"(b) Critical Value Comparison","steps":[{"type":"text","order":0,"title":"State the hypotheses","content":"To test for negative correlation between the distance from the city center ($D$) and the office rental price ($P$), we state our hypotheses in terms of the population Spearman rank correlation coefficient, $\\rho$.\n\n- The null hypothesis ($H_0$) represents no rank correlation.\n- The alternative hypothesis ($H_1$) represents the suspected negative rank correlation.\n\n$$\\begin{aligned} H_0 &: \\rho = 0 \\\\ H_1 &: \\rho < 0 \\end{aligned}$$\n\nHere, $\\rho$ is the population Spearman rank correlation coefficient.","highlights":[{"text":"Stating your hypotheses clearly, test at the 5% level of significance whether there is evidence of a negative correlation","location":"specification"}]},{"type":"text","order":1,"title":"Calculate the sample correlation","content":"First rank the two variables in ascending order. In the original order of the office blocks, the ranks are\n\n$$R_D=\\{1,2,3,4,5,6,7,8\\}$$\n\nand\n\n$$R_P=\\{8,6,7,4,5,3,2,1\\}.$$\n\nUsing these ranks in Spearman's formula gives\n\n$$r_s=1-\\frac{6\\sum d^2}{n(n^2-1)}=1-\\frac{6(164)}{8(8^2-1)}\\approx -0.952.$$","calculator":[{"gdc":{"instructions":"1. Go to 'Statistics' or 'Lists & Spreadsheet'.\n2. Input $D$ into list 1 (or column A) and $P$ into list 2 (or column B).\n3. Perform a 'Linear Regression' or '2-Variable Statistics' calculation to find the value of $r$."},"ti84":{"instructions":"1. Press [STAT] and select '1:Edit...'.\n2. Enter the $D$ values into L1 and the $P$ values into L2.\n3. Press [STAT], arrow over to 'CALC', and select '4:LinReg(ax+b)'.\n4. Ensure Xlist is L1, Ylist is L2, then select 'Calculate'. (If $r$ does not show, press [2nd][CATALOG], scroll to 'DiagnosticOn', and press [ENTER] twice)."},"desmos":{"instructions":"1. Add a table and input the $D$ values into $x_1$ and $P$ values into $y_1$.\n2. In a new expression line, type: `y_1 ~ m*x_1 + c`.\n3. Desmos will display the correlation coefficient $r$ under the statistics section."},"description":"Enter the bivariate data into lists to find the correlation coefficient $r$."}]},{"type":"text","order":2,"title":"Determine the critical value","content":"Since we are testing for a negative Spearman rank correlation ($H_1: \\rho < 0$), we perform a **one-tailed test** at the **5% significance level** with a sample size of $n=8$.\n\nLooking up the critical value for Spearman's rank correlation coefficient for $n=8$ at the $5\\%$ one-tailed level gives $0.6429$ in magnitude. Since the test is for a negative correlation,\n\n$$\\text{Critical Value}=-0.6429.$$","highlights":[{"text":"test at the 5% level of significance","location":"specification"},{"text":"State the critical value used.","location":"specification"}]},{"type":"text","order":3,"title":"Compare and conclude","content":"We now compare the sample Spearman rank correlation coefficient $r_s=-0.952$ to the critical value $-0.6429$.\n\nSince\n\n$$-0.952<-0.6429,$$\n\nthe test statistic lies inside the critical region. Therefore, we **reject $H_0$**.\n\nThere is **sufficient evidence** at the 5% significance level to conclude that there is a significant negative rank correlation between the distance from the city center and the office rental price."}]},{"id":"da0cbe91-6f7b-4efc-b1db-3d8f7f136a5a:p_value","label":"(b) P-value Comparison","steps":[{"type":"text","order":0,"title":"State the Hypotheses","content":"We begin by stating our hypotheses. We want to test if there is a negative correlation between the distance from the city center ($D$) and the rental price ($P$). \n\nLet $\\rho$ represent the population correlation coefficient between distance and office rental price.\n\n$$\\begin{aligned} H_0 &: \\rho = 0 \\\\ H_1 &: \\rho < 0 \\end{aligned}$$\n\nWhere:\n- $H_0$ is the null hypothesis (there is no correlation).\n- $H_1$ is the alternative hypothesis (there is a negative correlation). This is a **one-tailed test**.","highlights":[{"text":"Stating your hypotheses clearly","location":"specification"},{"text":"negative correlation","location":"specification"}]},{"type":"text","order":1,"title":"Calculate the P-value","content":"Using the ranked data rather than the original measurements, we obtain\n\n$$r_s\\approx -0.952.$$ \n\nFor an exact one-tailed Spearman test with $n=8$, there are $8!$ equally likely rank permutations under $H_0$. There are 23 permutations with a Spearman coefficient at least as negative as the observed value, so\n\n$$p=\\frac{23}{8!}=\\frac{23}{40320}\\approx 0.000570.$$","calculator":[{"gdc":{"instructions":"1. Go to the Statistics application.\n2. Enter the Distance values in List 1 and Price values in List 2.\n3. Choose Calc / Test / Reg / Linear Reg t-test.\n4. Set the alternative hypothesis to < 0.\n5. Execute the calculation to find the p-value and correlation coefficient r."},"ti84":{"instructions":"1. Press [STAT] and select 1:Edit...\n2. Enter the Distance values in L1: {1.5, 2.4, 3.6, 4.2, 5.9, 6.8, 8.1, 9.5}\n3. Enter the Price values in L2: {94, 82, 88, 70, 75, 60, 52, 48}\n4. Press [STAT], arrow over to [TESTS], and select F:LinRegTTest...\n5. Set Xlist: L1, Ylist: L2, Freq: 1\n6. For $\\beta$ & $\\rho$:, select <0 (since we are testing for a negative correlation)\n7. Highlight Calculate and press [ENTER].\n8. Read the value for p and r."},"desmos":{"instructions":"1. Create a table with columns x1 (Distance) and y1 (Price).\n2. Enter the data points into the table.\n3. In a new cell, type the regression model: y1 ~ m*x1 + c\n4. Desmos will output the correlation coefficient r. To find the p-value for the t-test, use the formula for the t-statistic: t = r * sqrt((n-2)/(1-r^2)) and then calculate the CDF of a t-distribution with df = 6."},"description":"Perform a Linear Regression $t$-test using list data."}]},{"type":"text","order":2,"title":"Find the Critical Value","content":"For a sample size of $n=8$, at a $5\\%$ level of significance for a one-tailed test of negative Spearman rank correlation, the critical value from the Spearman rank correlation tables is\n\n$$\\text{Critical Value}=-0.6429.$$ \n\nAny Spearman coefficient $r_s$ less than $-0.6429$ lies in the critical region.","highlights":[{"text":"State the critical value used.","location":"specification"},{"text":"5% level of significance","location":"specification"}]},{"type":"text","order":3,"title":"Compare and Conclude","content":"Now compare the exact one-tailed $p$-value to the significance level $\\alpha=0.05$:\n\n$$0.000570<0.05.$$ \n\nSince the $p$-value is less than $0.05$, we reject the null hypothesis $H_0$.\n\nAlternatively, comparing the Spearman statistic with the critical value:\n\n$$-0.952<-0.6429.$$ \n\nSince $r_s$ is in the critical region, we reject $H_0$.\n\n**Conclusion:**\nThere is sufficient evidence at the $5\\%$ level of significance to suggest a significant negative rank correlation between the distance from the city center and the office rental price."}]},{"id":"838e855f-d4e1-499c-9def-874d3b3c88e3:statistical_interpretation","label":"(c) Statistical Interpretation","steps":[{"type":"text","order":0,"title":"Analyze the claim","content":"The developer claims that moving an existing office building 2 km closer to the city center will **automatically** increase its market value.\n\nTo evaluate this, we need to look at the statistical relationship between distance $D$ and rental price $P$. Although we observe a negative correlation (as distance from the center increases, price generally decreases), we must be careful about how we interpret this relationship.","highlights":[{"text":"moving an existing office building 2 km closer to the city center will automatically increase its market value.","location":"specification"}]},{"type":"text","order":1,"title":"Correlation vs. causation","content":"A fundamental rule of statistics is that **correlation does not imply causation**.\n\nJust because there is a strong correlation between being closer to the city center and having a higher rental price, it does not mean that changing the location of a building will *cause* its value to increase. \n\nThis key insight earns the first mark C1."},{"type":"text","order":2,"title":"Consider confounding factors","content":"In reality, the market value of an office building is influenced by many other confounding factors. Even if a building is physically moved closer to the city center, its price might not increase if these other factors are unfavourable.\n\nSome of these other context-specific factors include:\n- **Age and condition** of the building\n- **Local amenities** and nearby transport links\n- **Size and facilities** of the office space\n\nSuggesting other realistic factors in context earns the second mark 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2 Statistics

Question 32

2 Statistics Questions

2 Statistics

Question 32

2 Statistics Questions