🧠:<think>Alright, so I have this problem about Alex, an aspiring actor who is working on independent film projects without getting paid. He works exactly 40 hours a week on acting, and he wants to divide his time across different projects. The goal is to figure out how he can allocate his time given certain constraints and then compute the expected number of contacts he can make based on that allocation.Let me start with part 1. The problem says that Alex wants to maximize the number of projects he works on. So, he wants to have as many projects as possible, but each project must receive a minimum of 6 hours per week. Also, he can't work on more than five projects. So, the number of projects, n, is between 2 and 5 inclusive. But since he wants to maximize n, he should aim for the maximum number of projects, which is 5.Wait, but hold on. If he wants to maximize n, then n should be 5. But I need to make sure that the total hours add up to 40, with each project getting at least 6 hours. So, if he has 5 projects, each with at least 6 hours, that would be 5*6=30 hours. But he works 40 hours, so that leaves 10 extra hours to distribute among the 5 projects. So, he can distribute these 10 hours in any way, but each project must have at least 6.But the question is asking for all possible combinations of hours he can allocate to each project when he is maximizing n, which is 5. So, n=5. So, we need to find all possible tuples (x1, x2, x3, x4, x5) such that each xi >=6, and the sum of xi is 40.But the problem is, how do we represent this as a system of inequalities? Let me think.First, the variables are x1, x2, x3, x4, x5, each representing the hours spent on each project. The constraints are:1. Each xi >=6, because each project must receive a minimum of 6 hours.2. The total hours sum up to 40: x1 + x2 + x3 + x4 + x5 = 40.3. The number of projects, n, is 5.So, the system of inequalities would be:x1 >=6x2 >=6x3 >=6x4 >=6x5 >=6x1 + x2 + x3 + x4 + x5 = 40That's the system. Now, to find all possible combinations, we can think of this as an integer solution problem where each xi is an integer greater than or equal to 6, and their sum is 40.But wait, the problem doesn't specify that the hours have to be integers. It just says "number of hours," so they could be real numbers. Hmm, but in reality, you can't really work a fraction of an hour, but since it's a math problem, maybe they allow real numbers. The problem doesn't specify, so I might have to assume they can be real numbers.But let me check the original problem. It says "the number of hours," but doesn't specify if they have to be integers. So, perhaps we can treat them as real numbers.But in any case, the problem is to determine all possible combinations of hours. So, if we consider real numbers, the solutions would be all 5-tuples where each xi >=6 and their sum is 40.But to represent all possible combinations, it's an uncountably infinite set. So, perhaps the problem is expecting a description rather than listing all possible combinations.Wait, the question says: "Determine all the possible combinations of hours Alex can allocate to each project, assuming he wants to maximize n."So, since n is fixed at 5, we need to find all possible allocations where each project gets at least 6 hours, and the total is 40. So, the possible combinations are all 5-tuples (x1, x2, x3, x4, x5) where each xi >=6 and x1+x2+x3+x4+x5=40.But since the problem is asking to "determine all the possible combinations," and given that it's a math problem, perhaps they want the number of possible combinations or a way to describe them.Alternatively, maybe they want the integer solutions. Let me think again.If we assume that the hours must be integers, then each xi is an integer >=6, and the sum is 40. So, the number of solutions is the number of integer solutions to x1 + x2 + x3 + x4 + x5 = 40, where each xi >=6.This is a classic stars and bars problem. The formula for the number of non-negative integer solutions to x1 + x2 + ... + xn = k is C(k-1, n-1). But in this case, each xi >=6, so we can make a substitution: let yi = xi -6, so yi >=0. Then, the equation becomes y1 + y2 + y3 + y4 + y5 = 40 - 5*6 = 40 -30=10.So, the number of non-negative integer solutions is C(10 +5 -1, 5 -1)=C(14,4)=1001.But the problem is asking for all possible combinations, not just the number. So, if we have to list them, that's 1001 different combinations, which is impractical. So, perhaps the problem is expecting a description rather than listing all.Alternatively, maybe the problem is expecting the general solution in terms of variables, but I'm not sure.Wait, the problem says "Formulate a system of inequalities to represent the constraints on x1, x2, ..., xn. Determine all the possible combinations of hours Alex can allocate to each project, assuming he wants to maximize n."So, first, formulate the system of inequalities, which I did above.Then, determine all possible combinations, which, as I thought, is an infinite set if we consider real numbers, or 1001 if we consider integers.But maybe the problem is expecting to express the general solution, like in terms of variables.Wait, perhaps the problem is expecting to describe the possible combinations as the set of all 5-tuples where each component is at least 6 and the sum is 40.But I'm not sure. Maybe I should proceed to part 2, and see if that gives me any clues.Part 2 says: Alex is interested in the network of connections he can build through these projects. For each hour spent on a project, the probability that he makes a beneficial industry contact is p=0.05. Assuming the number of contacts made follows a Poisson distribution, compute the expected number of contacts Alex makes in a week if he works on the maximum number of projects determined in part 1.So, in part 1, we determined that the maximum number of projects is 5. So, in part 2, we need to compute the expected number of contacts.But the number of contacts per hour is a Poisson distribution with parameter lambda, which is the expected number per hour. But wait, the probability of making a contact per hour is p=0.05. So, is this a Bernoulli trial per hour, where each hour has a 5% chance of a contact? Then, the number of contacts in a week would be a binomial distribution with parameters n=40 and p=0.05.But the problem says it follows a Poisson distribution. Hmm, maybe they approximate the binomial distribution with a Poisson distribution when n is large and p is small. Since 40 is not extremely large, but 0.05 is small, so the Poisson approximation might be used.But actually, the number of contacts per hour is a Bernoulli trial, so the total number of contacts in 40 hours would be a binomial(40, 0.05). The expected number of contacts would be 40*0.05=2.But if we model it as a Poisson distribution, the expected value is lambda=40*0.05=2 as well. So, regardless, the expected number is 2.Wait, but the problem says "the number of contacts made follows a Poisson distribution." So, perhaps they are directly telling us to model it as Poisson with lambda=40*0.05=2.So, the expected number is 2.But wait, let me think again. If each hour is an independent trial with probability p=0.05 of success (contact), then the number of contacts in t hours is a binomial(t, p). For t=40, p=0.05, the expected number is t*p=2.If we model it as Poisson, the lambda is equal to the expected number, so lambda=2. So, the expected number is still 2.Therefore, regardless of whether it's binomial or Poisson, the expected number is 2.But the problem says "the number of contacts made follows a Poisson distribution," so maybe they just want us to compute lambda=40*0.05=2.So, the expected number is 2.But let me make sure. The Poisson distribution is often used to model the number of events occurring in a fixed interval of time or space, given a constant mean rate of occurrence. So, if the average rate is lambda=2 per week, then the expected number is 2.Alternatively, if we think of each hour as a Bernoulli trial, then the total number is binomial, but the expectation is still 2.So, in either case, the expected number is 2.But wait, the problem says "for each hour spent on a project, the probability that he makes a beneficial industry contact is p=0.05." So, per hour, the probability is 0.05. So, over 40 hours, the expected number is 40*0.05=2.Yes, that makes sense.So, putting it all together.For part 1, the system of inequalities is:x1 >=6x2 >=6x3 >=6x4 >=6x5 >=6x1 + x2 + x3 + x4 + x5 =40And the possible combinations are all 5-tuples where each xi is at least 6 and their sum is 40.For part 2, the expected number of contacts is 2.Wait, but in part 1, the problem says "assuming he wants to maximize n," which is 5. So, in part 2, we are to compute the expected number of contacts when he is working on 5 projects, which is the maximum.But regardless of how he分配时间, the total hours are 40, so the expected number of contacts is always 2, regardless of how he分配时间. So, whether he works on 2 projects or 5 projects, as long as he works 40 hours, the expected number of contacts is 2.But maybe I'm missing something. Let me think again.Wait, no, because the number of contacts is based on the total hours, not the number of projects. So, regardless of how he分配时间 across projects, the total hours are 40, so the expected number of contacts is 40*0.05=2.Therefore, the answer is 2.But let me make sure. Suppose he works on 5 projects, each with at least 6 hours. So, the total is 40. The number of contacts is a Poisson random variable with lambda=40*0.05=2. So, the expectation is 2.Yes, that seems correct.So, summarizing:1. The system of inequalities is as above, and the possible combinations are all 5-tuples with each xi >=6 and sum 40.2. The expected number of contacts is 2.But wait, the problem says "compute the expected number of contacts Alex makes in a week if he works on the maximum number of projects determined in part 1." So, since the maximum number of projects is 5, and regardless of how he分配时间, the total hours are 40, so the expected number is 2.Therefore, the answer is 2.But just to make sure, let me think about whether the number of projects affects the contacts. The problem says "for each hour spent on a project," so each hour, regardless of the project, has a 5% chance of a contact. So, the total expected contacts are 40*0.05=2, regardless of the number of projects.Therefore, the expected number is 2.Final Answer1. The system of inequalities is ( x_1 geq 6 ), ( x_2 geq 6 ), ( x_3 geq 6 ), ( x_4 geq 6 ), ( x_5 geq 6 ), and ( x_1 + x_2 + x_3 + x_4 + x_5 = 40 ). All possible combinations are the 5-tuples where each ( x_i ) is at least 6 and their sum is 40.2. The expected number of contacts Alex makes in a week is boxed{2}.</think>

🧠:<think>Okay, so I need to help this AI ethics researcher with their statistical analysis. They have consumer satisfaction ratings before and after some corporate responsibility initiatives. The ratings are on a scale from 1 to 5. They want to see if there's a significant difference in satisfaction levels before and after the initiatives. First, the problem is divided into two parts. The first part is about deriving the test statistic for comparing the means of two normal distributions with unequal variances. The second part is about setting up a hypothesis test to verify the company's claim that the initiatives improved satisfaction by at least 0.5 on average, and then interpreting the results ethically.Starting with the first part: deriving the test statistic. I remember that when comparing two means from independent samples, if the variances are unknown and possibly unequal, we use the Welch's t-test. The formula for the test statistic is something like the difference in sample means divided by the standard error, which accounts for both sample variances.So, let me recall the formula. The test statistic t is calculated as:t = (Ȳ - X̄) / sqrt[(s_X²/n) + (s_Y²/m)]Where:- X̄ and Ȳ are the sample means of X and Y respectively.- s_X² and s_Y² are the sample variances.- n and m are the sample sizes.But wait, I should make sure that this is indeed the correct formula. Yes, because when variances are unequal, we can't assume a pooled variance, so each variance is divided by their respective sample sizes and then added together under the square root.Okay, so that's the test statistic for part 1.Moving on to part 2: Formulating the hypothesis test. The company claims that the initiatives improve satisfaction by at least 0.5 on average. So, the alternative hypothesis should reflect that the mean after is at least 0.5 higher than before.But wait, in hypothesis testing, the alternative hypothesis is what we're trying to support. So if the company claims an improvement of at least 0.5, that would be a one-tailed test. The null hypothesis would be that the improvement is less than 0.5, or that the difference is zero or negative.Wait, actually, let me think. The company is claiming that μ_Y - μ_X ≥ 0.5. So, the alternative hypothesis should be μ_Y - μ_X ≥ 0.5, and the null hypothesis would be μ_Y - μ_X < 0.5.But sometimes, people set the null as the status quo or no effect, which in this case might be μ_Y - μ_X = 0, but the company is making a specific claim. So perhaps the null should be μ_Y - μ_X ≤ 0.5, and the alternative is μ_Y - μ_X > 0.5. Wait, no, because the company is saying it's at least 0.5, so the alternative is μ_Y - μ_X ≥ 0.5, and the null is μ_Y - μ_X < 0.5.But in standard hypothesis testing, the null is usually a specific value or a range that includes the status quo. So maybe the null is μ_Y - μ_X ≤ 0.5, and the alternative is μ_Y - μ_X > 0.5. But actually, the company is claiming that the improvement is at least 0.5, so the alternative is that the difference is greater than or equal to 0.5. However, in hypothesis testing, the alternative is usually a statement that can be rejected, so we set the null as the complement.Wait, perhaps it's better to set the null as μ_Y - μ_X ≤ 0.5 and the alternative as μ_Y - μ_X > 0.5. But actually, the company is making a claim that the difference is at least 0.5, so the alternative should be μ_Y - μ_X ≥ 0.5, and the null would be μ_Y - μ_X < 0.5. But in practice, we often set the null as the equality, so maybe the null is μ_Y - μ_X = 0.5, and the alternative is μ_Y - μ_X > 0.5. Hmm, I'm a bit confused here.Wait, no. Let's think carefully. The company's claim is that the initiatives improve satisfaction by at least 0.5. So, they are saying that the mean after is at least 0.5 higher than before. So, the alternative hypothesis should be μ_Y - μ_X ≥ 0.5. The null hypothesis would then be μ_Y - μ_X < 0.5. But in hypothesis testing, we typically set the null as the statement that includes equality, so perhaps the null is μ_Y - μ_X ≤ 0.5, and the alternative is μ_Y - μ_X > 0.5. But that might not capture the company's exact claim.Alternatively, maybe the null is μ_Y - μ_X = 0.5, and the alternative is μ_Y - μ_X > 0.5. But that would be testing if the improvement is more than 0.5, which is slightly different from the company's claim of at least 0.5.Wait, perhaps the correct approach is to set the null as μ_Y - μ_X ≤ 0.5, and the alternative as μ_Y - μ_X > 0.5. Because the company is claiming that the improvement is at least 0.5, so we want to test if the improvement is greater than 0.5. But actually, if the improvement is exactly 0.5, that would satisfy the company's claim as well. So maybe the null should be μ_Y - μ_X < 0.5, and the alternative is μ_Y - μ_X ≥ 0.5. But in standard testing, we can't have a composite null with a less than. So perhaps we set the null as μ_Y - μ_X = 0.5, and the alternative as μ_Y - μ_X > 0.5. But that might not cover all cases where the improvement is exactly 0.5.Alternatively, maybe the null is μ_Y - μ_X ≤ 0.5, and the alternative is μ_Y - μ_X > 0.5. That way, if we reject the null, we have evidence that the improvement is more than 0.5, which would support the company's claim. But if the improvement is exactly 0.5, we wouldn't reject the null, which might not fully align with the company's claim.Wait, perhaps the correct way is to set the null as μ_Y - μ_X ≤ 0.5, and the alternative as μ_Y - μ_X > 0.5. Because the company is saying the improvement is at least 0.5, so we want to see if the data supports that the improvement is more than 0.5. If the improvement is exactly 0.5, the test wouldn't reject the null, but that's still in line with the company's claim. Hmm, maybe I'm overcomplicating.Alternatively, perhaps the null is μ_Y - μ_X = 0, and the alternative is μ_Y - μ_X > 0. But that's not considering the 0.5 threshold. The company's claim is about a minimum improvement of 0.5, so we need to incorporate that into the hypotheses.Wait, perhaps the correct approach is to set the null as μ_Y - μ_X ≤ 0.5, and the alternative as μ_Y - μ_X > 0.5. Because we want to test if the improvement is more than 0.5, which would support the company's claim of at least 0.5. If the test statistic falls into the rejection region, we can conclude that the improvement is more than 0.5, thus supporting the company's claim. If not, we fail to reject the null, meaning the improvement is 0.5 or less, which might not fully support the claim.But I'm not entirely sure. Let me think again. The company claims that the initiatives improve satisfaction by at least 0.5. So, the alternative hypothesis should be that the mean difference is greater than or equal to 0.5. However, in hypothesis testing, the alternative is usually a statement that can be tested, which is typically a strict inequality. So, perhaps the alternative is μ_Y - μ_X > 0.5, and the null is μ_Y - μ_X ≤ 0.5.Yes, that makes sense. So, the null hypothesis is that the improvement is 0.5 or less, and the alternative is that it's more than 0.5. If we reject the null, we have evidence that the improvement is more than 0.5, supporting the company's claim. If we fail to reject, we don't have sufficient evidence to support that the improvement is more than 0.5, but it could still be exactly 0.5.But wait, the company's claim is "at least 0.5," which includes exactly 0.5. So, if the true difference is exactly 0.5, the test would not reject the null, which is set as ≤0.5. So, in that case, we wouldn't be able to confirm the company's claim, but we also wouldn't reject it. Hmm, this is tricky.Alternatively, maybe the null should be μ_Y - μ_X < 0.5, and the alternative is μ_Y - μ_X ≥ 0.5. But in standard testing, the null is usually a single value or a range that includes the equality. So, perhaps setting the null as μ_Y - μ_X = 0.5 and the alternative as μ_Y - μ_X > 0.5. But then, if the true difference is exactly 0.5, we wouldn't reject the null, which is fine, but if it's more than 0.5, we would reject the null, supporting the company's claim.But the company's claim is that the improvement is at least 0.5, so the alternative should include 0.5 and above. Therefore, the alternative is μ_Y - μ_X ≥ 0.5, and the null is μ_Y - μ_X < 0.5. However, in practice, we often set the null as a single value for simplicity, so maybe μ_Y - μ_X = 0.5, and the alternative is μ_Y - μ_X > 0.5. But that might not capture the entire range of the company's claim.I think the correct approach is to set the null as μ_Y - μ_X ≤ 0.5, and the alternative as μ_Y - μ_X > 0.5. This way, the null covers all scenarios where the improvement is 0.5 or less, and the alternative is where it's more than 0.5. If we reject the null, we have evidence that the improvement is more than 0.5, which supports the company's claim. If we fail to reject, we can't conclude that the improvement is more than 0.5, but it could still be exactly 0.5.But wait, in hypothesis testing, the null is typically a specific value or a range that includes the equality. So, perhaps the null is μ_Y - μ_X = 0.5, and the alternative is μ_Y - μ_X > 0.5. That way, we're testing if the improvement is more than 0.5, which would support the company's claim of at least 0.5. If the true difference is exactly 0.5, we wouldn't reject the null, which is acceptable because the company's claim includes that scenario.However, if the true difference is exactly 0.5, the test might not have enough power to detect it as significantly different from 0.5, but that's a separate issue.In any case, I think the null hypothesis should be μ_Y - μ_X ≤ 0.5, and the alternative is μ_Y - μ_X > 0.5. This way, we're testing whether the improvement exceeds 0.5, which would support the company's claim.Now, regarding the interpretation. If the test results in rejecting the null hypothesis, it means there's sufficient evidence to support the company's claim that the initiatives improved satisfaction by more than 0.5. This would have positive implications for corporate responsibility, showing that the initiatives had a meaningful impact on consumer satisfaction, which aligns with ethical practices aimed at benefiting consumers.If the null is not rejected, it means there's not enough evidence to support that the improvement was more than 0.5. This could imply that the initiatives didn't have the intended effect, or that the sample size wasn't large enough to detect a difference. From an ethical standpoint, this might raise concerns about whether the company's initiatives truly benefited consumers as claimed.So, summarizing:1. The test statistic is derived using Welch's t-test formula, accounting for unequal variances.2. The hypothesis test is set up with null as μ_Y - μ_X ≤ 0.5 and alternative as μ_Y - μ_X > 0.5. Rejecting the null supports the company's claim, which is ethically positive. Failing to reject suggests the initiatives may not have met the claimed improvement, raising ethical questions about the company's transparency and effectiveness.I think that's the approach. Let me just double-check the test statistic formula. Yes, it's (Ȳ - X̄) divided by the square root of (s_X²/n + s_Y²/m). That's correct for unequal variances.For the hypothesis, I think the key is that the company's claim is a minimum improvement, so the alternative should reflect that the improvement is at least 0.5, but in testing, we usually set the alternative as a strict inequality to have a testable hypothesis. So, the alternative is μ_Y - μ_X > 0.5, and the null is μ_Y - μ_X ≤ 0.5.Yes, that makes sense.</think>

🧠:<think>Okay, so I have these two problems to solve about the growth of pharmaceutical companies. Let me take them one at a time.Starting with the first problem: The company's growth is modeled by the function ( G(t) = A e^{kt} ). They gave me that the initial valuation ( A ) is 100 million, and after 5 years, the valuation is expected to be 500 million. I need to find the constant growth rate ( k ).Alright, so let's write down what we know. At time ( t = 0 ), ( G(0) = 100 ) million. At ( t = 5 ), ( G(5) = 500 ) million. The formula is ( G(t) = A e^{kt} ). Since ( A = 100 ), the formula becomes ( G(t) = 100 e^{kt} ).So, plugging in ( t = 5 ), we have:( 500 = 100 e^{5k} ).Hmm, okay, so I can divide both sides by 100 to simplify:( 5 = e^{5k} ).Now, to solve for ( k ), I need to take the natural logarithm of both sides. Remember, the natural log is the inverse of the exponential function with base ( e ).Taking ln of both sides:( ln(5) = ln(e^{5k}) ).Simplify the right side:( ln(5) = 5k ).So, solving for ( k ):( k = frac{ln(5)}{5} ).Let me compute that. I know that ( ln(5) ) is approximately 1.6094. So,( k approx frac{1.6094}{5} approx 0.3219 ).So, ( k ) is approximately 0.3219 per year. Let me check if that makes sense. If I plug ( k ) back into the equation:( G(5) = 100 e^{0.3219 * 5} ).Calculating the exponent: 0.3219 * 5 = 1.6095.( e^{1.6095} ) is approximately ( e^{1.6094} ), which is 5. So, ( G(5) = 100 * 5 = 500 ). That checks out.Okay, so the first part seems solved. ( k ) is ( ln(5)/5 ), which is approximately 0.3219.Moving on to the second problem: Another company's growth is modeled by ( H(t) = B t^2 + C ). The initial valuation is 200 million, so ( H(0) = 200 ). In 10 years, the valuation is expected to be 800 million. I need to find constants ( B ) and ( C ).Alright, let's break this down. At ( t = 0 ), ( H(0) = B*(0)^2 + C = C = 200 ). So, ( C = 200 ).Now, at ( t = 10 ), ( H(10) = B*(10)^2 + C = 100B + C = 800 ).We already know ( C = 200 ), so plugging that in:( 100B + 200 = 800 ).Subtract 200 from both sides:( 100B = 600 ).Divide both sides by 100:( B = 6 ).So, ( B = 6 ) and ( C = 200 ).Let me verify that. Plugging back into ( H(t) ):At ( t = 0 ), ( H(0) = 6*0 + 200 = 200 ). Correct.At ( t = 10 ), ( H(10) = 6*100 + 200 = 600 + 200 = 800 ). Correct.So, that seems straightforward.Wait, hold on. The function is ( H(t) = B t^2 + C ). So, is it ( B t^2 + C ) or ( B t^2 + C t )? The problem says ( H(t) = B t^2 + C ), so only the quadratic term and a constant term. So, no linear term. So, my solution is correct.Therefore, the constants are ( B = 6 ) and ( C = 200 ).I think that's all. Let me recap.Problem 1: Exponential growth model. Given initial value and future value, solved for ( k ) using natural logarithm. Got ( k = ln(5)/5 approx 0.3219 ).Problem 2: Quadratic growth model. Given initial value and future value, set up two equations, solved for ( B ) and ( C ). Got ( B = 6 ) and ( C = 200 ).I don't see any mistakes in my reasoning. The calculations check out when plugging back into the original equations. So, I think I'm confident with these answers.Final Answer1. The constant growth rate ( k ) is boxed{dfrac{ln 5}{5}}.2. The constants are ( B = boxed{6} ) and ( C = boxed{200} ).</think>

🧠:<think>Okay, so I have this problem where I need to analyze the impact of fracking on groundwater quality. I'm given data from 10 fracking sites over 5 years. Each site has the annual volume of water used for fracking and the concentration of a specific contaminant in nearby wells. My tasks are to construct a multivariate regression model and then use it to make a prediction.Starting with part 1, I need to construct a multivariate regression model. The model is given as:[ C = beta_0 + beta_1 V + beta_2 D + epsilon ]Where:- ( C ) is the contaminant concentration,- ( V ) is the volume of water used,- ( D ) is the distance from the fracking site,- ( beta_0, beta_1, beta_2 ) are coefficients,- ( epsilon ) is the error term.I remember that in regression analysis, the normal equations are derived from minimizing the sum of squared residuals. In matrix notation, the normal equations are:[ mathbf{X}^T mathbf{X} mathbf{beta} = mathbf{X}^T mathbf{y} ]Where:- ( mathbf{X} ) is the design matrix,- ( mathbf{beta} ) is the vector of coefficients,- ( mathbf{y} ) is the vector of observed responses.So, I need to express this in matrix form. Let me think about how the design matrix ( mathbf{X} ) would look. Since we have 10 sites, each with data on V and D, the matrix will have 10 rows and 3 columns (including the intercept term ( beta_0 )).Each row in ( mathbf{X} ) will be [1, V_i, D_i] for each site i. So, the first column is all ones for the intercept, the second column is the volume of water used, and the third column is the distance.Therefore, the design matrix ( mathbf{X} ) is a 10x3 matrix:[ mathbf{X} = begin{bmatrix}1 & V_1 & D_1 1 & V_2 & D_2 vdots & vdots & vdots 1 & V_{10} & D_{10}end{bmatrix} ]The vector ( mathbf{y} ) is a 10x1 column vector of the contaminant concentrations:[ mathbf{y} = begin{bmatrix}C_1 C_2 vdots C_{10}end{bmatrix} ]The vector ( mathbf{beta} ) is:[ mathbf{beta} = begin{bmatrix}beta_0 beta_1 beta_2end{bmatrix} ]So, the normal equations are:[ mathbf{X}^T mathbf{X} mathbf{beta} = mathbf{X}^T mathbf{y} ]I need to write this out explicitly. Let's denote ( mathbf{X}^T mathbf{X} ) as a 3x3 matrix and ( mathbf{X}^T mathbf{y} ) as a 3x1 vector.Calculating ( mathbf{X}^T mathbf{X} ):- The (1,1) element is the sum of the first column of ( mathbf{X} ) squared, which is 10 (since there are 10 ones).- The (1,2) element is the sum of the first column times the second column, which is the sum of V_i.- The (1,3) element is the sum of the first column times the third column, which is the sum of D_i.- The (2,1) element is the same as (1,2) because it's symmetric.- The (2,2) element is the sum of V_i squared.- The (2,3) element is the sum of V_i times D_i.- The (3,1) element is the same as (1,3).- The (3,2) element is the same as (2,3).- The (3,3) element is the sum of D_i squared.Similarly, ( mathbf{X}^T mathbf{y} ) will have three elements:- The first element is the sum of C_i.- The second element is the sum of V_i times C_i.- The third element is the sum of D_i times C_i.So, putting it all together, the normal equations are:[ begin{bmatrix}10 & sum V_i & sum D_i sum V_i & sum V_i^2 & sum V_i D_i sum D_i & sum V_i D_i & sum D_i^2end{bmatrix}begin{bmatrix}beta_0 beta_1 beta_2end{bmatrix}=begin{bmatrix}sum C_i sum V_i C_i sum D_i C_iend{bmatrix} ]This is the system of equations I need to solve to find the coefficients ( beta_0, beta_1, beta_2 ).Moving on to part 2, I have the coefficients already: ( beta_0 = 4.5 ), ( beta_1 = 0.002 ), and ( beta_2 = -0.1 ). I need to predict the concentration ( C ) for a well that is 1.5 km away from a fracking site that used 5000 cubic meters of water.Plugging into the regression equation:[ C = 4.5 + 0.002 times 5000 + (-0.1) times 1.5 ]Let me compute each term step by step.First, ( 0.002 times 5000 ). 0.002 is 2 thousandths, so 5000 times that is 10. Because 5000 * 0.001 = 5, so 0.002 is 10.Next, ( -0.1 times 1.5 ). That's -0.15.So, adding them up:4.5 + 10 - 0.15 = 14.35 mg/L.Wait, let me double-check:4.5 + 10 = 14.514.5 - 0.15 = 14.35Yes, that seems correct.So, the predicted concentration is 14.35 mg/L.Just to make sure I didn't make a calculation error:0.002 * 5000 = 10-0.1 * 1.5 = -0.154.5 + 10 = 14.514.5 - 0.15 = 14.35Yep, that's right.I think that's all. I don't see any mistakes in my reasoning.Final AnswerThe predicted concentration of the contaminant is boxed{14.35} mg/L.</think>