M21-5-MATHX-HP2-ENG-TZ2-XX

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Keycap Digit Two

Paper 2

Question 1 - MaximumMarks - 6

Consider a caf&eacute; where the duration from placing an order to receiving a coffee depends on how many people are ahead in the queue. Sarah, who visits this caf&eacute; regularly, recorded her waiting times over five consecutive days. The table provides the number of customers x, who placed orders before her on each day, and her respective waiting times y, in minutes.

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The task is to establish a mathematical relationship between x and y using a linear regression model. The model takes the form y = ax + b, where &#39;a&#39; and &#39;b&#39; are constants to be determined based on the data.

Question 1 - Table

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Question 1 - A - i

Find the values of a and b.

Question 1 - A - i - Answer

The linear regression line that models the relationship between the number of customers ahead of Sarah ((x)) and the waiting time ((&nbsp;y)) is given by the equation (&nbsp;y = ax + b). To determine the values of (a) and (b), we first calculate the mean of (x) (the number of customers) and (y) (Sarah&#39;s waiting time).

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The mean of (x) is \((\frac{3 + 9 + 11 + 10 + 5}{5} = 7.6)\) and the mean of (y) is \((\frac{6 + 10 + 12 + 11 + 6}{5} = 9.0)\). We then use these means to find the slope (a) of the regression line using the formula:

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\((a = \frac{\sum((x - \bar{x}) \cdot (y - \bar{y}))}{\sum((x - \bar{x})^2)} )\).

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After performing the calculations, we find that \((a \approx 0.805)\). The intercept (b) can be calculated using the formula \((b = \bar{y} - a \cdot \bar{x})\), which gives us \((b \approx 2.881)\). Therefore, the regression line is approximately (&nbsp;y = 0.805x + 2.881).

Question 1 - A - i - Answer - EC

The candidate has correctly identified the means of both (x) and (&nbsp;y) and used them to calculate the slope (a) and the intercept (b) with appropriate formulas. The arithmetic is accurate, and the resulting regression line equation (&nbsp;y = 0.805x + 2.881) is presented clearly, demonstrating a solid understanding of linear regression analysis as per the IB Mathematics AA HL syllabus. This answer would receive full marks as it follows the marking scheme precisely, providing the required calculations and final answer with correct significant figures and units of measure.

Question 1 - A - ii

Calculate the value of Pearson&rsquo;s product-moment correlation coefficient, (r).

Question 1 - A - ii - Answer

To determine Pearson&#39;s product-moment correlation coefficient, (r), we first compute the mean of the x-values and y-values. The means are calculated as \((\bar{x})\) and \((\bar{y})\) respectively. Subsequently, we find the standard deviations of the x-values, \((\sigma_x)\), and y-values, \(( \sigma_y)\), ensuring the use of Bessel&#39;s correction for an unbiased estimate. Following that, the covariance between the x and y variables, cov(x, y), is computed. The coefficient (r) is then found by dividing the covariance by the product of the standard deviations of x and y variables.

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The formula used for the calculation is:

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\((r = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sqrt{\sum (x - \bar{x})^2 \sum (y - \bar{y})^2}} )\).

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Upon applying the values, the computed coefficient (r) is 0.9778, rounded to four decimal places.

Question 1 - A - ii - Answer - EC

The student has correctly calculated Pearson&#39;s correlation coefficient, utilizing appropriate statistical formulas and rounding conventions. The response demonstrates a clear understanding of the relationship between the variables and the correct application of statistical methods. The calculation of means, standard deviations, and covariance is methodically performed, and the final result is consistent with the expected value given the data provided. The accuracy and precision in the calculations reflect a high level of mathematical understanding appropriate for IB Mathematics HL.

Question 1 - B

In part (a)(i), we determined the value of &#39;a&#39;. Could you explain what this value signifies in relation to the scenario described?

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Question 1 - B - Answer

To interpret the value of a in the given context, we must understand what the coefficient represents in the regression equation y = ax + b. Here, y is the waiting time for Sarah to receive her coffee, and x is the number of customers ahead of her. The coefficient a indicates the change in the dependent variable, Sarah&#39;s waiting time, for each additional unit change in the independent variable, the number of customers.

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In this scenario, the value of a = 0.805 minutes. This means that for each additional customer ahead of Sarah in the queue, her waiting time increases by an average of 0.805 minutes. This linear relationship shows a direct proportionality between the number of customers and the waiting time, where the slope of the line, represented by a, is the rate of increase in waiting time per additional customer.

Question 1 - B - Answer - EC

The answer correctly identifies the role of a as the slope of the regression line, which in the context of the scenario, is the average increase in waiting time per additional customer. The student has accurately interpreted the slope a as representing the minute increase in waiting time, thus applying the concept of slope in a linear model to a real-world situation. The clarity in explaining the direct relationship and the precision in stating the value of a demonstrates a strong understanding of the mathematical concept and its application, adhering to the standards expected at the level of IB Mathematics AA HL.

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The use of the term &quot;average&quot; is appropriate as it accounts for the variability in data, indicating that a is not a fixed increase but an average value, which could vary slightly in different situations. The explanation is succinct and directly addresses the requirement of the question, without unnecessary elaboration or deviation from the mathematical interpretation needed.

Question 1 - C

On another occasion, Sarah goes to a caf&eacute; to order coffee. There are already seven people ahead of her waiting for their coffee orders.

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Based on the calculation from part (a)(i), could you estimate how long Sarah might have to wait to receive her coffee?

Question 1 - C - Video - Coming soon

Question 1 - C - Answer

The task requires us to use linear regression to model the relationship between the number of customers ahead of Sarah (x) and the time she waits to receive her coffee (y). Using the least squares method, we calculate the regression line with the equation (y = ax + b).

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Firstly, we find the slope (a) and y-intercept (b) of the line. The slope is the ratio of the change in y to the change in x, which is the coefficient that indicates how much y changes for each unit change in x. The y-intercept is the value of y when x is 0, representing the time Sarah would wait if there were no other customers ahead of her.

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Using the data points provided:

<ul>
	<li>Number of customers (x): 3, 9, 11, 10, 5</li>
	<li>Sarah&rsquo;s waiting time&nbsp;(y): 6, 10, 12, 11, 6</li>
</ul>

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We calculate the slope (a) as approximately 0.805 and the y-intercept (b) as approximately 2.881. Therefore, the regression line equation is (&nbsp;y = 0.805x + 2.881).

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To estimate Sarah&#39;s waiting time when 7 customers are ahead, we substitute x with 7 in our regression equation:

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\(( y = 0.805 \times 7 + 2.881 )\)

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Which gives us:

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\(( y \approx 8.52 ) (min)\)

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Therefore, based on the regression model, we estimate that Sarah&#39;s waiting time to receive her coffee would be approximately 8.52 minutes when seven customers have already ordered before her.

Question 1 - C - Answer - EC

The candidate has correctly applied the principles of linear regression to estimate the dependent variable using a given set of data points. The calculations for the slope and y-intercept of the regression line are accurate, following the method of least squares. The subsequent prediction for Sarah&#39;s waiting time is also precise, with the correct substitution of x = 7 into the regression equation. The answer is presented in a clear, logical manner, demonstrating a strong understanding of the concept and its application. The final estimate of the waiting time is rounded appropriately, in accordance with the significant figures presented in the data. This response would meet the criteria for full marks in the IB Mathematics AA HL examination.

Question 2 - MaximumMarks - 5

An arithmetic sequence begins with a first term of 60, and each subsequent term decreases by 2.5, referred to as the common difference.

Question 2 - A

In the context of a given sequence, if we know that a certain term, denoted as the (k)-th term, is zero, how can we find the value of (k)

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IB MATH AA HL, Paper 2, May, 2021, TZ2, Solved Past Paper