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Actual Maturation Periods:** * $D$ ($62$ months) $\rightarrow$ Rank $1$ * $A$ ($54$ months) $\rightarrow$ Rank $2$ * $F$ ($45$ months) $\rightarrow$ Rank $3$ * $C$ ($38$ months) $\rightarrow$ Rank $4$ * $H$ ($30$ months) $\rightarrow$ Rank $5$ * $E$ ($24$ months) $\rightarrow$ Rank $6$ * $G$ ($18$ months) $\rightarrow$ Rank $7$ * $B$ ($12$ months) $\rightarrow$ Rank $8$ **2. Sommelier's Estimates:** Since the sommelier's order is already descending ($D, F, C, A, E, H, G, B$): * $D \rightarrow$ Rank $1$ * $F \rightarrow$ Rank $2$ * $C \rightarrow$ Rank $3$ * $A \rightarrow$ Rank $4$ * $E \rightarrow$ Rank $5$ * $H \rightarrow$ Rank $6$ * $G \rightarrow$ Rank $7$ * $B \rightarrow$ Rank $8$ Let's pair these ranks for each wine: $$\begin{array}{|l|c|c|c|c|c|c|c|c|} \hline \text{Wine} & A & B & C & D & E & F & G & H \\ \hline \text{Actual Rank } (x) & 2 & 8 & 4 & 1 & 6 & 3 & 7 & 5 \\ \hline \text{Estimated Rank } (y) & 4 & 8 & 3 & 1 & 5 & 2 & 7 & 6 \\ \hline \end{array}$$94:T4c3,To calculate Spearman's rank correlation coefficient using a GDC, we first assign ranks from $1$ to $8$ to both datasets in **descending order** (from oldest to youngest, where rank $1$ represents the longest maturation period). **1. Actual Maturation Periods:** * $D$ ($62$ months) $\rightarrow$ Rank $1$ * $A$ ($54$ months) $\rightarrow$ Rank $2$ * $F$ ($45$ months) $\rightarrow$ Rank $3$ * $C$ ($38$ months) $\rightarrow$ Rank $4$ * $H$ ($30$ months) $\rightarrow$ Rank $5$ * $E$ ($24$ months) $\rightarrow$ Rank $6$ * $G$ ($18$ months) $\rightarrow$ Rank $7$ * $B$ ($12$ months) $\rightarrow$ Rank $8$ **2. Sommelier's Estimates:** Since the sommelier's order is already descending ($D, F, C, A, E, H, G, B$): * $D \rightarrow$ Rank $1$ * $F \rightarrow$ Rank $2$ * $C \rightarrow$ Rank $3$ * $A \rightarrow$ Rank $4$ * $E \rightarrow$ Rank $5$ * $H \rightarrow$ Rank $6$ * $G \rightarrow$ Rank $7$ * $B \rightarrow$ Rank $8$ Let's pair these ranks for each wine: $$\begin{array}{|l|c|c|c|c|c|c|c|c|} \hline \text{Wine} & A & B & C & D & E & F & G & H \\ \hline \text{Actual Rank } (x) & 2 & 8 & 4 & 1 & 6 & 3 & 7 & 5 \\ \hline \text{Estimated Rank } (y) & 4 & 8 & 3 & 1 & 5 & 2 & 7 & 6 \\ \hline \end{array}$$72:["$","$L8a",null,{"bundle":{"topicIds":["f5df8ac7-231b-4c91-a66c-6df3f975fe72","7d228e5d-8eea-47a6-9f82-66bffcdba53e","6c6ffb21-3ffd-4e54-a8a3-c5d1373db668","67894de2-ce27-4892-bef7-934cf5cedf07","7c2d2967-c1bd-4bb5-ad4a-dba1494405ba","c8b7219d-34d4-4a1b-aadf-1f80ac2cd223"],"paperIds":[],"stageIds":[],"stageTopicIds":[],"stageAssessmentIds":[],"topicNoteIds":[]},"children":[["$","$L8b",null,{"feature":"question-bank"}],"$L8c",["$","$L8e",null,{"subjectCode":"a_level_maths__ocr","subjectName":"Maths","topicSlug":"statistics","topicTitle":"2 Statistics","sectionTitle":null,"activity":"worksheet","resourceTitle":"2 Statistics Questions","resourceHref":"/a-level/maths/ocr/statistics/questions/q1","saveTrail":false}],["$","$L90",null,{"href":"/a-level/maths/ocr/statistics/questions/q1","label":"2 Statistics","subjectCode":"a_level_maths__ocr","surface":"questions"}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"LearningResource\",\"name\":\"Question 391 – 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/statistics/questions/q391\",\"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/questions","basePath":"/a-level/maths/ocr/statistics/questions","formulaUrl":null,"initialIndex":390,"initialQuestionWindow":[{"id":"313e4581-90fa-4464-b04b-60ab5f462b27","revisionId":"7780db36-4a4a-4e95-b7a4-d0bcf3f3f5fd","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"medium","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"af69c4d5-91d1-4400-87a2-8757d83696d6","label":null,"content":"The daily rainfall in a certain region during the monsoon season is modeled by a normal distribution with a mean of 22 mm and a standard deviation of 6.8 mm. A researcher believes that recent climate shifts have caused the mean daily rainfall to increase. A random sample of 40 days during the current monsoon season is taken, and the mean daily rainfall is found to be 24.5 mm. Carry out a hypothesis test at the 5% significance level to investigate the researcher's claim.","metadata":null,"markscheme":"- States $H_0: \\mu = 22$ and $H_1: \\mu > 22$ B1\n- Identifies that under $H_0$, the sample mean follows $\\bar{X} \\sim N\\left(22, \\frac{6.8^2}{40}\\right)$ or calculates standard error $\\sigma_{\\bar{x}} = \\frac{6.8}{\\sqrt{40}} \\approx 1.075$ B1\n- Attempts to calculate the test statistic $z = \\frac{24.5 - 22}{1.075}$ or the $p$-value $P(\\bar{X} > 24.5)$ M1\n- Obtains $z \\approx 2.325$ or $p$-value $\\approx 0.0100$ A1\n- Compares the $p$-value to the significance level (e.g. $0.0100 < 0.05$) or the $z$-score to the critical value (e.g. $2.325 > 1.645$) M1\n- Rejects $H_0$ and concludes that there is sufficient evidence at the 5% level to suggest that the mean daily rainfall has increased A1","marks":6,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"pvalue","label":"P-Value Approach","steps":[{"type":"text","order":0,"title":"State the hypotheses","content":"To begin, we define the population mean daily rainfall during the current monsoon season as $\\mu$.\n\nWe set up our null hypothesis ($H_0$), which represents the status quo (no change in rainfall), and our alternative hypothesis ($H_1$), representing the researcher's claim that the mean rainfall has increased:\n\n$$\\begin{aligned} H_0: \\mu &= 22 \\\\ H_1: \\mu &> 22 \\end{aligned}$$\n\nThis is a one-tailed test because we are testing specifically for an increase in the mean.","highlights":[{"text":"mean of 22 mm","location":"specification"},{"text":"mean daily rainfall to increase","location":"specification"}]},{"type":"text","order":1,"title":"Calculate the standard error","content":"Since the individual daily rainfall is modeled by a normal distribution with standard deviation $\\sigma = 6.8$ mm, the sample mean daily rainfall, $\\bar{X}$, for a random sample of size $n = 40$ is distributed as:\n\n$$\\bar{X} \\sim N\\left(\\mu, \\frac{\\sigma^2}{n}\\right)$$\n\nUnder $H_0$ where $\\mu = 22$, we calculate the standard error (the standard deviation of the sample mean distribution):\n\n$$\\text{Standard Error } (\\sigma_{\\bar{x}}) = \\frac{\\sigma}{\\sqrt{n}} = \\frac{6.8}{\\sqrt{40}} \\approx 1.0752\\text{ mm}$$","highlights":[{"text":"standard deviation of 6.8 mm","location":"specification"},{"text":"random sample of 40 days","location":"specification"}]},{"type":"text","order":2,"title":"Calculate the z-score","content":"To find the probability of obtaining a sample mean as extreme as the observed value, we standardise the observed sample mean $\\bar{x} = 24.5$ mm to find its z-score:\n\n$$z = \\frac{\\bar{x} - 22}{\\sigma_{\\bar{x}}} = \\frac{24.5 - 22}{\\frac{6.8}{\\sqrt{40}}}$$\n\nEvaluating this, we get:\n\n$$z \\approx \\frac{2.5}{1.0752} \\approx 2.325$$","highlights":[{"text":"mean daily rainfall is found to be 24.5 mm","location":"specification"}]},{"type":"text","order":3,"title":"Find the p-value","content":"The p-value is the probability of observing a sample mean of $24.5$ mm or more under the assumption that the null hypothesis is true:\n\n$$\\text{p-value} = P(\\bar{X} \\ge 24.5) = P(Z \\ge 2.325)$$\n\nUsing standard normal distribution tables or a calculator, we find:\n\n$$\\text{p-value} \\approx 0.0100 \\text{ (or } 1.00\\% \\text{)}$$","calculator":[{"gdc":{"expression":"NormCD(2.3251, 1E99, 1, 0)","instructions":"1. Go to the Statistics menu and choose Dist.\n2. Select Normal, then Ncd.\n3. Set Lower: 2.3251, Upper: 1E99, $\\sigma$: 1, $\\mu$: 0.\n4. Execute to calculate."},"ti84":{"expression":"normalcdf(2.3251, 1E99, 0, 1)","instructions":"1. Press [2nd] [VARS] (DISTR).\n2. Select 2: normalcdf(.\n3. Enter Lower: 2.3251, Upper: 1E99, $\\mu$: 0, $\\sigma$: 1.\n4. Highlight Paste and press [ENTER] twice."},"desmos":{"expression":"normaldist(0, 1).cdf(2.3251, \\infty)","instructions":"In a command line, enter the normal distribution CDF and specify the lower bound."},"description":"Calculate the p-value $P(Z \\ge 2.325)$ on a graphing calculator."}]},{"type":"text","order":4,"title":"Compare to significance level","content":"Now, we compare our calculated p-value to the significance level $\\alpha = 0.05$:\n\n$$0.0100 < 0.05$$\n\nSince the p-value is less than $0.05$, the result is statistically significant. This means the probability of obtaining such an extreme sample mean by pure chance is very low (only $1.00\\%$).","highlights":[{"text":"hypothesis test at the 5% significance level","location":"specification"}]},{"type":"text","order":5,"title":"State the conclusion","content":"Because the p-value ($0.0100$) is less than the significance level ($0.05$), we reject the null hypothesis $H_0$.\n\nThere is sufficient evidence at the $5\\%$ significance level to support the researcher's claim that the mean daily rainfall has increased."}]},{"id":"criticalz","label":"Critical Z-Value Approach","steps":[{"type":"text","order":0,"title":"State the hypotheses","content":"To test the researcher's claim that the mean daily rainfall has increased, we first formulate the null hypothesis ($H_0$) and the alternative hypothesis ($H_1$).\n\nLet $\\mu$ represent the population mean daily rainfall during the current monsoon season.\n\n$$\\begin{aligned} H_0: \\mu &= 22 \\\\ H_1: \\mu &> 22 \\end{aligned}$$\n\nThis is a one-tailed test because we are testing specifically for an increase in the mean daily rainfall at a $5\\%$ level of significance.","highlights":[{"text":"mean of 22 mm","location":"specification"},{"text":"mean daily rainfall to increase","location":"specification"}]},{"type":"text","order":1,"title":"Define sample distribution","content":"Under the null hypothesis $H_0$, the sample mean $\\bar{X}$ for a sample size of $n = 40$ follows a normal distribution:\n\n$$\\bar{X} \\sim N\\left(\\mu, \\frac{\\sigma^2}{n}\\right)$$\n\nUsing the given parameters:\n- Population mean, $\\mu = 22$\n- Standard deviation, $\\sigma = 6.8$\n- Sample size, $n = 40$\n\nWe calculate the standard error of the mean ($\\sigma_{\\bar{x}}$):\n\n$$\\sigma_{\\bar{x}} = \\frac{\\sigma}{\\sqrt{n}} = \\frac{6.8}{\\sqrt{40}} \\approx 1.0752\\text{ mm}$$","highlights":[{"text":"standard deviation of 6.8 mm","location":"specification"},{"text":"random sample of 40 days","location":"specification"}]},{"type":"text","order":2,"title":"Calculate test statistic","content":"Next, we calculate the test statistic (z-score) for our observed sample mean, $\\bar{x} = 24.5\\text{ mm}$.\n\nThe formula for the standard z-score of a sample mean is:\n\n$$z = \\frac{\\bar{x} - \\mu}{\\sigma_{\\bar{x}}}$$\n\nSubstituting our values in:\n\n$$z = \\frac{24.5 - 22}{1.0752} \\approx 2.325$$","highlights":[{"text":"mean daily rainfall is found to be 24.5 mm","location":"specification"}]},{"type":"text","order":3,"title":"Determine critical value","content":"For a one-tailed test at the $5\\%$ significance level ($\\alpha = 0.05$) in the upper tail, the critical value $z_{\\text{crit}}$ is the value such that:\n\n$$P(Z > z_{\\text{crit}}) = 0.05$$\n\nUsing the standard normal distribution table or a calculator, we find:\n\n$$z_{\\text{crit}} = 1.645$$\n\nThe critical region (rejection region) for this test is $z \\ge 1.645$.","highlights":[{"text":"5% significance level","location":"specification"}]},{"type":"text","order":4,"title":"Compare and conclude","content":"We compare our calculated test statistic $z$ with the critical value $z_{\\text{crit}}$:\n\n$$z \\approx 2.325 > 1.645$$\n\nSince our calculated z-score lies in the critical region ($2.325 > 1.645$), we reject the null hypothesis $H_0$.\n\nThere is sufficient evidence at the $5\\%$ significance level to support the researcher's claim that the mean daily rainfall has increased."}]},{"id":"criticalmean","label":"Critical Sample Mean Approach","steps":[{"type":"text","order":0,"title":"State the hypotheses","content":"First, we must state the null hypothesis $H_0$ and the alternative hypothesis $H_1$ to define the claim we are testing.\n\nLet $\\mu$ be the true mean daily rainfall in mm.\n\n* **Null Hypothesis ($H_0$):** The mean daily rainfall has not changed:\n $$H_0: \\mu = 22$$\n* **Alternative Hypothesis ($H_1$):** The mean daily rainfall has increased:\n $$H_1: \\mu > 22$$\n\nThis is a one-tailed test at the $5\\%$ significance level ($\\alpha = 0.05$).","highlights":[{"text":"mean of 22 mm","location":"specification"},{"text":"mean daily rainfall to increase","location":"specification"},{"text":"5% significance level","location":"specification"}]},{"type":"text","order":1,"title":"Find the distribution of sample mean","content":"Under the null hypothesis $H_0$, the sample mean $\\bar{X}$ for a sample size of $n = 40$ follows a normal distribution with mean $\\mu = 22$ and standard error $\\sigma_{\\bar{x}} = \\frac{\\sigma}{\\sqrt{n}}$:\n\n$$\\sigma_{\\bar{x}} = \\frac{6.8}{\\sqrt{40}} \\approx 1.0751\\text{ mm}$$\n\nThus, under $H_0$:\n\n$$\\bar{X} \\sim N\\left(22, \\frac{6.8^2}{40}\\right)$$","highlights":[{"text":"standard deviation of 6.8 mm","location":"specification"},{"text":"sample of 40 days","location":"specification"}]},{"type":"text","order":2,"title":"Determine the critical value","content":"Using the Critical Sample Mean Approach, we find the critical value $c$ such that the probability of rejecting $H_0$ when it is actually true is $0.05$.\n\nSince this is a right-tailed test, we want to find $c$ such that:\n\n$$P(\\bar{X} > c) = 0.05$$\n\nUsing the standard normal distribution $Z \\sim N(0, 1)$, the critical z-value for a $5\\%$ significance level (one-tailed) is $z_{\\text{crit}} = 1.6449$.\n\nWe convert this to our sample mean distribution:\n\n$$\\begin{aligned} c &= \\mu + z_{\\text{crit}} \\times \\sigma_{\\bar{x}} \\\\ c &= 22 + 1.6449 \\times 1.0751 \\\\ c &\\approx 22 + 1.7684 \\\\ c &\\approx 23.77\\text{ mm} \\end{aligned}$$\n\nSo, the critical region is $\\bar{X} \\ge 23.77\\text{ mm}$."},{"type":"text","order":3,"title":"Compare and conclude","content":"Now we compare our observed sample mean, $\\bar{x} = 24.5\\text{ mm}$, to the critical value $c = 23.77\\text{ mm}$.\n\nSince:\n\n$$24.5 > 23.77$$\n\nThe observed sample mean falls within the critical region.\n\nTherefore, we reject $H_0$.\n\nThere is sufficient evidence at the $5\\%$ significance level to support the researcher's claim that the mean daily rainfall has increased.","highlights":[{"text":"mean daily rainfall is found to be 24.5 mm","location":"specification"}]}]}],"generatedAt":"2026-06-09T04:39:42.627Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"af69c4d5-91d1-4400-87a2-8757d83696d6:pvalue","label":"P-Value Approach","steps":[{"type":"text","order":0,"title":"State the hypotheses","content":"To begin, we define the population mean daily rainfall during the current monsoon season as $\\mu$.\n\nWe set up our null hypothesis ($H_0$), which represents the status quo (no change in rainfall), and our alternative hypothesis ($H_1$), representing the researcher's claim that the mean rainfall has increased:\n\n$$\\begin{aligned} H_0: \\mu &= 22 \\\\ H_1: \\mu &> 22 \\end{aligned}$$\n\nThis is a one-tailed test because we are testing specifically for an increase in the mean.","highlights":[{"text":"mean of 22 mm","location":"specification"},{"text":"mean daily rainfall to increase","location":"specification"}]},{"type":"text","order":1,"title":"Calculate the standard error","content":"Since the individual daily rainfall is modeled by a normal distribution with standard deviation $\\sigma = 6.8$ mm, the sample mean daily rainfall, $\\bar{X}$, for a random sample of size $n = 40$ is distributed as:\n\n$$\\bar{X} \\sim N\\left(\\mu, \\frac{\\sigma^2}{n}\\right)$$\n\nUnder $H_0$ where $\\mu = 22$, we calculate the standard error (the standard deviation of the sample mean distribution):\n\n$$\\text{Standard Error } (\\sigma_{\\bar{x}}) = \\frac{\\sigma}{\\sqrt{n}} = \\frac{6.8}{\\sqrt{40}} \\approx 1.0752\\text{ mm}$$","highlights":[{"text":"standard deviation of 6.8 mm","location":"specification"},{"text":"random sample of 40 days","location":"specification"}]},{"type":"text","order":2,"title":"Calculate the z-score","content":"To find the probability of obtaining a sample mean as extreme as the observed value, we standardise the observed sample mean $\\bar{x} = 24.5$ mm to find its z-score:\n\n$$z = \\frac{\\bar{x} - 22}{\\sigma_{\\bar{x}}} = \\frac{24.5 - 22}{\\frac{6.8}{\\sqrt{40}}}$$\n\nEvaluating this, we get:\n\n$$z \\approx \\frac{2.5}{1.0752} \\approx 2.325$$","highlights":[{"text":"mean daily rainfall is found to be 24.5 mm","location":"specification"}]},{"type":"text","order":3,"title":"Find the p-value","content":"The p-value is the probability of observing a sample mean of $24.5$ mm or more under the assumption that the null hypothesis is true:\n\n$$\\text{p-value} = P(\\bar{X} \\ge 24.5) = P(Z \\ge 2.325)$$\n\nUsing standard normal distribution tables or a calculator, we find:\n\n$$\\text{p-value} \\approx 0.0100 \\text{ (or } 1.00\\% \\text{)}$$","calculator":[{"gdc":{"expression":"NormCD(2.3251, 1E99, 1, 0)","instructions":"1. Go to the Statistics menu and choose Dist.\n2. Select Normal, then Ncd.\n3. Set Lower: 2.3251, Upper: 1E99, $\\sigma$: 1, $\\mu$: 0.\n4. Execute to calculate."},"ti84":{"expression":"normalcdf(2.3251, 1E99, 0, 1)","instructions":"1. Press [2nd] [VARS] (DISTR).\n2. Select 2: normalcdf(.\n3. Enter Lower: 2.3251, Upper: 1E99, $\\mu$: 0, $\\sigma$: 1.\n4. Highlight Paste and press [ENTER] twice."},"desmos":{"expression":"normaldist(0, 1).cdf(2.3251, \\infty)","instructions":"In a command line, enter the normal distribution CDF and specify the lower bound."},"description":"Calculate the p-value $P(Z \\ge 2.325)$ on a graphing calculator."}]},{"type":"text","order":4,"title":"Compare to significance level","content":"Now, we compare our calculated p-value to the significance level $\\alpha = 0.05$:\n\n$$0.0100 < 0.05$$\n\nSince the p-value is less than $0.05$, the result is statistically significant. This means the probability of obtaining such an extreme sample mean by pure chance is very low (only $1.00\\%$).","highlights":[{"text":"hypothesis test at the 5% significance level","location":"specification"}]},{"type":"text","order":5,"title":"State the conclusion","content":"Because the p-value ($0.0100$) is less than the significance level ($0.05$), we reject the null hypothesis $H_0$.\n\nThere is sufficient evidence at the $5\\%$ significance level to support the researcher's claim that the mean daily rainfall has increased."}]},{"id":"af69c4d5-91d1-4400-87a2-8757d83696d6:criticalz","label":"Critical Z-Value Approach","steps":[{"type":"text","order":0,"title":"State the hypotheses","content":"To test the researcher's claim that the mean daily rainfall has increased, we first formulate the null hypothesis ($H_0$) and the alternative hypothesis ($H_1$).\n\nLet $\\mu$ represent the population mean daily rainfall during the current monsoon season.\n\n$$\\begin{aligned} H_0: \\mu &= 22 \\\\ H_1: \\mu &> 22 \\end{aligned}$$\n\nThis is a one-tailed test because we are testing specifically for an increase in the mean daily rainfall at a $5\\%$ level of significance.","highlights":[{"text":"mean of 22 mm","location":"specification"},{"text":"mean daily rainfall to increase","location":"specification"}]},{"type":"text","order":1,"title":"Define sample distribution","content":"Under the null hypothesis $H_0$, the sample mean $\\bar{X}$ for a sample size of $n = 40$ follows a normal distribution:\n\n$$\\bar{X} \\sim N\\left(\\mu, \\frac{\\sigma^2}{n}\\right)$$\n\nUsing the given parameters:\n- Population mean, $\\mu = 22$\n- Standard deviation, $\\sigma = 6.8$\n- Sample size, $n = 40$\n\nWe calculate the standard error of the mean ($\\sigma_{\\bar{x}}$):\n\n$$\\sigma_{\\bar{x}} = \\frac{\\sigma}{\\sqrt{n}} = \\frac{6.8}{\\sqrt{40}} \\approx 1.0752\\text{ mm}$$","highlights":[{"text":"standard deviation of 6.8 mm","location":"specification"},{"text":"random sample of 40 days","location":"specification"}]},{"type":"text","order":2,"title":"Calculate test statistic","content":"Next, we calculate the test statistic (z-score) for our observed sample mean, $\\bar{x} = 24.5\\text{ mm}$.\n\nThe formula for the standard z-score of a sample mean is:\n\n$$z = \\frac{\\bar{x} - \\mu}{\\sigma_{\\bar{x}}}$$\n\nSubstituting our values in:\n\n$$z = \\frac{24.5 - 22}{1.0752} \\approx 2.325$$","highlights":[{"text":"mean daily rainfall is found to be 24.5 mm","location":"specification"}]},{"type":"text","order":3,"title":"Determine critical value","content":"For a one-tailed test at the $5\\%$ significance level ($\\alpha = 0.05$) in the upper tail, the critical value $z_{\\text{crit}}$ is the value such that:\n\n$$P(Z > z_{\\text{crit}}) = 0.05$$\n\nUsing the standard normal distribution table or a calculator, we find:\n\n$$z_{\\text{crit}} = 1.645$$\n\nThe critical region (rejection region) for this test is $z \\ge 1.645$.","highlights":[{"text":"5% significance level","location":"specification"}]},{"type":"text","order":4,"title":"Compare and conclude","content":"We compare our calculated test statistic $z$ with the critical value $z_{\\text{crit}}$:\n\n$$z \\approx 2.325 > 1.645$$\n\nSince our calculated z-score lies in the critical region ($2.325 > 1.645$), we reject the null hypothesis $H_0$.\n\nThere is sufficient evidence at the $5\\%$ significance level to support the researcher's claim that the mean daily rainfall has increased."}]},{"id":"af69c4d5-91d1-4400-87a2-8757d83696d6:criticalmean","label":"Critical Sample Mean Approach","steps":[{"type":"text","order":0,"title":"State the hypotheses","content":"First, we must state the null hypothesis $H_0$ and the alternative hypothesis $H_1$ to define the claim we are testing.\n\nLet $\\mu$ be the true mean daily rainfall in mm.\n\n* **Null Hypothesis ($H_0$):** The mean daily rainfall has not changed:\n $$H_0: \\mu = 22$$\n* **Alternative Hypothesis ($H_1$):** The mean daily rainfall has increased:\n $$H_1: \\mu > 22$$\n\nThis is a one-tailed test at the $5\\%$ significance level ($\\alpha = 0.05$).","highlights":[{"text":"mean of 22 mm","location":"specification"},{"text":"mean daily rainfall to increase","location":"specification"},{"text":"5% significance level","location":"specification"}]},{"type":"text","order":1,"title":"Find the distribution of sample mean","content":"Under the null hypothesis $H_0$, the sample mean $\\bar{X}$ for a sample size of $n = 40$ follows a normal distribution with mean $\\mu = 22$ and standard error $\\sigma_{\\bar{x}} = \\frac{\\sigma}{\\sqrt{n}}$:\n\n$$\\sigma_{\\bar{x}} = \\frac{6.8}{\\sqrt{40}} \\approx 1.0751\\text{ mm}$$\n\nThus, under $H_0$:\n\n$$\\bar{X} \\sim N\\left(22, \\frac{6.8^2}{40}\\right)$$","highlights":[{"text":"standard deviation of 6.8 mm","location":"specification"},{"text":"sample of 40 days","location":"specification"}]},{"type":"text","order":2,"title":"Determine the critical value","content":"Using the Critical Sample Mean Approach, we find the critical value $c$ such that the probability of rejecting $H_0$ when it is actually true is $0.05$.\n\nSince this is a right-tailed test, we want to find $c$ such that:\n\n$$P(\\bar{X} > c) = 0.05$$\n\nUsing the standard normal distribution $Z \\sim N(0, 1)$, the critical z-value for a $5\\%$ significance level (one-tailed) is $z_{\\text{crit}} = 1.6449$.\n\nWe convert this to our sample mean distribution:\n\n$$\\begin{aligned} c &= \\mu + z_{\\text{crit}} \\times \\sigma_{\\bar{x}} \\\\ c &= 22 + 1.6449 \\times 1.0751 \\\\ c &\\approx 22 + 1.7684 \\\\ c &\\approx 23.77\\text{ mm} \\end{aligned}$$\n\nSo, the critical region is $\\bar{X} \\ge 23.77\\text{ mm}$."},{"type":"text","order":3,"title":"Compare and conclude","content":"Now we compare our observed sample mean, $\\bar{x} = 24.5\\text{ mm}$, to the critical value $c = 23.77\\text{ mm}$.\n\nSince:\n\n$$24.5 > 23.77$$\n\nThe observed sample mean falls within the critical region.\n\nTherefore, we reject $H_0$.\n\nThere is sufficient evidence at the $5\\%$ significance level to support the researcher's claim that the mean daily rainfall has increased.","highlights":[{"text":"mean daily rainfall is found to be 24.5 mm","location":"specification"}]}]}],"generatedAt":"2026-06-09T04:39:59.082Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"32d051f0-43ec-41e8-955f-73fcd6b17469","revisionId":"0656d722-362d-4b27-ad54-00f655cb3df0","questionNumber":null,"questionType":"LA","specification":"The events $A$ and $B$ are such that $P(A) = 0.3$ and $P(A \\cup B) = 0.7$.","difficultyBand":"medium","tier":null,"totalMarks":7,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"3f78bb27-6f1b-416b-b816-62b7bd721bbc","label":"a","content":"Find $P(A' \\cap B')$.","metadata":null,"markscheme":"- $P(A' \\cap B') = 1 - P(A \\cup B) = 1 - 0.7$ M1\n- $= 0.3$ A1","marks":2,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]},{"id":"cc8ae882-ad4b-48bb-9d7d-0763ead8cfa2","label":"b","content":"Find $P(A' \\cap B)$.","metadata":null,"markscheme":"- $P(A' \\cap B) = P(A \\cup B) - P(A) = 0.7 - 0.3$ M1\n- $= 0.4$ A1","marks":2,"order":1,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]},{"id":"a63077aa-9bd0-4a09-b67e-03433e5af373","label":"c","content":"Given also that $P(A\\mid B)=\\dfrac{1}{4}$, find $P(B)$.","metadata":null,"markscheme":"- $P(A\\cap B)=P(A\\mid B)P(B)=\\dfrac14P(B)$, and the addition rule gives $P(A\\cap B)=0.3+P(B)-0.7$ M1\n- $\\dfrac14P(B)=P(B)-0.4 \\Rightarrow \\dfrac34P(B)=0.4$ M1\n- $P(B)=\\dfrac{8}{15}$ A1\n\nAccept $0.533$ or better. Follow through a correct equation using the given conditional probability.","marks":3,"order":2,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"330203ba-0666-4974-84ae-ff2b3efa7547","revisionId":"0b0cda30-642e-49c9-8dca-e20e2e8524ce","questionNumber":null,"questionType":"LA","specification":"A master sommelier is judging the maturation periods of 8 unique vintage wines from a vineyard. The actual maturation periods, in months, for the 8 wines are given in the following table:\n\n\\begin{array}{|l|c|c|c|c|c|c|c|c|} \\hline \\text{Wine} & A & B & C & D & E & F & G & H \\\\ \\hline \\text{Maturation (months)} & 54 & 12 & 38 & 62 & 24 & 45 & 18 & 30 \\\\ \\hline \\end{array}\n\nThe sommelier is asked to list the wines in descending order of their maturation period (from the oldest to the youngest). She provides the following sequence:\n\\[ D, F, C, A, E, H, G, B \\]","difficultyBand":"medium","tier":null,"totalMarks":7,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"68d4f1f5-a502-40d8-bd3d-833c47159733","label":"a","content":"Calculate Spearman’s rank correlation coefficient between the sommelier's perceived order and the actual order, giving your answer to 3 decimal places.","metadata":null,"markscheme":"- Correctly ranks actual maturation periods (from longest to shortest): $D=1, A=2, F=3, C=4, H=5, E=6, G=7, B=8$ B1\n- Correctly identifies ranks assigned by the sommelier: $D=1, F=2, C=3, A=4, E=5, H=6, G=7, B=8$ B1\n- Calculates $\\sum d^2 = (1-1)^2 + (2-4)^2 + (3-2)^2 + (4-3)^2 + (5-6)^2 + (6-5)^2 + (7-7)^2 + (8-8)^2 = 0 + 4 + 1 + 1 + 1 + 1 + 0 + 0 = 8$ M1\n- Calculates $r_s = 1 - \\dfrac{6(8)}{8(63)} = 1 - \\dfrac{48}{504} = \\dfrac{19}{21} \\approx 0.905$ A1","marks":4,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"manual_formula","label":"Manual Calculation using Formula","steps":[{"type":"text","order":0,"title":"Rank the actual periods","content":"To calculate Spearman's rank correlation coefficient ($r_s$), we first need to rank the actual maturation periods in descending order (from longest to shortest, so $1$ is the longest maturation period and $8$ is the shortest).\n\nLet's list the actual maturation periods:\n- **A**: 54 months\n- **B**: 12 months\n- **C**: 38 months\n- **D**: 62 months\n- **E**: 24 months\n- **F**: 45 months\n- **G**: 18 months\n- **H**: 30 months\n\nSorting these values from largest to smallest:\n$$\\text{D (62) > A (54) > F (45) > C (38) > H (30) > E (24) > G (18) > B (12)}$$\n\nThis gives us the **Actual Rank ($R_{\\text{actual}}$)** for each wine:\n\n$$\\begin{array}{|c|c|c|c|c|c|c|c|c|} \\hline \\text{Wine} & A & B & C & D & E & F & G & H \\\\ \\hline R_{\\text{actual}} & 2 & 8 & 4 & 1 & 6 & 3 & 7 & 5 \\\\ \\hline \\end{array}$$"},{"type":"text","order":1,"title":"Rank the sommelier's order","content":"Next, we find the ranks assigned by the sommelier. The sommelier provided the sequence in descending order of maturation: \n$$D, F, C, A, E, H, G, B$$\n\nAssigning ranks $1$ to $8$ from left to right gives the **Sommelier's Rank ($R_{\\text{somm}}$)**:\n- **D** is 1st $\\rightarrow 1$\n- **F** is 2nd $\\rightarrow 2$\n- **C** is 3rd $\\rightarrow 3$\n- **A** is 4th $\\rightarrow 4$\n- **E** is 5th $\\rightarrow 5$\n- **H** is 6th $\\rightarrow 6$\n- **G** is 7th $\\rightarrow 7$\n- **B** is 8th $\\rightarrow 8$\n\nSummarizing the sommelier's ranks for each wine alphabetically:\n\n$$\\begin{array}{|c|c|c|c|c|c|c|c|c|} \\hline \\text{Wine} & A & B & C & D & E & F & G & H \\\\ \\hline R_{\\text{somm}} & 4 & 8 & 3 & 1 & 5 & 2 & 7 & 6 \\\\ \\hline \\end{array}$$"},{"type":"text","order":2,"title":"Calculate rank differences","content":"Now, we find the difference in ranks $d = R_{\\text{actual}} - R_{\\text{somm}}$ and calculate $d^2$ for each wine:\n\n$$\\begin{array}{|c|c|c|c|c|} \\hline \\text{Wine} & R_{\\text{actual}} & R_{\\text{somm}} & d & d^2 \\\\ \\hline A & 2 & 4 & -2 & 4 \\\\ B & 8 & 8 & 0 & 0 \\\\ C & 4 & 3 & 1 & 1 \\\\ D & 1 & 1 & 0 & 0 \\\\ E & 6 & 5 & 1 & 1 \\\\ F & 3 & 2 & 1 & 1 \\\\ G & 7 & 7 & 0 & 0 \\\\ H & 5 & 6 & -1 & 1 \\\\ \\hline \\end{array}$$\n\nSumming the $d^2$ values:\n$$\\sum d^2 = 4 + 0 + 1 + 0 + 1 + 1 + 0 + 1 = 8$$"},{"type":"text","order":3,"title":"Apply Spearman's formula","content":"With $n = 8$ wines and $\\sum d^2 = 8$, we use the Spearman's rank correlation coefficient formula:\n\n$$r_s = 1 - \\frac{6\\sum d^2}{n(n^2 - 1)}$$\n\nSubstitute our values into the formula:\n\n$$\\begin{aligned} r_s &= 1 - \\frac{6(8)}{8(8^2 - 1)} \\\\ &= 1 - \\frac{48}{8(63)} \\\\ &= 1 - \\frac{48}{504} \\\\ &= 1 - \\frac{2}{21} \\\\ &= \\frac{19}{21} \\approx 0.905 \\end{aligned}$$\n\nTo 3 decimal places, the Spearman's rank correlation coefficient is **$0.905$**."}]},{"id":"gdc_calculation","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 (GDC), we first need to assign ranks to both sets of data. \n\nOnce both sets of data are ranked, we can enter these ranks as lists into our GDC and perform a linear regression calculation. The Pearson correlation coefficient ($r$) of these ranks is mathematically equivalent to Spearman's rank correlation coefficient ($r_s$)."},{"type":"text","order":1,"title":"Rank actual maturation periods","content":"Let's first rank the actual maturation periods in descending order (from longest to shortest, meaning $1$ is the oldest and $8$ is the youngest).\n\nLet's list the wines and their actual maturation periods in months:\n- $D$: $62$ months $\\rightarrow$ Rank **$1$**\n- $A$: $54$ months $\\rightarrow$ Rank **$2$**\n- $F$: $45$ months $\\rightarrow$ Rank **$3$**\n- $C$: $38$ months $\\rightarrow$ Rank **$4$**\n- $H$: $30$ months $\\rightarrow$ Rank **$5$**\n- $E$: $24$ months $\\rightarrow$ Rank **$6$**\n- $G$: $18$ months $\\rightarrow$ Rank **$7$**\n- $B$: $12$ months $\\rightarrow$ Rank **$8$**\n\nSorting them by wine name, the actual ranks are:\n$$\\begin{array}{|l|c|c|c|c|c|c|c|c|} \\hline \\text{Wine} & A & B & C & D & E & F & G & H \\\\ \\hline \\text{Actual Rank} & 2 & 8 & 4 & 1 & 6 & 3 & 7 & 5 \\\\ \\hline \\end{array}$$","highlights":[{"text":"The actual maturation periods, in months, for the 8 wines are given in the following table","location":"specification"}]},{"type":"text","order":2,"title":"Rank perceived periods","content":"Next, let's identify the ranks from the sommelier's sequence. She listed the wines in descending order of her perceived maturation periods as:\n$$\\text{D, F, C, A, E, H, G, B}$$\n\nThis means she assigned the following ranks directly:\n- $D$ is $1^{\\text{st}}$ $\\rightarrow$ Rank **$1$**\n- $F$ is $2^{\\text{nd}}$ $\\rightarrow$ Rank **$2$**\n- $C$ is $3^{\\text{rd}}$ $\\rightarrow$ Rank **$3$**\n- $A$ is $4^{\\text{th}}$ $\\rightarrow$ Rank **$4$**\n- $E$ is $5^{\\text{th}}$ $\\rightarrow$ Rank **$5$**\n- $H$ is $6^{\\text{th}}$ $\\rightarrow$ Rank **$6$**\n- $G$ is $7^{\\text{th}}$ $\\rightarrow$ Rank **$7$**\n- $B$ is $8^{\\text{th}}$ $\\rightarrow$ Rank **$8$**\n\nSorting these by wine name to line up with our actual ranks table:\n$$\\begin{array}{|l|c|c|c|c|c|c|c|c|} \\hline \\text{Wine} & A & B & C & D & E & F & G & H \\\\ \\hline \\text{Sommelier Rank} & 4 & 8 & 3 & 1 & 5 & 2 & 7 & 6 \\\\ \\hline \\end{array}$$","highlights":[{"text":"descending order of their maturation period (from the oldest to the youngest)","location":"specification"},{"text":"D, F, C, A, E, H, G, B","location":"specification"}]},{"type":"text","order":3,"title":"Use your graphing calculator","content":"Now, enter these two sets of ranks into your graphing calculator as lists: \n\n$$\\begin{aligned} L_1 & = \\{2, 8, 4, 1, 6, 3, 7, 5\\} \\\\ L_2 & = \\{4, 8, 3, 1, 5, 2, 7, 6\\} \\end{aligned}$$\n\nRun a 2-variable statistics or a linear regression calculation ($ax+b$) to find the correlation coefficient $r$.","calculator":[{"gdc":{"instructions":"1. Go to the Statistics / List editor application.\n2. In List 1, enter: 2, 8, 4, 1, 6, 3, 7, 5.\n3. In List 2, enter: 4, 8, 3, 1, 5, 2, 7, 6.\n4. Perform a Linear Regression calculation (usually under Calc -> Reg -> X).\n5. Read the value for 'r'."},"ti84":{"instructions":"1. Press [STAT] and select 1:Edit...\n2. In L1, enter the actual ranks: 2, 8, 4, 1, 6, 3, 7, 5.\n3. In L2, enter the sommelier's ranks: 4, 8, 3, 1, 5, 2, 7, 6.\n4. Press [STAT], arrow over to CALC, and select 4:LinReg(ax+b).\n5. Set Xlist to L1, Ylist to L2, and press Calculate.\n(Note: If 'r' does not appear, press [2nd] [CATALOG], scroll to 'DiagnosticOn', press [ENTER] twice, and rerun the regression)."},"desmos":{"instructions":"1. Click the plus sign and add a table.\n2. In the x1 column, enter: 2, 8, 4, 1, 6, 3, 7, 5.\n3. In the y1 column, enter: 4, 8, 3, 1, 5, 2, 7, 6.\n4. In a new expression line, type: y1 ~ m * x1 + c\n5. Look at the output value for r."},"description":"Enter ranks into lists and calculate the correlation coefficient."}]},{"type":"text","order":4,"title":"State the final answer","content":"From your calculator, the correlation coefficient is:\n\n$$r \\approx 0.9047619...$$\n\nRounding to $3$ decimal places as requested, we obtain:\n\n$$r_s \\approx 0.905$$","highlights":[{"text":"giving your answer to 3 decimal places.","location":"specification"}]}]}],"generatedAt":"2026-06-11T03:04:57.388Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"313220ba-9e81-4571-81bc-079d73af6a2a","label":"b","content":"Test, at the 5% significance level, whether there is evidence of a positive correlation between the sommelier's estimates and the true maturation periods. State your hypotheses clearly.","metadata":null,"markscheme":"- States null hypothesis $H_0: \\rho_s = 0$ and alternative hypothesis $H_1: \\rho_s > 0$ B1\n- Identifies critical value for $n=8$ at the 5% level of significance for a one-tailed test: $0.6429$ B1\n- Compares $0.905 > 0.6429$ and concludes that there is sufficient evidence to reject $H_0$. There is significant evidence of a positive correlation between the sommelier's estimates and the actual maturation periods A1","marks":3,"order":1,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"formula_critical_value","label":"Analytical Calculation and Table Comparison","steps":[{"type":"text","order":0,"title":"State the Hypotheses","content":"To test whether there is a positive correlation between the sommelier's estimates and the true maturation periods, we formulate the null and alternative hypotheses in terms of the population Spearman rank correlation coefficient, $\\rho_s$:\n\n* **Null Hypothesis ($H_0$)**: $\\rho_s = 0$ (There is no correlation between the sommelier's estimates and the actual maturation periods.)\n* **Alternative Hypothesis ($H_1$)**: $\\rho_s > 0$ (There is a positive correlation between the sommelier's estimates and the actual maturation periods.)","highlights":[{"text":"positive correlation","location":"specification"},{"text":"State your hypotheses clearly.","location":"specification"}]},{"type":"text","order":1,"title":"Rank the Data","content":"To calculate the Spearman's rank correlation coefficient $r_s$, we assign ranks to both the actual maturation periods and the sommelier's estimated order. We will rank from oldest (1) to youngest (8):\n\n**Actual Maturation Ranks ($R_x$):**\n* $D$ (62 months) $\\rightarrow 1$\n* $A$ (54 months) $\\rightarrow 2$\n* $F$ (45 months) $\\rightarrow 3$\n* $C$ (38 months) $\\rightarrow 4$\n* $H$ (30 months) $\\rightarrow 5$\n* $E$ (24 months) $\\rightarrow 6$\n* $G$ (18 months) $\\rightarrow 7$\n* $B$ (12 months) $\\rightarrow 8$\n\n**Sommelier's Ranks ($R_y$):**\nBased on her list $D, F, C, A, E, H, G, B$:\n* $D \\rightarrow 1$\n* $F \\rightarrow 2$\n* $C \\rightarrow 3$\n* $A \\rightarrow 4$\n* $E \\rightarrow 5$\n* $H \\rightarrow 6$\n* $G \\rightarrow 7$\n* $B \\rightarrow 8$"},{"type":"text","order":2,"title":"Calculate Rank Differences","content":"We calculate the difference in ranks, $d = R_x - R_y$, and its square, $d^2$, for each wine:\n\n$$\\begin{array}{|c|c|c|c|c|} \\hline \\text{Wine} & R_x & R_y & d & d^2 \\\\ \\hline A & 2 & 4 & -2 & 4 \\\\ B & 8 & 8 & 0 & 0 \\\\ C & 4 & 3 & 1 & 1 \\\\ D & 1 & 1 & 0 & 0 \\\\ E & 6 & 5 & 1 & 1 \\\\ F & 3 & 2 & 1 & 1 \\\\ G & 7 & 7 & 0 & 0 \\\\ H & 5 & 6 & -1 & 1 \\\\ \\hline \\end{array}$$\n\nSumming the squared differences:\n$$\\sum d^2 = 4 + 0 + 1 + 0 + 1 + 1 + 0 + 1 = 8$$"},{"type":"text","order":3,"title":"Calculate Spearman's Coefficient","content":"We substitute $\\sum d^2 = 8$ and $n = 8$ into the Spearman's rank correlation coefficient formula:\n\n$$r_s = 1 - \\frac{6\\sum d^2}{n(n^2 - 1)}$$\n\n$$\\begin{aligned} r_s &= 1 - \\frac{6(8)}{8(8^2 - 1)} \\\\ r_s &= 1 - \\frac{48}{8(63)} \\\\ r_s &= 1 - \\frac{48}{504} \\\\ r_s &= 1 - \\frac{2}{21} \\\\ r_s &\\approx 0.905 \\end{aligned}$$"},{"type":"text","order":4,"title":"Compare with Critical Value","content":"Now we determine the critical value from the statistical tables for a **one-tailed test** at the **5% level of significance** with a sample size of $n = 8$:\n\n$$\\text{Critical Value} = 0.6429$$\n\nWe compare our test statistic $r_s$ against this critical value:\n\n$$0.905 > 0.6429$$\n\nSince our calculated value of $r_s$ is greater than the critical value, we reject the null hypothesis $H_0$.","highlights":[{"text":"5% significance level","location":"specification"}]},{"type":"text","order":5,"title":"State the Conclusion","content":"There is **sufficient evidence** at the 5% significance level to reject $H_0$.\n\nTherefore, we conclude that there is significant evidence of a positive correlation between the sommelier's estimates and the actual maturation periods of the wines."}]},{"id":"gdc_ranks","label":"Using Graphic Display Calculator (GDC)","steps":[{"type":"text","order":0,"title":"State the Hypotheses","content":"To test for a positive correlation between the sommelier's estimates and the actual maturation periods, we use Spearman's rank correlation coefficient, denoted by $\\rho_s$ (rho).\n\n* **Null Hypothesis ($H_0$):** There is no correlation between the sommelier's estimates and the true maturation periods.\n $$H_0: \\rho_s = 0$$\n* **Alternative Hypothesis ($H_1$):** There is a positive correlation between the sommelier's estimates and the true maturation periods.\n $$H_1: \\rho_s > 0$$\n\nThis is a **one-tailed test** at the **$5\\%$ significance level**.","highlights":[{"text":"evidence of a positive correlation","location":"specification"},{"text":"5% significance level","location":"specification"}]},{"type":"text","order":1,"title":"Rank the Data","content":"$93"},{"type":"text","order":2,"title":"Calculate r_s using GDC","content":"Now, enter these ranks into your GDC to directly compute the correlation coefficient, $r_s$.","calculator":[{"gdc":{"instructions":"1. Open the Statistics application.\n2. In List 1 (or List A), enter: `2, 8, 4, 1, 6, 3, 7, 5`.\n3. In List 2 (or List B), enter: `4, 8, 3, 1, 5, 2, 7, 6`.\n4. Select `CALC` (or `Regression`), then choose `Linear Regression` (usually $X$ or $a+bx$).\n5. The correlation coefficient $r$ will be displayed as $r_s \\approx 0.905$."},"ti84":{"instructions":"1. Press [STAT] and select `1:Edit...`.\n2. In `L1`, enter the Actual Ranks: `2, 8, 4, 1, 6, 3, 7, 5`.\n3. In `L2`, enter the Estimated Ranks: `4, 8, 3, 1, 5, 2, 7, 6`.\n4. Press [STAT], arrow over to `CALC`, and select `4:LinReg(ax+b)`.\n5. Ensure `Xlist` is `L1` and `Ylist` is `L2`, then select `Calculate`.\n6. The value of $r$ shown is the Spearman's rank correlation coefficient $r_s \\approx 0.905$."},"desmos":{"expression":"y_1 ~ m x_1 + c","instructions":"1. Click the '+' button and select 'Table'.\n2. In $x_1$, enter the Actual Ranks: `2, 8, 4, 1, 6, 3, 7, 5`.\n3. In $y_1$, enter the Estimated Ranks: `4, 8, 3, 1, 5, 2, 7, 6`.\n4. In a new expression line, type: `y_1 ~ m x_1 + c`.\n5. Look at the value listed next to $r$ under 'Statistics'. It displays $r = 0.90476...$, which rounds to $0.905$."},"description":"Enter the paired ranks into List 1 and List 2 to perform a Linear Regression calculation."}]},{"type":"text","order":3,"title":"Compare with Critical Value","content":"From statistical tables, the critical value for Spearman's rank correlation coefficient with sample size $n = 8$ at the $5\\%$ level of significance (one-tailed) is:\n\n$$\\text{Critical Value} = 0.6429$$\n\nNow, compare the calculated $r_s$ value to the critical value:\n\n$$0.905 > 0.6429$$\n\nSince $0.905$ is greater than the critical value, our result lies in the critical region."},{"type":"text","order":4,"title":"Draw Conclusion","content":"Because the calculated $r_s$ exceeds the critical value, we **reject $H_0$**.\n\nThere is significant evidence, at the $5\\%$ level of significance, to suggest a positive correlation between the sommelier's estimates and the actual maturation periods."}]}],"generatedAt":"2026-06-11T03:05:19.478Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"68d4f1f5-a502-40d8-bd3d-833c47159733:manual_formula","label":"(a) Manual Calculation using Formula","steps":[{"type":"text","order":0,"title":"Rank the actual periods","content":"To calculate Spearman's rank correlation coefficient ($r_s$), we first need to rank the actual maturation periods in descending order (from longest to shortest, so $1$ is the longest maturation period and $8$ is the shortest).\n\nLet's list the actual maturation periods:\n- **A**: 54 months\n- **B**: 12 months\n- **C**: 38 months\n- **D**: 62 months\n- **E**: 24 months\n- **F**: 45 months\n- **G**: 18 months\n- **H**: 30 months\n\nSorting these values from largest to smallest:\n$$\\text{D (62) > A (54) > F (45) > C (38) > H (30) > E (24) > G (18) > B (12)}$$\n\nThis gives us the **Actual Rank ($R_{\\text{actual}}$)** for each wine:\n\n$$\\begin{array}{|c|c|c|c|c|c|c|c|c|} \\hline \\text{Wine} & A & B & C & D & E & F & G & H \\\\ \\hline R_{\\text{actual}} & 2 & 8 & 4 & 1 & 6 & 3 & 7 & 5 \\\\ \\hline \\end{array}$$"},{"type":"text","order":1,"title":"Rank the sommelier's order","content":"Next, we find the ranks assigned by the sommelier. The sommelier provided the sequence in descending order of maturation: \n$$D, F, C, A, E, H, G, B$$\n\nAssigning ranks $1$ to $8$ from left to right gives the **Sommelier's Rank ($R_{\\text{somm}}$)**:\n- **D** is 1st $\\rightarrow 1$\n- **F** is 2nd $\\rightarrow 2$\n- **C** is 3rd $\\rightarrow 3$\n- **A** is 4th $\\rightarrow 4$\n- **E** is 5th $\\rightarrow 5$\n- **H** is 6th $\\rightarrow 6$\n- **G** is 7th $\\rightarrow 7$\n- **B** is 8th $\\rightarrow 8$\n\nSummarizing the sommelier's ranks for each wine alphabetically:\n\n$$\\begin{array}{|c|c|c|c|c|c|c|c|c|} \\hline \\text{Wine} & A & B & C & D & E & F & G & H \\\\ \\hline R_{\\text{somm}} & 4 & 8 & 3 & 1 & 5 & 2 & 7 & 6 \\\\ \\hline \\end{array}$$"},{"type":"text","order":2,"title":"Calculate rank differences","content":"Now, we find the difference in ranks $d = R_{\\text{actual}} - R_{\\text{somm}}$ and calculate $d^2$ for each wine:\n\n$$\\begin{array}{|c|c|c|c|c|} \\hline \\text{Wine} & R_{\\text{actual}} & R_{\\text{somm}} & d & d^2 \\\\ \\hline A & 2 & 4 & -2 & 4 \\\\ B & 8 & 8 & 0 & 0 \\\\ C & 4 & 3 & 1 & 1 \\\\ D & 1 & 1 & 0 & 0 \\\\ E & 6 & 5 & 1 & 1 \\\\ F & 3 & 2 & 1 & 1 \\\\ G & 7 & 7 & 0 & 0 \\\\ H & 5 & 6 & -1 & 1 \\\\ \\hline \\end{array}$$\n\nSumming the $d^2$ values:\n$$\\sum d^2 = 4 + 0 + 1 + 0 + 1 + 1 + 0 + 1 = 8$$"},{"type":"text","order":3,"title":"Apply Spearman's formula","content":"With $n = 8$ wines and $\\sum d^2 = 8$, we use the Spearman's rank correlation coefficient formula:\n\n$$r_s = 1 - \\frac{6\\sum d^2}{n(n^2 - 1)}$$\n\nSubstitute our values into the formula:\n\n$$\\begin{aligned} r_s &= 1 - \\frac{6(8)}{8(8^2 - 1)} \\\\ &= 1 - \\frac{48}{8(63)} \\\\ &= 1 - \\frac{48}{504} \\\\ &= 1 - \\frac{2}{21} \\\\ &= \\frac{19}{21} \\approx 0.905 \\end{aligned}$$\n\nTo 3 decimal places, the Spearman's rank correlation coefficient is **$0.905$**."}]},{"id":"68d4f1f5-a502-40d8-bd3d-833c47159733:gdc_calculation","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 (GDC), we first need to assign ranks to both sets of data. \n\nOnce both sets of data are ranked, we can enter these ranks as lists into our GDC and perform a linear regression calculation. The Pearson correlation coefficient ($r$) of these ranks is mathematically equivalent to Spearman's rank correlation coefficient ($r_s$)."},{"type":"text","order":1,"title":"Rank actual maturation periods","content":"Let's first rank the actual maturation periods in descending order (from longest to shortest, meaning $1$ is the oldest and $8$ is the youngest).\n\nLet's list the wines and their actual maturation periods in months:\n- $D$: $62$ months $\\rightarrow$ Rank **$1$**\n- $A$: $54$ months $\\rightarrow$ Rank **$2$**\n- $F$: $45$ months $\\rightarrow$ Rank **$3$**\n- $C$: $38$ months $\\rightarrow$ Rank **$4$**\n- $H$: $30$ months $\\rightarrow$ Rank **$5$**\n- $E$: $24$ months $\\rightarrow$ Rank **$6$**\n- $G$: $18$ months $\\rightarrow$ Rank **$7$**\n- $B$: $12$ months $\\rightarrow$ Rank **$8$**\n\nSorting them by wine name, the actual ranks are:\n$$\\begin{array}{|l|c|c|c|c|c|c|c|c|} \\hline \\text{Wine} & A & B & C & D & E & F & G & H \\\\ \\hline \\text{Actual Rank} & 2 & 8 & 4 & 1 & 6 & 3 & 7 & 5 \\\\ \\hline \\end{array}$$","highlights":[{"text":"The actual maturation periods, in months, for the 8 wines are given in the following table","location":"specification"}]},{"type":"text","order":2,"title":"Rank perceived periods","content":"Next, let's identify the ranks from the sommelier's sequence. She listed the wines in descending order of her perceived maturation periods as:\n$$\\text{D, F, C, A, E, H, G, B}$$\n\nThis means she assigned the following ranks directly:\n- $D$ is $1^{\\text{st}}$ $\\rightarrow$ Rank **$1$**\n- $F$ is $2^{\\text{nd}}$ $\\rightarrow$ Rank **$2$**\n- $C$ is $3^{\\text{rd}}$ $\\rightarrow$ Rank **$3$**\n- $A$ is $4^{\\text{th}}$ $\\rightarrow$ Rank **$4$**\n- $E$ is $5^{\\text{th}}$ $\\rightarrow$ Rank **$5$**\n- $H$ is $6^{\\text{th}}$ $\\rightarrow$ Rank **$6$**\n- $G$ is $7^{\\text{th}}$ $\\rightarrow$ Rank **$7$**\n- $B$ is $8^{\\text{th}}$ $\\rightarrow$ Rank **$8$**\n\nSorting these by wine name to line up with our actual ranks table:\n$$\\begin{array}{|l|c|c|c|c|c|c|c|c|} \\hline \\text{Wine} & A & B & C & D & E & F & G & H \\\\ \\hline \\text{Sommelier Rank} & 4 & 8 & 3 & 1 & 5 & 2 & 7 & 6 \\\\ \\hline \\end{array}$$","highlights":[{"text":"descending order of their maturation period (from the oldest to the youngest)","location":"specification"},{"text":"D, F, C, A, E, H, G, B","location":"specification"}]},{"type":"text","order":3,"title":"Use your graphing calculator","content":"Now, enter these two sets of ranks into your graphing calculator as lists: \n\n$$\\begin{aligned} L_1 & = \\{2, 8, 4, 1, 6, 3, 7, 5\\} \\\\ L_2 & = \\{4, 8, 3, 1, 5, 2, 7, 6\\} \\end{aligned}$$\n\nRun a 2-variable statistics or a linear regression calculation ($ax+b$) to find the correlation coefficient $r$.","calculator":[{"gdc":{"instructions":"1. Go to the Statistics / List editor application.\n2. In List 1, enter: 2, 8, 4, 1, 6, 3, 7, 5.\n3. In List 2, enter: 4, 8, 3, 1, 5, 2, 7, 6.\n4. Perform a Linear Regression calculation (usually under Calc -> Reg -> X).\n5. Read the value for 'r'."},"ti84":{"instructions":"1. Press [STAT] and select 1:Edit...\n2. In L1, enter the actual ranks: 2, 8, 4, 1, 6, 3, 7, 5.\n3. In L2, enter the sommelier's ranks: 4, 8, 3, 1, 5, 2, 7, 6.\n4. Press [STAT], arrow over to CALC, and select 4:LinReg(ax+b).\n5. Set Xlist to L1, Ylist to L2, and press Calculate.\n(Note: If 'r' does not appear, press [2nd] [CATALOG], scroll to 'DiagnosticOn', press [ENTER] twice, and rerun the regression)."},"desmos":{"instructions":"1. Click the plus sign and add a table.\n2. In the x1 column, enter: 2, 8, 4, 1, 6, 3, 7, 5.\n3. In the y1 column, enter: 4, 8, 3, 1, 5, 2, 7, 6.\n4. In a new expression line, type: y1 ~ m * x1 + c\n5. Look at the output value for r."},"description":"Enter ranks into lists and calculate the correlation coefficient."}]},{"type":"text","order":4,"title":"State the final answer","content":"From your calculator, the correlation coefficient is:\n\n$$r \\approx 0.9047619...$$\n\nRounding to $3$ decimal places as requested, we obtain:\n\n$$r_s \\approx 0.905$$","highlights":[{"text":"giving your answer to 3 decimal places.","location":"specification"}]}]},{"id":"313220ba-9e81-4571-81bc-079d73af6a2a:formula_critical_value","label":"(b) Analytical Calculation and Table Comparison","steps":[{"type":"text","order":0,"title":"State the Hypotheses","content":"To test whether there is a positive correlation between the sommelier's estimates and the true maturation periods, we formulate the null and alternative hypotheses in terms of the population Spearman rank correlation coefficient, $\\rho_s$:\n\n* **Null Hypothesis ($H_0$)**: $\\rho_s = 0$ (There is no correlation between the sommelier's estimates and the actual maturation periods.)\n* **Alternative Hypothesis ($H_1$)**: $\\rho_s > 0$ (There is a positive correlation between the sommelier's estimates and the actual maturation periods.)","highlights":[{"text":"positive correlation","location":"specification"},{"text":"State your hypotheses clearly.","location":"specification"}]},{"type":"text","order":1,"title":"Rank the Data","content":"To calculate the Spearman's rank correlation coefficient $r_s$, we assign ranks to both the actual maturation periods and the sommelier's estimated order. We will rank from oldest (1) to youngest (8):\n\n**Actual Maturation Ranks ($R_x$):**\n* $D$ (62 months) $\\rightarrow 1$\n* $A$ (54 months) $\\rightarrow 2$\n* $F$ (45 months) $\\rightarrow 3$\n* $C$ (38 months) $\\rightarrow 4$\n* $H$ (30 months) $\\rightarrow 5$\n* $E$ (24 months) $\\rightarrow 6$\n* $G$ (18 months) $\\rightarrow 7$\n* $B$ (12 months) $\\rightarrow 8$\n\n**Sommelier's Ranks ($R_y$):**\nBased on her list $D, F, C, A, E, H, G, B$:\n* $D \\rightarrow 1$\n* $F \\rightarrow 2$\n* $C \\rightarrow 3$\n* $A \\rightarrow 4$\n* $E \\rightarrow 5$\n* $H \\rightarrow 6$\n* $G \\rightarrow 7$\n* $B \\rightarrow 8$"},{"type":"text","order":2,"title":"Calculate Rank Differences","content":"We calculate the difference in ranks, $d = R_x - R_y$, and its square, $d^2$, for each wine:\n\n$$\\begin{array}{|c|c|c|c|c|} \\hline \\text{Wine} & R_x & R_y & d & d^2 \\\\ \\hline A & 2 & 4 & -2 & 4 \\\\ B & 8 & 8 & 0 & 0 \\\\ C & 4 & 3 & 1 & 1 \\\\ D & 1 & 1 & 0 & 0 \\\\ E & 6 & 5 & 1 & 1 \\\\ F & 3 & 2 & 1 & 1 \\\\ G & 7 & 7 & 0 & 0 \\\\ H & 5 & 6 & -1 & 1 \\\\ \\hline \\end{array}$$\n\nSumming the squared differences:\n$$\\sum d^2 = 4 + 0 + 1 + 0 + 1 + 1 + 0 + 1 = 8$$"},{"type":"text","order":3,"title":"Calculate Spearman's Coefficient","content":"We substitute $\\sum d^2 = 8$ and $n = 8$ into the Spearman's rank correlation coefficient formula:\n\n$$r_s = 1 - \\frac{6\\sum d^2}{n(n^2 - 1)}$$\n\n$$\\begin{aligned} r_s &= 1 - \\frac{6(8)}{8(8^2 - 1)} \\\\ r_s &= 1 - \\frac{48}{8(63)} \\\\ r_s &= 1 - \\frac{48}{504} \\\\ r_s &= 1 - \\frac{2}{21} \\\\ r_s &\\approx 0.905 \\end{aligned}$$"},{"type":"text","order":4,"title":"Compare with Critical Value","content":"Now we determine the critical value from the statistical tables for a **one-tailed test** at the **5% level of significance** with a sample size of $n = 8$:\n\n$$\\text{Critical Value} = 0.6429$$\n\nWe compare our test statistic $r_s$ against this critical value:\n\n$$0.905 > 0.6429$$\n\nSince our calculated value of $r_s$ is greater than the critical value, we reject the null hypothesis $H_0$.","highlights":[{"text":"5% significance level","location":"specification"}]},{"type":"text","order":5,"title":"State the Conclusion","content":"There is **sufficient evidence** at the 5% significance level to reject $H_0$.\n\nTherefore, we conclude that there is significant evidence of a positive correlation between the sommelier's estimates and the actual maturation periods of the wines."}]},{"id":"313220ba-9e81-4571-81bc-079d73af6a2a:gdc_ranks","label":"(b) Using Graphic Display Calculator (GDC)","steps":[{"type":"text","order":0,"title":"State the Hypotheses","content":"To test for a positive correlation between the sommelier's estimates and the actual maturation periods, we use Spearman's rank correlation coefficient, denoted by $\\rho_s$ (rho).\n\n* **Null Hypothesis ($H_0$):** There is no correlation between the sommelier's estimates and the true maturation periods.\n $$H_0: \\rho_s = 0$$\n* **Alternative Hypothesis ($H_1$):** There is a positive correlation between the sommelier's estimates and the true maturation periods.\n $$H_1: \\rho_s > 0$$\n\nThis is a **one-tailed test** at the **$5\\%$ significance level**.","highlights":[{"text":"evidence of a positive correlation","location":"specification"},{"text":"5% significance level","location":"specification"}]},{"type":"text","order":1,"title":"Rank the Data","content":"$94"},{"type":"text","order":2,"title":"Calculate r_s using GDC","content":"Now, enter these ranks into your GDC to directly compute the correlation coefficient, $r_s$.","calculator":[{"gdc":{"instructions":"1. Open the Statistics application.\n2. In List 1 (or List A), enter: `2, 8, 4, 1, 6, 3, 7, 5`.\n3. In List 2 (or List B), enter: `4, 8, 3, 1, 5, 2, 7, 6`.\n4. Select `CALC` (or `Regression`), then choose `Linear Regression` (usually $X$ or $a+bx$).\n5. The correlation coefficient $r$ will be displayed as $r_s \\approx 0.905$."},"ti84":{"instructions":"1. Press [STAT] and select `1:Edit...`.\n2. In `L1`, enter the Actual Ranks: `2, 8, 4, 1, 6, 3, 7, 5`.\n3. In `L2`, enter the Estimated Ranks: `4, 8, 3, 1, 5, 2, 7, 6`.\n4. Press [STAT], arrow over to `CALC`, and select `4:LinReg(ax+b)`.\n5. Ensure `Xlist` is `L1` and `Ylist` is `L2`, then select `Calculate`.\n6. The value of $r$ shown is the Spearman's rank correlation coefficient $r_s \\approx 0.905$."},"desmos":{"expression":"y_1 ~ m x_1 + c","instructions":"1. Click the '+' button and select 'Table'.\n2. In $x_1$, enter the Actual Ranks: `2, 8, 4, 1, 6, 3, 7, 5`.\n3. In $y_1$, enter the Estimated Ranks: `4, 8, 3, 1, 5, 2, 7, 6`.\n4. In a new expression line, type: `y_1 ~ m x_1 + c`.\n5. Look at the value listed next to $r$ under 'Statistics'. It displays $r = 0.90476...$, which rounds to $0.905$."},"description":"Enter the paired ranks into List 1 and List 2 to perform a Linear Regression calculation."}]},{"type":"text","order":3,"title":"Compare with Critical Value","content":"From statistical tables, the critical value for Spearman's rank correlation coefficient with sample size $n = 8$ at the $5\\%$ level of significance (one-tailed) is:\n\n$$\\text{Critical Value} = 0.6429$$\n\nNow, compare the calculated $r_s$ value to the critical value:\n\n$$0.905 > 0.6429$$\n\nSince $0.905$ is greater than the critical value, our result lies in the critical region."},{"type":"text","order":4,"title":"Draw Conclusion","content":"Because the calculated $r_s$ exceeds the critical value, we **reject $H_0$**.\n\nThere is significant evidence, at the $5\\%$ level of significance, to suggest a positive correlation between the sommelier's estimates and the actual maturation 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2 Statistics

Question 207

2 Statistics Questions

2 Statistics

Question 207

2 Statistics Questions