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In a series of 50 flight trials, the product moment correlation coefficient was calculated to be $r = 0.354$. Assuming the data is drawn from a population follows a bivariate normal distribution, perform a hypothesis test at the 1% level of significance to determine whether there is evidence of a positive correlation between vibration frequency and cabin noise. The critical value for the product moment correlation coefficient at the 1% significance level for a sample size of 50 is $0.3281$.","markscheme":"- States $H_0: \\rho = 0$ and $H_1: \\rho > 0$, where $\\rho$ is the population correlation coefficient B1\n- Compares the test statistic $r = 0.354$ with the critical value $0.3281$ (e.g., $0.354 > 0.3281$) M1\n- Makes the decision to reject $H_0$ (since the test statistic falls in the critical region) A1\n- Concludes in context that there is sufficient evidence at the 1% significance level to suggest that a positive correlation exists between wing-tip vibration frequency and internal cabin noise B1","marks":4,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"critical_value_comparison","label":"Critical Value Comparison","steps":[{"type":"text","order":0,"title":"State the Hypotheses","content":"To perform a hypothesis test for the population correlation coefficient, we must first define our null and alternative hypotheses. Let $\\rho$ represent the population correlation coefficient between wing-tip vibration frequency and internal cabin noise level.\n\nWe want to test if there is a positive correlation, which means we are conducting a one-tailed test:\n\n$$H_0: \\rho = 0$$\n$$H_1: \\rho > 0$$\n\n* **$H_0$ (Null Hypothesis):** There is no correlation between wing-tip vibration frequency and internal cabin noise in the population.\n* **$H_1$ (Alternative Hypothesis):** There is a positive correlation between wing-tip vibration frequency and internal cabin noise in the population.","highlights":[{"text":"positive correlation between vibration frequency and cabin noise","location":"specification"}]},{"type":"text","order":1,"title":"Compare the Coefficients","content":"In a critical value comparison, we directly compare our calculated sample correlation coefficient, $r$, with the given critical value for the test.\n\nFrom the question, we have:\n* Calculated sample correlation coefficient: $r = 0.354$\n* Critical value at the 1% significance level (for $n = 50$): $0.3281$\n\nComparing these two values:\n\n$$0.354 > 0.3281$$","highlights":[{"text":"product moment correlation coefficient was calculated to be $r = 0.354$","location":"specification"},{"text":"critical value for the product moment correlation coefficient at the 1% significance level for a sample size of 50 is $0.3281$","location":"specification"}]},{"type":"text","order":2,"title":"Make a Decision","content":"Since our calculated correlation coefficient $r = 0.354$ is greater than the critical value $0.3281$, the test statistic falls inside the critical region (the rejection region).\n\nTherefore, we **reject $H_0$**."},{"type":"text","order":3,"title":"Conclude in Context","content":"We conclude by writing a final sentence that links our mathematical decision back to the context of the problem.\n\nThere is **sufficient evidence** at the 1% significance level to suggest that there is a **positive correlation** between wing-tip vibration frequency and internal cabin noise level of the prototype drone."}]},{"id":"p_value_t_test","label":"Using p-value / t-test","steps":[{"type":"text","order":0,"title":"State the hypotheses","content":"To test for a positive correlation between wing-tip vibration frequency and internal cabin noise level, we first state our null and alternative hypotheses. \n\nLet $\\rho$ represent the population correlation coefficient.\n\n$$H_0: \\rho = 0 \\quad \\text{(no correlation)}$$\n$$H_1: \\rho > 0 \\quad \\text{(positive correlation)}$$\n\nThis is a **one-tailed test** at the 1% (0.01) level of significance.","highlights":[{"text":"positive correlation","location":"specification"}]},{"type":"text","order":1,"title":"Calculate the t-statistic","content":"For a sample size of $n$, we can test the significance of a product moment correlation coefficient $r$ by converting it into a t-statistic using the formula:\n\n$$t = r\\sqrt{\\frac{n-2}{1-r^2}}$$\n\nGiven $n = 50$ and $r = 0.354$, we substitute these values into the formula:\n\n$$\\begin{aligned} t &= 0.354\\sqrt{\\frac{50-2}{1-0.354^2}} \\\\ t &= 0.354\\sqrt{\\frac{48}{0.874684}} \\\\ t &\\approx 2.6186 \\end{aligned}$$\n\nThis test statistic follows a t-distribution with $df = n - 2 = 48$ degrees of freedom.","highlights":[{"text":"50 flight trials","location":"specification"},{"text":"r = 0.354","location":"specification"}]},{"type":"text","order":2,"title":"Find the p-value","content":"Using a graphing calculator, we can find the p-value corresponding to our test statistic $t \\approx 2.6186$ with 48 degrees of freedom for a one-tailed upper test:\n\n$$\\text{P}(T > 2.6186) \\approx 0.0058$$\n\nAlternatively, modern graphing calculators can compute the p-value directly from the correlation coefficient using a Linear Regression t-test.","calculator":[{"gdc":{"expression":"tCdf(2.6186, 1E99, 48)","instructions":"1. Go to the Probability/Statistics menu.\n2. Select Tests, then select Linear Regression t-Test or t-Distribution Cumulative Density (t-CDF).\n3. Input the lower bound 2.6186, upper bound as 1E99, and degrees of freedom 48."},"ti84":{"expression":"tcdf(2.6186, 1E99, 48)","instructions":"1. Press [STAT], arrow over to [TESTS], and select [F:LinRegTTest].\n2. Set [Xlist] and [Ylist] to empty lists if you don't have raw data, or enter mock data to match $r=0.354, n=50$. Alternatively, use the cumulative distribution function: \nPress [2nd] [VARS] (DISTR), select [6:tcdf(].\n3. Enter the lower bound as 2.6186, upper bound as 1E99, and df as 48.\n4. Paste and press [ENTER]."},"desmos":{"expression":"1 - tdist(48).cdf(2.6186)","instructions":"1. In an empty cell, use the t-distribution command to define the distribution with 48 degrees of freedom: `T = tdist(48)`\n2. To find the cumulative probability above 2.6186, calculate: `1 - T.cdf(2.6186)`"},"description":"Calculate the p-value from a correlation test."}]},{"type":"text","order":3,"title":"Compare with significance level","content":"Now, we compare our calculated p-value to the significance level $\\alpha = 0.01$:\n\n$$0.0058 < 0.01$$\n\nSince the p-value ($0.0058$) is less than the 1% significance level, the result is statistically significant. Therefore, we **reject $H_0$**.","highlights":[{"text":"1% level of significance","location":"specification"}]},{"type":"text","order":4,"title":"State the final conclusion","content":"Finally, we state our conclusion in the context of the problem:\n\nThere is sufficient evidence at the 1% significance level to suggest that there is a **positive correlation** between the wing-tip vibration frequency and the internal cabin noise level of the prototype drone."}]}],"generatedAt":"2026-06-09T04:22:53.662Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"4f48f814-eba1-496c-9cbc-018c3a2489b0:critical_value_comparison","label":"Critical Value Comparison","steps":[{"type":"text","order":0,"title":"State the Hypotheses","content":"To perform a hypothesis test for the population correlation coefficient, we must first define our null and alternative hypotheses. Let $\\rho$ represent the population correlation coefficient between wing-tip vibration frequency and internal cabin noise level.\n\nWe want to test if there is a positive correlation, which means we are conducting a one-tailed test:\n\n$$H_0: \\rho = 0$$\n$$H_1: \\rho > 0$$\n\n* **$H_0$ (Null Hypothesis):** There is no correlation between wing-tip vibration frequency and internal cabin noise in the population.\n* **$H_1$ (Alternative Hypothesis):** There is a positive correlation between wing-tip vibration frequency and internal cabin noise in the population.","highlights":[{"text":"positive correlation between vibration frequency and cabin noise","location":"specification"}]},{"type":"text","order":1,"title":"Compare the Coefficients","content":"In a critical value comparison, we directly compare our calculated sample correlation coefficient, $r$, with the given critical value for the test.\n\nFrom the question, we have:\n* Calculated sample correlation coefficient: $r = 0.354$\n* Critical value at the 1% significance level (for $n = 50$): $0.3281$\n\nComparing these two values:\n\n$$0.354 > 0.3281$$","highlights":[{"text":"product moment correlation coefficient was calculated to be $r = 0.354$","location":"specification"},{"text":"critical value for the product moment correlation coefficient at the 1% significance level for a sample size of 50 is $0.3281$","location":"specification"}]},{"type":"text","order":2,"title":"Make a Decision","content":"Since our calculated correlation coefficient $r = 0.354$ is greater than the critical value $0.3281$, the test statistic falls inside the critical region (the rejection region).\n\nTherefore, we **reject $H_0$**."},{"type":"text","order":3,"title":"Conclude in Context","content":"We conclude by writing a final sentence that links our mathematical decision back to the context of the problem.\n\nThere is **sufficient evidence** at the 1% significance level to suggest that there is a **positive correlation** between wing-tip vibration frequency and internal cabin noise level of the prototype drone."}]},{"id":"4f48f814-eba1-496c-9cbc-018c3a2489b0:p_value_t_test","label":"Using p-value / t-test","steps":[{"type":"text","order":0,"title":"State the hypotheses","content":"To test for a positive correlation between wing-tip vibration frequency and internal cabin noise level, we first state our null and alternative hypotheses. \n\nLet $\\rho$ represent the population correlation coefficient.\n\n$$H_0: \\rho = 0 \\quad \\text{(no correlation)}$$\n$$H_1: \\rho > 0 \\quad \\text{(positive correlation)}$$\n\nThis is a **one-tailed test** at the 1% (0.01) level of significance.","highlights":[{"text":"positive correlation","location":"specification"}]},{"type":"text","order":1,"title":"Calculate the t-statistic","content":"For a sample size of $n$, we can test the significance of a product moment correlation coefficient $r$ by converting it into a t-statistic using the formula:\n\n$$t = r\\sqrt{\\frac{n-2}{1-r^2}}$$\n\nGiven $n = 50$ and $r = 0.354$, we substitute these values into the formula:\n\n$$\\begin{aligned} t &= 0.354\\sqrt{\\frac{50-2}{1-0.354^2}} \\\\ t &= 0.354\\sqrt{\\frac{48}{0.874684}} \\\\ t &\\approx 2.6224 \\end{aligned}$$\n\nThis test statistic follows a t-distribution with $df = n - 2 = 48$ degrees of freedom.","highlights":[{"text":"50 flight trials","location":"specification"},{"text":"r = 0.354","location":"specification"}]},{"type":"text","order":2,"title":"Find the p-value","content":"Using a graphing calculator, we can find the p-value corresponding to our test statistic $t \\approx 2.6224$ with 48 degrees of freedom for a one-tailed upper test:\n\n$$\\text{P}(T > 2.6224) \\approx 0.0058$$\n\nAlternatively, modern graphing calculators can compute the p-value directly from the correlation coefficient using a Linear Regression t-test.","calculator":[{"gdc":{"expression":"tCdf(2.6186, 1E99, 48)","instructions":"1. Go to the Probability/Statistics menu.\n2. Select Tests, then select Linear Regression t-Test or t-Distribution Cumulative Density (t-CDF).\n3. Input the lower bound 2.6186, upper bound as 1E99, and degrees of freedom 48."},"ti84":{"expression":"tcdf(2.6186, 1E99, 48)","instructions":"1. Press [STAT], arrow over to [TESTS], and select [F:LinRegTTest].\n2. Set [Xlist] and [Ylist] to empty lists if you don't have raw data, or enter mock data to match $r=0.354, n=50$. Alternatively, use the cumulative distribution function: \nPress [2nd] [VARS] (DISTR), select [6:tcdf(].\n3. Enter the lower bound as 2.6186, upper bound as 1E99, and df as 48.\n4. Paste and press [ENTER]."},"desmos":{"expression":"1 - tdist(48).cdf(2.6186)","instructions":"1. In an empty cell, use the t-distribution command to define the distribution with 48 degrees of freedom: `T = tdist(48)`\n2. To find the cumulative probability above 2.6186, calculate: `1 - T.cdf(2.6186)`"},"description":"Calculate the p-value from a correlation test."}]},{"type":"text","order":3,"title":"Compare with significance level","content":"Now, we compare our calculated p-value to the significance level $\\alpha = 0.01$:\n\n$$0.0058 < 0.01$$\n\nSince the p-value ($0.0058$) is less than the 1% significance level, the result is statistically significant. Therefore, we **reject $H_0$**.","highlights":[{"text":"1% level of significance","location":"specification"}]},{"type":"text","order":4,"title":"State the final conclusion","content":"Finally, we state our conclusion in the context of the problem:\n\nThere is sufficient evidence at the 1% significance level to suggest that there is a **positive correlation** between the wing-tip vibration frequency and the internal cabin noise level of the prototype drone."}]}],"generatedAt":"2026-06-09T04:23:20.947Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"3be2911f-bc9b-484b-9f63-1a62aa806886","revisionId":"adee81c4-0673-4f46-964c-a342f8b2a18c","questionNumber":null,"questionType":"LA","specification":"A botanist records leaf length and leaf width for a random sample of 20 plants of one species. The sample product moment correlation coefficient is $0.431$. For a two-tailed test at the $5\\%$ significance level, the critical values are $\\pm0.4438$.","difficulty":null,"difficultyBand":"medium","tier":null,"totalMarks":4,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"18aca9a8-1c07-4e26-8b9a-f8706fd6a1c5","label":null,"content":"Carry out the test and interpret the result.","markscheme":"- $H_0:\\rho=0$ and $H_1:\\rho\\ne0$ B1\n- $|0.431|<0.4438$ M1\n- There is insufficient evidence to reject $H_0$ A1\n- There is insufficient evidence at the $5\\%$ significance level of correlation between leaf length and leaf width in this species A1","marks":4,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"3c2b2eab-ca63-46ea-a663-2cb8c60a4c98","revisionId":"ca50fe5a-b3c6-4639-92d8-58efb6cea16a","questionNumber":null,"questionType":"LA","specification":"A chemical engineer is investigating the relationship between the temperature of a solvent, \$T\$ (in \$^\\circ\\text{C}\$), and the mass of a specific solute that dissolves in it, \$d\$ (in grams). A random sample of 15 trials was conducted, and the following summary statistics were recorded:\n\n\\[\\sum T = 975 \\quad \\sum T^2 = 64125 \\quad \\sum d = 420 \\quad S_{dd} = 560 \\quad \\sum Td = 27930\\]","difficulty":4,"difficultyBand":"medium","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"ae182c64-8188-4851-8b7a-abbd058cd1fa","label":"a","content":"Find the values of \$S_{TT}\$ and \$S_{Td}\$.","markscheme":"- Calculates $S_{TT} = 64125 - \\frac{975^2}{15} = 750$ B1\n- Calculates $S_{Td} = 27930 - \\frac{975 \\times 420}{15} = 630$ B1","marks":2,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"direct_formula","label":"Direct Formula Application","steps":[{"type":"text","order":0,"title":"Identify given statistics","content":"To find the sum of squares $S_{TT}$ and the sum of products $S_{Td}$, we first list the summary statistics given in the question:\n\n* Number of trials, $n = 15$\n* $\\sum T = 975$\n* $\\sum T^2 = 64125$\n* $\\sum d = 420$\n* $\\sum Td = 27930$","highlights":[{"text":"15 trials","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"text":"We use the standard formula for the sum of squares:"},{"latex":"$$S_{TT} = \\sum T^2 - \\frac{\\left(\\sum T\\right)^2}{n}$"},{"text":"Substitute the given values into the formula:"},{"latex":"$$S_{TT} = \\htmlData{anchor=sum-t2}{\\color{@sky}{64125}} - \\frac{\\left(\\htmlData{anchor=sum-t}{\\color{@amber}{975}}\\right)^2}{\\htmlData{anchor=n-val}{\\color{@purple}{15}}}$"},{"latex":"$$S_{TT} = 64125 - 63375 = 750$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sum-t2"},{"color":"amber","style":"text-color","anchor":"sum-t"},{"color":"purple","style":"text-color","anchor":"n-val"}],"connections":[]},"order":1,"title":"Calculate S_TT","description":"Calculate $S_{TT}$ by substituting $\\sum T^2$, $\\sum T$, and $n$ directly into the formula."},{"type":"hyper_latex","block":{"nodes":[{"text":"Now, we use the standard formula for the sum of products:"},{"latex":"$$S_{Td} = \\sum Td - \\frac{\\left(\\sum T\\right)\\left(\\sum d\\right)}{n}$"},{"text":"Substitute the given values into the formula:"},{"latex":"$$S_{Td} = \\htmlData{anchor=sum-td}{\\color{@sky}{27930}} - \\frac{\\left(\\htmlData{anchor=sum-t-2}{\\color{@amber}{975}}\\right)\\left(\\htmlData{anchor=sum-d}{\\color{@teal}{420}}\\right)}{\\htmlData{anchor=n-val-2}{\\color{@purple}{15}}}$"},{"latex":"$$S_{Td} = 27930 - 27300 = 630$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sum-td"},{"color":"amber","style":"text-color","anchor":"sum-t-2"},{"color":"teal","style":"text-color","anchor":"sum-d"},{"color":"purple","style":"text-color","anchor":"n-val-2"}],"connections":[]},"order":2,"title":"Calculate S_Td","description":"Calculate $S_{Td}$ by substituting $\\sum Td$, $\\sum T$, $\\sum d$, and $n$ directly into the formula."},{"type":"text","order":3,"title":"State the final answers","content":"The calculated values are:\n\n* $S_{TT} = 750$\n* $S_{Td} = 630$\n\nEach calculation earns 1 mark in this 2-mark question."}]}],"generatedAt":"2026-06-09T04:36:55.338Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"b7d087f9-e244-4a77-8c98-a874eb02a71b","label":"b","content":"Calculate, to 3 significant figures, the product moment correlation coefficient between \$T\$ and \$d\$.","markscheme":"- Uses $r = \\frac{S_{Td}}{\\sqrt{S_{TT} S_{dd}}} = \\frac{630}{\\sqrt{750 \\times 560}}$ M1\n- Calculates $r = \\frac{630}{\\sqrt{420000}} \\approx 0.9721...$ which is $0.972$ (to 3 s.f.) A1","marks":2,"order":1,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"formulaic","label":"Standard Formula Approach","steps":[{"type":"text","order":0,"title":"Identify given statistics","content":"First, let's identify all the given values from the question that we will need for our calculations:\n\n* Sample size $n = 15$\n* $\\sum T = 975$\n* $\\sum T^2 = 64125$\n* $\\sum d = 420$\n* $S_{dd} = 560$\n* $\\sum Td = 27930$","highlights":[{"text":"15 trials","location":"specification"},{"text":"\\sum T = 975","location":"specification"},{"text":"\\sum T^2 = 64125","location":"specification"},{"text":"\\sum d = 420","location":"specification"},{"text":"S_{dd} = 560","location":"specification"},{"text":"\\sum Td = 27930","location":"specification"}]},{"type":"text","order":1,"title":"Calculate $S_{TT}$","content":"To find the product moment correlation coefficient, we first need to calculate the sum of squares for temperature, $S_{TT}$. \n\nThe standard formula for $S_{TT}$ from our formula booklet is:\n\n$$S_{TT} = \\sum T^2 - \\frac{(\\sum T)^2}{n}$$\n\nSubstituting our given values into the formula:\n\n$$\\begin{aligned} S_{TT} &= 64125 - \\frac{975^2}{15} \\\\ S_{TT} &= 64125 - \\frac{950625}{15} \\\\ S_{TT} &= 64125 - 63375 \\\\ S_{TT} &= 750 \\end{aligned}$$"},{"type":"text","order":2,"title":"Calculate $S_{Td}$","content":"Next, we calculate the sum of products, $S_{Td}$, using the formula:\n\n$$S_{Td} = \\sum Td - \\frac{(\\sum T)(\\sum d)}{n}$$\n\nSubstituting our values:\n\n$$\\begin{aligned} S_{Td} &= 27930 - \\frac{975 \\times 420}{15} \\\\ S_{Td} &= 27930 - \\frac{409500}{15} \\\\ S_{Td} &= 27930 - 27300 \\\\ S_{Td} &= 630 \\end{aligned}$$"},{"type":"text","order":3,"title":"Substitute into correlation formula","content":"The product moment correlation coefficient, $r$, is calculated using the formula:\n\n$$r = \\frac{S_{Td}}{\\sqrt{S_{TT} S_{dd}}}$$ \n\nWe are given $S_{dd} = 560$, and we calculated $S_{TT} = 750$ and $S_{Td} = 630$. Substituting these values gives:\n\n$$r = \\frac{630}{\\sqrt{750 \\times 560}}$$\n\n$$r = \\frac{630}{\\sqrt{420000}}$$\n\n$$r \\approx 0.972169...$$"},{"type":"text","order":4,"title":"State the final answer","content":"Rounding our result to 3 significant figures, we get:\n\n$$r = 0.972$$\n\nThis strong positive correlation indicates that as temperature increases, the mass of the solute that dissolves also increases significantly."}]}],"generatedAt":"2026-06-09T04:37:41.967Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"be57813f-3fde-4210-9b6d-9b73f6e3e141","label":"c","content":"The engineer also measured the viscosity of the solvent, \$v\$, during these 15 trials. The product moment correlation coefficient between \$T\$ and \$v\$ was found to be \$-0.552\$.\n\nDescribe, giving a reason, the nature of the correlation you would expect to find between \$v\$ and \$d\$.","markscheme":"- Identifies negative correlation B1\n- Justifies using the relationship: temperature is positively correlated with the amount of dissolved solute ($r = 0.972$), and viscosity is negatively correlated with temperature ($r = -0.552$). Therefore, as viscosity increases, temperature decreases, which corresponds to a decrease in the amount of solute dissolved. C1","marks":2,"order":2,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"pmcc_calculation","label":"PMCC Calculation and Sign Reasoning","steps":[{"type":"text","order":0,"title":"Recall PMCC formula components","content":"To understand the relationship between temperature $T$ and amount dissolved $d$, we must first calculate the product-moment correlation coefficient (PMCC), denoted by $r$, between these two variables.\n\nThe formulas for the sum of products $S_{Td}$ and sum of squares $S_{TT}$ from our summary statistics are:\n\n$$S_{Td} = \\sum Td - \\frac{\\left(\\sum T\\right)\\left(\\sum d\\right)}{n}$$\n\n$$S_{TT} = \\sum T^2 - \\frac{\\left(\\sum T\\right)^2}{n}$$\n\nwhere $n = 15$ is the number of trials.","highlights":[{"text":"15 trials","location":"specification"}]},{"type":"text","order":1,"title":"Calculate STd and STT","content":"Now substitute the given summary statistics into these formulas:\n\n$$\\begin{aligned} S_{Td} &= 27930 - \\frac{975 \\times 420}{15} \\\\ &= 27930 - 27300 \\\\ &= 630 \\end{aligned}$$\n\n$$\\begin{aligned} S_{TT} &= 64125 - \\frac{975^2}{15} \\\\ &= 64125 - 63375 \\\\ &= 750 \\end{aligned}$$"},{"type":"text","order":2,"title":"Calculate correlation coefficient r","content":"Using the calculated values and the given $S_{dd} = 560$, we can compute the PMCC $r_{Td}$:\n\n$$r_{Td} = \\frac{S_{Td}}{\\sqrt{S_{TT} \\times S_{dd}}}$$\n\n$$r_{Td} = \\frac{630}{\\sqrt{750 \\times 560}} = \\frac{630}{\\sqrt{420000}} = \\frac{630}{648.074...} \\approx 0.972$$"},{"type":"text","order":3,"title":"Analyse the signs of r","content":"Let's examine the relationships between the three variables:\n- Between $T$ and $d$: the PMCC is positive ($r_{Td} \\approx 0.972$). This means higher temperatures correspond to a greater amount of solute dissolved (positive correlation).\n- Between $T$ and $v$: the PMCC is negative ($r_{Tv} = -0.552$). This means higher temperatures correspond to lower viscosity (negative correlation).","highlights":[{"text":"-0.552","location":"specification"}]},{"type":"text","order":4,"title":"Determine final expected correlation","content":"By combining these two relationships:\n- As viscosity ($v$) increases, temperature ($T$) must decrease (due to negative correlation).\n- As temperature ($T$) decreases, the amount of dissolved solute ($d$) also decreases (due to positive correlation).\n\nTherefore, we expect a **negative correlation** between viscosity ($v$) and amount dissolved ($d$)."}]},{"id":"covariance_sign_analysis","label":"Sign of Covariance/S_Td Analysis","steps":[{"type":"text","order":0,"title":"Analyze the question task","content":"We need to describe and justify the correlation we would expect to find between the viscosity of the solvent ($v$) and the mass of dissolved solute ($d$).\n\nWe are given the following summary statistics from the sample of size $n = 15$:\n\n* $\\sum T = 975$\n* $\\sum d = 420$\n* $\\sum Td = 27930$\n\nAdditionally, the correlation coefficient between temperature ($T$) and viscosity ($v$) is $-0.552$."},{"type":"text","order":1,"title":"Determine the sign of S_Td","content":"To understand the relationship between temperature ($T$) and dissolved solute ($d$), let's find the sign of their covariance using the sum of products formula:\n\n$$S_{Td} = \\sum Td - \\frac{(\\sum T)(\\sum d)}{n}$$\n\nLet's substitute the given values into the formula:\n\n$$\\begin{aligned} S_{Td} &= 27930 - \\frac{(975)(420)}{15} \\\\ &= 27930 - 27300 \\\\ &= 630 \\end{aligned}$$\n\nSince $S_{Td} = 630 > 0$, the covariance is positive, meaning there is a **positive relationship** between temperature ($T$) and dissolved solute ($d$) (as temperature increases, dissolved solute increases)."},{"type":"text","order":2,"title":"Analyze relationship with viscosity","content":"Now let's examine how viscosity ($v$) relates to both variables:\n\n1. We established that temperature ($T$) and dissolved solute ($d$) have a **positive** correlation ($S_{Td} > 0$). This means that higher temperature generally corresponds to a higher mass of dissolved solute.\n2. We are given that temperature ($T$) and viscosity ($v$) have a **negative** correlation ($r = -0.552$). This means that higher temperature corresponds to lower viscosity, or conversely, higher viscosity corresponds to lower temperature."},{"type":"text","order":3,"title":"Deduce expected correlation","content":"Combining these two relationships:\n\n* As viscosity ($v$) increases, temperature ($T$) decreases (due to their negative correlation).\n* As temperature ($T$) decreases, the mass of dissolved solute ($d$) decreases (due to their positive correlation).\n\nTherefore, we expect a **negative correlation** between viscosity ($v$) and dissolved solute ($d$).\n\n### Markscheme Breakdown\n* **B1**: Correctly identifying that the correlation is **negative**.\n* **C1**: Justifying this by explaining that $T$ and $d$ are positively correlated, while $T$ and $v$ are negatively correlated, resulting in an overall negative relationship between $v$ and $d$."}]}],"generatedAt":"2026-06-09T04:52:24.026Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"ae182c64-8188-4851-8b7a-abbd058cd1fa:direct_formula","label":"(a) Direct Formula Application","steps":[{"type":"text","order":0,"title":"Identify given statistics","content":"To find the sum of squares $S_{TT}$ and the sum of products $S_{Td}$, we first list the summary statistics given in the question:\n\n* Number of trials, $n = 15$\n* $\\sum T = 975$\n* $\\sum T^2 = 64125$\n* $\\sum d = 420$\n* $\\sum Td = 27930$","highlights":[{"text":"15 trials","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"text":"We use the standard formula for the sum of squares:"},{"latex":"$$S_{TT} = \\sum T^2 - \\frac{\\left(\\sum T\\right)^2}{n}$"},{"text":"Substitute the given values into the formula:"},{"latex":"$$S_{TT} = \\htmlData{anchor=sum-t2}{\\color{@sky}{64125}} - \\frac{\\left(\\htmlData{anchor=sum-t}{\\color{@amber}{975}}\\right)^2}{\\htmlData{anchor=n-val}{\\color{@purple}{15}}}$"},{"latex":"$$S_{TT} = 64125 - 63375 = 750$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sum-t2"},{"color":"amber","style":"text-color","anchor":"sum-t"},{"color":"purple","style":"text-color","anchor":"n-val"}],"connections":[]},"order":1,"title":"Calculate S_TT","description":"Calculate $S_{TT}$ by substituting $\\sum T^2$, $\\sum T$, and $n$ directly into the formula."},{"type":"hyper_latex","block":{"nodes":[{"text":"Now, we use the standard formula for the sum of products:"},{"latex":"$$S_{Td} = \\sum Td - \\frac{\\left(\\sum T\\right)\\left(\\sum d\\right)}{n}$"},{"text":"Substitute the given values into the formula:"},{"latex":"$$S_{Td} = \\htmlData{anchor=sum-td}{\\color{@sky}{27930}} - \\frac{\\left(\\htmlData{anchor=sum-t-2}{\\color{@amber}{975}}\\right)\\left(\\htmlData{anchor=sum-d}{\\color{@teal}{420}}\\right)}{\\htmlData{anchor=n-val-2}{\\color{@purple}{15}}}$"},{"latex":"$$S_{Td} = 27930 - 27300 = 630$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sum-td"},{"color":"amber","style":"text-color","anchor":"sum-t-2"},{"color":"teal","style":"text-color","anchor":"sum-d"},{"color":"purple","style":"text-color","anchor":"n-val-2"}],"connections":[]},"order":2,"title":"Calculate S_Td","description":"Calculate $S_{Td}$ by substituting $\\sum Td$, $\\sum T$, $\\sum d$, and $n$ directly into the formula."},{"type":"text","order":3,"title":"State the final answers","content":"The calculated values are:\n\n* $S_{TT} = 750$\n* $S_{Td} = 630$\n\nEach calculation earns 1 mark in this 2-mark question."}]},{"id":"b7d087f9-e244-4a77-8c98-a874eb02a71b:formulaic","label":"(b) Standard Formula Approach","steps":[{"type":"text","order":0,"title":"Identify given statistics","content":"First, let's identify all the given values from the question that we will need for our calculations:\n\n* Sample size $n = 15$\n* $\\sum T = 975$\n* $\\sum T^2 = 64125$\n* $\\sum d = 420$\n* $S_{dd} = 560$\n* $\\sum Td = 27930$","highlights":[{"text":"15 trials","location":"specification"},{"text":"\\sum T = 975","location":"specification"},{"text":"\\sum T^2 = 64125","location":"specification"},{"text":"\\sum d = 420","location":"specification"},{"text":"S_{dd} = 560","location":"specification"},{"text":"\\sum Td = 27930","location":"specification"}]},{"type":"text","order":1,"title":"Calculate $S_{TT}$","content":"To find the product moment correlation coefficient, we first need to calculate the sum of squares for temperature, $S_{TT}$. \n\nThe standard formula for $S_{TT}$ from our formula booklet is:\n\n$$S_{TT} = \\sum T^2 - \\frac{(\\sum T)^2}{n}$$\n\nSubstituting our given values into the formula:\n\n$$\\begin{aligned} S_{TT} &= 64125 - \\frac{975^2}{15} \\\\ S_{TT} &= 64125 - \\frac{950625}{15} \\\\ S_{TT} &= 64125 - 63375 \\\\ S_{TT} &= 750 \\end{aligned}$$"},{"type":"text","order":2,"title":"Calculate $S_{Td}$","content":"Next, we calculate the sum of products, $S_{Td}$, using the formula:\n\n$$S_{Td} = \\sum Td - \\frac{(\\sum T)(\\sum d)}{n}$$\n\nSubstituting our values:\n\n$$\\begin{aligned} S_{Td} &= 27930 - \\frac{975 \\times 420}{15} \\\\ S_{Td} &= 27930 - \\frac{409500}{15} \\\\ S_{Td} &= 27930 - 27300 \\\\ S_{Td} &= 630 \\end{aligned}$$"},{"type":"text","order":3,"title":"Substitute into correlation formula","content":"The product moment correlation coefficient, $r$, is calculated using the formula:\n\n$$r = \\frac{S_{Td}}{\\sqrt{S_{TT} S_{dd}}}$$ \n\nWe are given $S_{dd} = 560$, and we calculated $S_{TT} = 750$ and $S_{Td} = 630$. Substituting these values gives:\n\n$$r = \\frac{630}{\\sqrt{750 \\times 560}}$$\n\n$$r = \\frac{630}{\\sqrt{420000}}$$\n\n$$r \\approx 0.972111...$$"},{"type":"text","order":4,"title":"State the final answer","content":"Rounding our result to 3 significant figures, we get:\n\n$$r = 0.972$$\n\nThis strong positive correlation indicates that as temperature increases, the mass of the solute that dissolves also increases significantly."}]},{"id":"be57813f-3fde-4210-9b6d-9b73f6e3e141:pmcc_calculation","label":"(c) PMCC Calculation and Sign Reasoning","steps":[{"type":"text","order":0,"title":"Recall PMCC formula components","content":"To understand the relationship between temperature $T$ and amount dissolved $d$, we must first calculate the product-moment correlation coefficient (PMCC), denoted by $r$, between these two variables.\n\nThe formulas for the sum of products $S_{Td}$ and sum of squares $S_{TT}$ from our summary statistics are:\n\n$$S_{Td} = \\sum Td - \\frac{\\left(\\sum T\\right)\\left(\\sum d\\right)}{n}$$\n\n$$S_{TT} = \\sum T^2 - \\frac{\\left(\\sum T\\right)^2}{n}$$\n\nwhere $n = 15$ is the number of trials.","highlights":[{"text":"15 trials","location":"specification"}]},{"type":"text","order":1,"title":"Calculate STd and STT","content":"Now substitute the given summary statistics into these formulas:\n\n$$\\begin{aligned} S_{Td} &= 27930 - \\frac{975 \\times 420}{15} \\\\ &= 27930 - 27300 \\\\ &= 630 \\end{aligned}$$\n\n$$\\begin{aligned} S_{TT} &= 64125 - \\frac{975^2}{15} \\\\ &= 64125 - 63375 \\\\ &= 750 \\end{aligned}$$"},{"type":"text","order":2,"title":"Calculate correlation coefficient r","content":"Using the calculated values and the given $S_{dd} = 560$, we can compute the PMCC $r_{Td}$:\n\n$$r_{Td} = \\frac{S_{Td}}{\\sqrt{S_{TT} \\times S_{dd}}}$$\n\n$$r_{Td} = \\frac{630}{\\sqrt{750 \\times 560}} = \\frac{630}{\\sqrt{420000}} = \\frac{630}{648.074...} \\approx 0.972$$"},{"type":"text","order":3,"title":"Analyse the signs of r","content":"Let's examine the relationships between the three variables:\n- Between $T$ and $d$: the PMCC is positive ($r_{Td} \\approx 0.972$). This means higher temperatures correspond to a greater amount of solute dissolved (positive correlation).\n- Between $T$ and $v$: the PMCC is negative ($r_{Tv} = -0.552$). This means higher temperatures correspond to lower viscosity (negative correlation).","highlights":[{"text":"-0.552","location":"specification"}]},{"type":"text","order":4,"title":"Determine final expected correlation","content":"By combining these two relationships:\n- As viscosity ($v$) increases, temperature ($T$) must decrease (due to negative correlation).\n- As temperature ($T$) decreases, the amount of dissolved solute ($d$) also decreases (due to positive correlation).\n\nTherefore, we expect a **negative correlation** between viscosity ($v$) and amount dissolved ($d$)."}]},{"id":"be57813f-3fde-4210-9b6d-9b73f6e3e141:covariance_sign_analysis","label":"(c) Sign of Covariance/S_Td Analysis","steps":[{"type":"text","order":0,"title":"Analyze the question task","content":"We need to describe and justify the correlation we would expect to find between the viscosity of the solvent ($v$) and the mass of dissolved solute ($d$).\n\nWe are given the following summary statistics from the sample of size $n = 15$:\n\n* $\\sum T = 975$\n* $\\sum d = 420$\n* $\\sum Td = 27930$\n\nAdditionally, the correlation coefficient between temperature ($T$) and viscosity ($v$) is $-0.552$."},{"type":"text","order":1,"title":"Determine the sign of S_Td","content":"To understand the relationship between temperature ($T$) and dissolved solute ($d$), let's find the sign of their covariance using the sum of products formula:\n\n$$S_{Td} = \\sum Td - \\frac{(\\sum T)(\\sum d)}{n}$$\n\nLet's substitute the given values into the formula:\n\n$$\\begin{aligned} S_{Td} &= 27930 - \\frac{(975)(420)}{15} \\\\ &= 27930 - 27300 \\\\ &= 630 \\end{aligned}$$\n\nSince $S_{Td} = 630 > 0$, the covariance is positive, meaning there is a **positive relationship** between temperature ($T$) and dissolved solute ($d$) (as temperature increases, dissolved solute increases)."},{"type":"text","order":2,"title":"Analyze relationship with viscosity","content":"Now let's examine how viscosity ($v$) relates to both variables:\n\n1. We established that temperature ($T$) and dissolved solute ($d$) have a **positive** correlation ($S_{Td} > 0$). This means that higher temperature generally corresponds to a higher mass of dissolved solute.\n2. We are given that temperature ($T$) and viscosity ($v$) have a **negative** correlation ($r = -0.552$). This means that higher temperature corresponds to lower viscosity, or conversely, higher viscosity corresponds to lower temperature."},{"type":"text","order":3,"title":"Deduce expected correlation","content":"Combining these two relationships:\n\n* As viscosity ($v$) increases, temperature ($T$) decreases (due to their negative correlation).\n* As temperature ($T$) decreases, the mass of dissolved solute ($d$) decreases (due to their positive correlation).\n\nTherefore, we expect a **negative correlation** between viscosity ($v$) and dissolved solute ($d$).\n\n### Markscheme Breakdown\n* **B1**: Correctly identifying that the correlation is **negative**.\n* **C1**: Justifying this by explaining that $T$ and $d$ are positively correlated, while $T$ and $v$ are negatively correlated, resulting in an overall negative relationship between $v$ and 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Statistical hypothesis testing (A-level only)","topicId":"ff483f04-f111-4369-be7f-780143754b2f","topicSidebar":"$L3d","topicSlug":"statistical-hypothesis-testing-7","topicTitle":"4.3.1 Statistical hypothesis testing (A-level only)"}]}],"$L3e","$L3f"]}]

4.3.1 Statistical hypothesis testing (A-level only)

Question 65

4.3.1 Statistical hypothesis testing (A-level only) Questions

4.3.1 Statistical hypothesis testing (A-level only)

Question 65

4.3.1 Statistical hypothesis testing (A-level only) Questions