N22-5-MATHY-HP3-ENG-TZ0-XX

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Keycap Digit Three

Paper 3

Question 1 - MaximumMarks - 29

In this scenario, you are tasked with exploring methods for predicting the outcomes of international football matches involving Argentina by using historical sports results.

 
Peter and Helen, two friends, are discussing potential prediction strategies. Peter proposes analyzing historical match data. He compiles the results of the last 240 matches that Argentina played, listed in chronological order. 
Peter then divides these matches into 60 blocks of four matches each. For each block, he records the number of times Argentina won. The table below summarizes his findings for these 60 blocks of matches.

Question 1 - Table

Question 1 - A - Video - Coming soon

Question 1 - A

Calculate the average number of wins per set of four matches for the Argentina team.

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In simpler terms, this question is asking you to determine how many wins Argentina typically achieves when looking at every four matches they play.

Question 1 - A - Answer

To determine the mean number of wins per block of four matches for Argentina, we calculate the weighted average of wins based on the frequency of each outcome. This involves multiplying the number of wins by their respective frequency, summing these products, and then dividing by the total number of blocks.

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We use the formula for the mean,\( μ = Σ(x⋅f)Σf\) where x represents the number of wins in a block, and f is the frequency of this outcome. Thus:

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Mean number of wins per block

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= [(0&sdot;0) + (1&sdot;11) + (2&sdot;21) + (3&sdot;21) + (4&sdot;7)]&nbsp; 60

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Calculating the numerator:

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(0&sdot;0) + (1&sdot;11) + (2&sdot;21) + (3&sdot;21) + (4&sdot;7)

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= 0 + 11 + 42 + 63 + 28 = 144

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Dividing the total by the number of blocks (60), we get:

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\(\frac{144}{ 60} = 2.4\)

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Thus, the mean number of wins per block of four matches for Argentina is 2.4.

Question 1 - A - Answer - EC

The response correctly identifies the requirement to use a weighted average to find the mean number of wins per block, which is key to solving this problem. The formula is appropriately defined, and each step of the calculation is accurately executed and clearly explained.

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The calculation of the numerator is correct, showing a detailed breakdown of each component contributing to the total sum of 144&nbsp;This is crucial for ensuring the accuracy of the final mean calculation.

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The division to find the mean is correctly performed, with the final answer, 2.4&nbsp;being prominently displayed. This step is essential as it gives the mean number of wins, which is the focus of the question.

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The student&#39;s approach adheres to the principles of clarity, accuracy, and relevance to the question posed, making the response an excellent example of how to address this type of problem in IB Mathematics AI HL

Question 1 - B

Peter believes that his data can be modeled using a binomial distribution with n = 4. To test this model, he decides to perform a \((\chi^2)\) goodness of fit test.

Using Peter&#39;s data, provide an estimate for the probability (p) in this binomial model.

This question requires calculating the probability parameter (p) for a binomial distribution based on the given data.

Question 1 - B - Video - Coming soon

Question 1 - B - Answer

To estimate the probability (p) for the binomial model using Peter&#39;s data, we start by recognizing that the binomial distribution (B(n, p)) describes the number of successes in (n) trials, with the probability of success in each trial being (p). In this context, (n = 4) since Peter considers blocks of four matches at a time.

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The table provided shows the frequency of wins in these blocks of four matches. We use this information to estimate (p), the probability of Argentina winning a single match. The expected value (E(X)) of a binomial distribution (B(n, p)) is given by \( np \). To find (p), we need to determine the total number of wins and the total number of trials.

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The total number of blocks is 60, each containing 4 matches, giving us (60 &times; 4 = 240) matches in total. Next, we calculate the total number of wins:

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\([ \text{Total wins} = (1 \times 11) + (2 \times 21) + (3 \times 21) + (4 \times 7) = 11 + 42 + 63 + 28 = 144]\)

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The estimated probability (p) of winning a single match is then given by the ratio of the total number of wins to the total number of matches:

 
\([p = \frac{\text{Total wins}}{\text{Total matches}} = \frac{144}{240} = 0.6]\)

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Thus, the estimate for the probability (p) for the binomial model is 0.6.

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Question 1 - B - Answer - EC

This answer correctly applies the method to estimate the probability (p) for the binomial model. The student has clearly outlined the steps taken to determine the total number of wins and has correctly computed the probability as (p = 0.6). The work is well-organized, showing an understanding of both the binomial distribution and the calculation of expected probabilities. The use of the formula for (E(X)) and the correct computation of totals from the given data demonstrate a strong grasp of the necessary concepts. The clear explanation of each step helps in understanding the process, and the final answer is both accurate and well-presented.

Question 1 - C - Video - Coming soon

Question 1 - C - i

Determine the likelihood that Argentina will not win any matches in a series of four games using the binomial distribution.

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In simpler terms, we want to calculate the probability that Argentina loses all four matches if we know their chances of winning each match.

Question 1 - C - i -  Answer

Given that the number of wins for Argentina per block of four matches follows a binomial distribution, we need to determine the probability that Argentina wins zero matches in a block of four matches. According to the problem, the binomial distribution parameters are (n = 4) and \(( p = \frac{3}{5})\).

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We use the binomial probability formula:

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\([ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}]\)

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Here, (k = 0), (n = 4), and \((p = \frac{3}{5})\). Substituting these values, we get:

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\([ P(X = 0) = \binom{4}{0} \left(\frac{3}{5}\right)^0 \left(\frac{2}{5}\right)^4 ]\)

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\([ = 1 \cdot 1 \cdot \left(\frac{2}{5}\right)^4 ]\)

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\([ = \left(\frac{2}{5}\right)^4]\)

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\([ = \frac{16}{625}]\)

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\([\approx 0.0256]\)

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Thus, the probability that Argentina wins zero matches in a block of four matches is approximately 0.0256.

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Allow follow-through from the use of their probability in part (b) but only when used with (n = 4).

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Question 1 - C - i - Answer - EC

The answer correctly uses the binomial probability formula to determine the probability of Argentina winning zero matches in a block of four. The parameters (n = 4) and \(( p = \frac{3}{5} )\) are identified correctly and substituted into the formula accurately. The steps are shown clearly, demonstrating the calculations needed to arrive at the final probability of approximately 0.0256. The notation and mathematical expressions are used appropriately, meeting the standard for an IB Mathematics AI HL level response.

Question 1 - C - ii

Determine the expected frequency for recording no wins in a given scenario.

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This question involves calculating the expected number of times an event (specifically, &#39;no wins&#39;) will occur under certain conditions or over a set period.

Question 1 - C - ii - Answer

To find the expected frequency for zero wins, we need to model the data using a binomial distribution with (n = 4). This is because Peter considers blocks of four matches at a time. The probability (p) of Argentina winning a match needs to be estimated from the given data.

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First, we calculate the total number of wins and the total number of blocks:

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Total number of wins:

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\(\text{Total wins} = 1 \times 11 + 2 \times 21 + 3 \times 21 + 4 \times 7 = 1 \cdot 11 + 2 \cdot 21 + 3 \cdot 21 + 4 \cdot 7 = 11 + 42 + 63 + 28 = 144\)


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Total number of blocks:

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\(\text{Total blocks} = 60\)


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The probability (p) of Argentina winning a match is:

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\(p = \frac{\text{Total wins}}{(\text{Total blocks} \times n)} = \frac{144}{60 \times 4} = \frac{144}{240} = 0.6\)


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Using the binomial distribution, the probability of zero wins in four matches (k = 0) is given by:

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\(P(X = 0) = \binom{4}{0} p^0 (1-p)^4 = (1-0.6)^4 = 0.4^4 = 0.0256\)


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The expected frequency for zero wins is then:

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\(\text{Expected frequency} = \text{Total number of blocks} \times P(X = 0) = 60 \times 0.0256 = 1.536\)


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Thus, the expected frequency for zero wins is 1.536.

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Question 1 - C - ii - Answer - EC

The student has correctly followed the necessary steps to find the expected frequency for zero wins using a binomial distribution. The total number of wins and total number of blocks were accurately calculated, leading to the correct determination of the probability (p). By applying the binomial formula for zero wins and multiplying by the total number of blocks, the student arrived at the expected frequency of 1.536. This clear and thorough explanation demonstrates a strong understanding of the concepts and calculations involved.

Question 1 - D

Since some expected frequencies are less than 5, Peter combines rows in his table to produce the following observed frequencies. He then uses his binomial model to find appropriate expected frequencies, correct to one decimal place.

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Peter uses this table to carry out a &chi;2 goodness of fit test, to test the hypothesis that the data follows a binomial distribution with (n = 4), at the 5% significance level.

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In simpler terms, Peter is testing if his data fits a binomial model with specific parameters. He adjusts the data because some values are too small, calculates expected values, and then performs a statistical test to see if the observed data matches the expected pattern.

Question 1 - D - Table

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Question 2 - MaximumMarks - 26

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IB MATH AI HL, Paper 3, November, 2022, TZ0, Solved Past Paper