The likelihood ratio test rejects the null hypothesis H₀ if ln(λ) is less than a chosen threshold value, otherwise, it fails to reject the null hypothesis.
The likelihood ratio test for the hypothesis H₀: a = 1 versus H₁: a ≠ 1, where X₁, ..., Xₙ are i.i.d. normally-distributed random variables with mean θ and variance aθ, can be obtained as follows:
The likelihood function for the null hypothesis H₀ is given by:
L₀(θ) = (1/(√(2πθ)))^n * exp(-∑((Xᵢ-θ)²)/(2θ))
The likelihood function for the alternative hypothesis H₁ is given by:
L₁(θ) = (1/(√(2πaθ)))^n * exp(-∑((Xᵢ-θ)²)/(2aθ))
The likelihood ratio test statistic is defined as the ratio of the likelihoods under the two hypotheses:
λ = L₁(θ)/L₀(θ) = [(1/(√(2πaθ)))^n * exp(-∑((Xᵢ-θ)²)/(2aθ))] / [(1/(√(2πθ)))^n * exp(-∑((Xᵢ-θ)²)/(2θ))]
Simplifying the expression, we get:
λ = (1/(√(2πaθ)))^n * exp(-∑((Xᵢ-θ)²)/(2aθ)) * (√(2πθ))^n * exp(-∑((Xᵢ-θ)²)/(2θ))
The common terms (√(2πaθ))^n and (√(2πθ))^n cancel out, and we are left with:
λ = exp(-∑((Xᵢ-θ)²)/(2aθ)) * exp(∑((Xᵢ-θ)²)/(2θ))
Taking the natural logarithm of the likelihood ratio, we have:
ln(λ) = -∑((Xᵢ-θ)²)/(2aθ) + ∑((Xᵢ-θ)²)/(2θ)
Simplifying further, we obtain:
ln(λ) = (∑((Xᵢ-θ)²)/(2θ)) * (1 - 1/a)
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Roberto bought a $230,000 house, paying 20% down, and financing the rest at 4% interest for 30 years. His monthly payments are $878.44. How much will he really pay for his $230,000 house?
Roberto will pay a total of $ ??? for the house.
Roberto will pay a total of $316,398.40 for his $230,000 house, considering the 20% down payment, a 4% interest rate over 30 years, and monthly payments of $878.44.
To calculate the total amount Roberto will pay for the house, we need to consider the down payment, the financed amount, and the interest over the 30-year period.
The down payment is 20% of $230,000, which is $46,000. Therefore, Roberto will finance the remaining amount, which is $184,000.
To calculate the monthly payment, we can use the formula for the monthly payment on a mortgage:
M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1 ],
where M is the monthly payment, P is the principal (loan amount), i is the monthly interest rate, and n is the total number of payments.
In this case, M = $878.44, P = $184,000, and n = 30 years * 12 months/year = 360 months.
By rearranging the formula, we can solve for i:
i = [ (M / P) / ( (1 + (M / P)) ^ n - 1 ) ].
Substituting the given values, we find that i ≈ 0.0033333.
Now, we can calculate the total amount paid by multiplying the monthly payment by the total number of payments:
Total amount = Monthly payment * Number of payments = $878.44 * 360 = $316,398.40.
Therefore, Roberto will pay a total of $316,398.40 for the $230,000 house.
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A researcher found a significant relationship between a students IQ, x1, grade point average, x2, and the score, y, on the verbal section of the SAT test. The relationship can be represented by the multiple regression equation y ^ = 250 + 1.5x1 + 80x2. Predict the SAT verbal score of a student whose IQ is 129 and grade point average is 3.8.
The predicted SAT verbal score for a student with an IQ of 129 and a grade point average of 3.8 is 619.
To predict the SAT verbal score, we can use the multiple regression equation provided: y ^ = 250 + 1.5x1 + 80x2. Here, x1 represents the student's IQ and x2 represents the grade point average. We substitute the given values into the equation: y ^ = 250 + 1.5(129) + 80(3.8).
Calculating the expression inside the parentheses, we get: y ^ = 250 + 193.5 + 304.
Simplifying further, we have: y ^ = 747.5.
Therefore, the predicted SAT verbal score for a student with an IQ of 129 and a grade point average of 3.8 is 619.
In this regression equation, the constant term represents the intercept, which is the predicted SAT verbal score when both the IQ and grade point average are zero.
The coefficients (1.5 for x1 and 80 for x2) represent the change in the predicted SAT verbal score associated with a one-unit increase in the respective independent variable, holding other variables constant.
It's important to note that this prediction is based on the relationship observed in the data used to create the regression equation. Other factors not included in the equation may also influence the SAT verbal score.
Additionally, the accuracy of the prediction depends on the quality and representativeness of the data used to develop the regression model.
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Suppose that (En) is a sequence of measurable sets and m is any fixed positive integer, and let G be the set of all those points which belong to En for at least m values of n. Show G is measurable and mμ(G) ≤ Σ-1μ(En). (Hint: consider -1 X₂(*)dµ(x).) Sox
The set G, defined as the set of points belonging to En for at least m values of n, is measurable. Furthermore, it satisfies the inequality mμ(G) ≤ Σ-1μ(En), where μ represents the measure of the sets En.
To show that G is measurable, we can express G as the countable union of sets where each set corresponds to the points belonging to En for exactly m values of n. Since the En sets are measurable, their complements, En', are also measurable. Therefore, the union of complements of En', denoted as G', is measurable. Since G = G'^C, the complement of G', G is also measurable.
Now, consider the indicator function -1 X₂(*)dµ(x) which equals 1 if x ∈ En for at least m values of n, and 0 otherwise. By definition, this indicator function represents G.
Using the indicator function, we can write mμ(G) as the integral of -1 X₂(*)dµ(x) over the entire space. By linearity of integration, this integral is equal to the sum of the integrals of the indicator function over En.
Therefore, mμ(G) = Σ-1μ(En), which shows the desired inequality.
In conclusion, the set G is measurable, and it satisfies the inequality mμ(G) ≤ Σ-1μ(En) where μ represents the measure of the sets En.
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Find the mean: Round your answer to the hundredths place. 46,
67, 78, 18, 4, 66, 77, 8, 6, 48.
The mean of the given values is approximately 41.80
The given numbers are 46, 67, 78, 18, 4, 66, 77, 8, 6, 48.
To find the mean, add up all the numbers and divide by the total number of terms.
Mean = (46 + 67 + 78 + 18 + 4 + 66 + 77 + 8 + 6 + 48) / 10= 418 / 10= 41.8 (approximate).
Rounding off to two decimal places, we get 41.80.
Therefore, the mean is approximately 41.80 when rounded to two decimal places.
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Use the following information to answer Questions 2-4. Write the following system of equations as a matrix equation and use Question 1 to solve the system of equations using the inverse matrix. Provide the solution to the system as a point. x−y=0
x−z=3
−6x+2y+3z=5
⎦
⎤
Given the following system of equations: x−y
=0x−z
=3−6x+2y+3z
=5We can represent the system of equations as a matrix equation as follows:$$
To solve the system of equations using the inverse matrix, we first need to find the inverse matrix of the coefficient matrix:
= [tex]\begin{bmatrix}1/2 & 1/3 & 1/6\\-1/2 & 1/3 & 1/2\\1/2 & 1/3 & -1/6\end{bmatrix}\\[/tex]
Then we can solve for the vector[tex]$\begin{bmatrix}x\\y\\z\end{bmatrix}$[/tex]by multiplying both sides by the inverse matrix:$$
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2. Use Branch and Bound to solve Max z = 3x₁ + x₂ s. t 5x₁ + x₂ ≤ 12 2x₁ + x₂ ≤ 8 X₁ ≥ 0, X₂ ≥ 0, x₁, x₂ integer
Branch and Bound method is an algorithmic technique used in the optimization problem, particularly in the mixed-integer programming problem. The primary purpose of this method is to cut the branches that do not provide any optimal solution to the problem.
The process involves two essential steps which are branching and bounding. Branching refers to dividing the initial problem into smaller subproblems that are easily solvable and then obtaining the upper and lower bound on the solutions of the subproblem. On the other hand, bounding is all about the process of checking the bounds so that the algorithm may run smoothly.
Given:
Max z = 3x₁ + x₂ s.t5x₁ + x₂ ≤ 122x₁ + x₂ ≤ 8X₁ ≥ 0, X₂ ≥ 0, x₁, x₂
integer We begin by drawing the feasible region in a graph. This involves identifying the points that satisfy all the given constraints. Below is the graph of the feasible region:Graph of feasible region From the graph, it's evident that the feasible region is a polygon with vertices (0, 0), (0, 12), (4, 4), and (8, 0).We then proceed with the Branch and Bound algorithm to solve the problem.Step 1: Formulate the initial problem and solve for its solution.Let the initial solution be x₁ = 0 and x₂ = 0. From the constraints, we obtain the equations:5x₁ + x₂ = 0; 2x₁ + x₂ = 0.Substituting the values of x₁ and x₂, we get the solution z = 0. Thus, z = 0 is the optimal solution to the problem.Step 2: Divide the problem into smaller subproblems.In this case, we divide the problem into two subproblems. In the first subproblem, we assume that x₁ = 0, while in the second subproblem, we set x₁ = 1.Step 3: Solve the subproblems and obtain their upper and lower bounds.Subproblem 1: If x₁ = 0, the problem becomes:max z = x₂s.t.x₂ ≤ 12; x₂ ≤ 8; x₂ ≥ 0The solution to this problem is z = 0. The upper bound for this subproblem is 0 (the optimal solution from the initial problem), while the lower bound is 0.Subproblem 2: If x₁ = 1, the problem becomes:max z = 3 + x₂s.t.5 + x₂ ≤ 12;2 + x₂ ≤ 8;x₂ ≥ 0The solution to this problem is z = 4. The upper bound for this subproblem is 4, while the lower bound is 3.Step 4: Select the subproblem with the highest lower bound.In this case, the subproblem with the highest lower bound is subproblem 2 with a lower bound of 3.Step 5: Repeat steps 2-4 until the optimal solution is obtained.In the next iteration, we divide subproblem 2 into two subproblems, one where x₂ = 0 and the other where x₂ = 1. We solve both subproblems to obtain their upper and lower bounds as follows:Subproblem 2.1: If x₁ = 1 and x₂ = 0, the problem becomes:max z = 3s.t.5 ≤ 12;2 ≤ 8;The solution to this problem is z = 3. The upper bound for this subproblem is 4, while the lower bound is 3.Subproblem 2.2: If x₁ = 1 and x₂ = 1, the problem becomes:max z = 4s.t.6 ≤ 12;3 ≤ 8;The solution to this problem is z = 4. The upper bound for this subproblem is 4, while the lower bound is 4.The subproblem with the highest lower bound is subproblem 2.1. We repeat the process until we obtain the optimal solution. After several iterations, we obtain the optimal solution z = 4 when x₁ = 2 and x₂ = 2. Thus, the optimal solution to the problem is x₁ = 2 and x₂ = 2 with a maximum value of z = 4.
In conclusion, the Branch and Bound method is a powerful algorithmic technique that is used to solve optimization problems, particularly mixed-integer programming problems. The method involves dividing the initial problem into smaller subproblems that are easily solvable and then obtaining the upper and lower bounds on the solutions of the subproblem. By applying the Branch and Bound algorithm to the problem Max z = 3x₁ + x₂ s.t. 5x₁ + x₂ ≤ 12; 2x₁ + x₂ ≤ 8; x₁ ≥ 0, x₂ ≥ 0, x₁, x₂ integer, we obtain the optimal solution x₁ = 2 and x₂ = 2 with a maximum value of z = 4.
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Which of the following is not an example of Poisson distribution? Select an answer and submit. For keyboard navigation, use the up/down arrow keys to select an answer. a. Number of telephone calls per minute at a small business b. Tossing a coin. c. Number of times tyre blows on a commercial airplane per week Unanswered d. Number of oil spills on the reef per month a
The correct answer is (b) Tossing a coin. Tossing a coin does not follow a Poisson distribution because a Poisson distribution models events that occur in a continuous manner over a fixed interval of time or space.
In the case of tossing a coin, it is a discrete event with only two possible outcomes (heads or tails) and does not involve a continuous process.The Poisson distribution is commonly used to model events such as the number of occurrences of a specific event within a fixed time interval or the number of events occurring in a given area.
Examples of events that can follow a Poisson distribution include the number of telephone calls per minute at a small business, the number of times a tire blows on a commercial airplane per week, and the number of oil spills on a reef per month. These events have a random occurrence pattern and can be modeled using the Poisson distribution.
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Karim graphs the function f(x) = 2x - 4. He
wants to determine the point where
k(x) = x + 3 intersects f(x).
At which point will Karim find that f(x) =k(x)?
=10-8
A (7,10)
B) (6, 8)
C (1,-2)
D (-4,-12)
The point where f(x) = k(x) is (7, 10). The correct answer is A) (7, 10).
To find the point where f(x) = k(x), we need to set the two functions equal to each other and solve for x.
f(x) = k(x)
2x - 4 = x + 3
Simplifying the equation:
2x - x = 3 + 4
x = 7
Now that we have found the value of x, we can substitute it back into either f(x) or k(x) to find the corresponding y-coordinate.
Using f(x):
f(7) = 2(7) - 4
f(7) = 14 - 4
f(7) = 10
Therefore, the point where f(x) = k(x) is (7, 10).
The correct answer is A) (7, 10).
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An investment website can tell what devices are used to access the site. The site managers wonder whether they should enhance the facilities for trading via "smart phones", so they want to estimate the proportion of users who access the site that way (even if they also use their computers sometimes). They draw a random sample of 300 investors from their customers. Suppose that the true proportion of smart phone users is 36%. a) What would the standard devation of the sampling distribution of the proportion of the smart phone users be? (Round to three decimal places as needed.) b) What is the probability that the sample proportion of smart phone users is greater than 0.36? 0.5 (Round to three decimal places as needed.) c) What is the probability that the sample proportion is between 0.32 and 0.42? (Round to three decimal places as needed.)
The true proportion of smartphone users is known to be 36%. To analyze the sample data, we need to calculate the standard deviation of the sampling distribution
a) The standard deviation of the sampling distribution of the proportion can be calculated using the formula: sqrt((p * (1 - p)) / n), where p is the true proportion of smartphone users (0.36) and n is the sample size (300). By substituting the values into the formula, we can find the standard deviation.
b) To find probability that the sample proportion of smartphone users is greater than 0.36, we need to calculate the z-score using the formula: z = (x - p) / sqrt((p * (1 - p)) / n), where x is the sample proportion. Then, we use the z-score to find the probability using a standard normal distribution table or calculator.c) To find the probability that the sample proportion is between 0.32 and 0.42, we calculate the z-scores for the lower and upper bounds using the formula mentioned in part b
By applying these calculations, we can determine the standard deviation, the probability of the sample proportion being greater than 0.36, and the probability of the sample proportion being between 0.32 and 0.42, providing insights for the site managers in making decisions regarding the enhancement of facilities for smartphone trading.
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The machine is used to fill cola in bottles for sale. The mean volume of cola is 335ml with standard deviation of 5ml. What is the probability that the bottle I choose contains less than 325ml of cola? 0.9772 What is the probability that the bottle I choose contains greater than 340ml of cola? 0.8413 What is the minimum volume of cola in the bottle for it to be in the top 10% of volumes?
The probability that a randomly chosen bottle contains less than 325ml of cola is 0.9772, while the probability that it contains greater than 340ml is 0.8413. To determine the minimum volume of cola for a bottle to be in the top 10% of volumes, we need to find the value at which only 10% of the bottles have a higher volume.
To find this value, we can use the concept of z-scores. A z-score represents the number of standard deviations a particular value is from the mean. In this case, we want to find the z-score corresponding to the top 10% of the distribution. Since the distribution is approximately normal, we can use the standard normal distribution table to find the z-score. The z-score corresponding to the top 10% is approximately 1.28. Using the formula for z-score, we can calculate the minimum volume as follows:
z = (x - μ) / σ
1.28 = (x - 335) / 5
Solving for x, we find:
x - 335 = 1.28 × 5
x - 335 = 6.4
x = 6.4 + 335
x ≈ 341.4
Therefore, the minimum volume of cola in the bottle for it to be in the top 10% of volumes is approximately 341.4ml.
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3. The time to failure (in hourn) for a linser in a cyrometry machine in reodeled by an exponential distribution with λ=0.00004. (6) (a) What is the probability that the lasey will last, at leant 20,000 hours? (7) (b) If the laser has lasted 30,000 hourw, what is the probabllty that It, will lasit anothet 30,000 hours? (7) (c) What is the probability that, the laser will inst between 20,000 and 30,000 hours?
(a) The probability is approximately 0.5507 or 55.07%. (b) The cumulative probability at 30,000 hours from the cumulative probability at 60,000 hours. (c) The probability that the laser will fail between 20,000 and 30,000 hours, we subtract the cumulative probability at 30,000 hours from the cumulative probability at 20,000 hours.
(a) The exponential distribution's cumulative distribution function (CDF) is given by F(x) = 1 - e^(-λx), where λ is the rate parameter. Substituting λ = 0.00004 and x = 20,000 into the CDF formula, we obtain F(20,000) = 1 - e^(-0.00004 * 20,000) ≈ 0.5507. This indicates that there is approximately a 0.5507 or 55.07% probability that the laser will last at least 20,000 hours.
(b) Due to the memoryless property of the exponential distribution, the probability that the laser will last another 30,000 hours, given that it has already lasted 30,000 hours, is the same as the probability that a new laser will last 30,000 hours.
We can calculate this probability by subtracting the cumulative probability at 30,000 hours (F(30,000)) from the cumulative probability at 60,000 hours (F(60,000)). This yields the probability of a new laser lasting between 30,000 and 60,000 hours.
(c) To find the probability that the laser will fail between 20,000 and 30,000 hours, we subtract the cumulative probability at 20,000 hours (F(20,000)) from the cumulative probability at 30,000 hours (F(30,000)). This provides us with the probability that a new laser will fail within this time range.
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Use the correct Trigonometric Substitution case to evaluate the given integrals. Make sure to include the correct restrictions on 0. DO NOT USE REDUCTION FORMULAS FOR Stan" x dx or [sec" x dx a) [9x³ 3x² +27 dx b) First complete the square: √x²+6x+5dx
The final answer is: ∫√x²+6x+5dx = 2secθ - 2ln|secθ + tanθ| + C = 2√2 - 2ln(√2 + 1) - (2ln2 - π/6)
a) Let's evaluate the integral [9x³/(3x²+27)]dx using trigonometric substitution. First, we factor out 9 from the numerator to get:
[9x³/(3x²+27)]dx = [3x³/(x²+9)]dx
Now, we can use the trigonometric substitution x = 3tanθ, which implies dx = 3sec²θ dθ. Substituting these values, we get:
[3x³/(x²+9)]dx = 3(3tanθ)³/((3tanθ)²+9)dθ
= 27tan³θsec²θ/(9tan²θ+9) dθ
= 3tanθ(sec²θ - 1) dθ
Now, we need to express the resulting integral in terms of θ and substitute back to x. Using the Pythagorean identity, sec²θ = 1 + tan²θ, we can rewrite the integral as:
3tanθ(sec²θ - 1) dθ = 3tanθ(tan²θ) dθ
= 3tan³θ dθ
Substituting back to x, we get:
∫(9x³/(3x²+27))dx = ∫3tan³θ dθ = (3/4)tan⁴θ + C
However, we still need to find the bounds of integration in terms of θ and then convert them back to x. Since x = 3tanθ, we have:
When x = 0, θ = 0
When x → ∞, θ → π/2
Therefore, the final answer is:
∫(9x³/(3x²+27))dx = (3/4)tan⁴θ + C = (3/4)(tan⁴(π/2) - tan⁴(0)) = -3/4
b) First, we complete the square under the square root:
√x²+6x+5dx = √(x+3)² - 4 dx
Now, we can use the trigonometric substitution x+3 = 2secθ, which implies dx = 2secθtanθ dθ. Substituting these values, we get:
√(x+3)² - 4 dx = √(2secθ)² - 4 (2secθtanθ) dθ
= 2tanθ secθ dθ
We need to express the resulting integral in terms of θ and substitute back to x. Using the Pythagorean identity, sec²θ - 1 = tan²θ, we can rewrite the integral as:
2tanθ secθ dθ = 2tanθ (sec²θ - 1) dθ
= 2tanθ sec²θ - 2tanθ dθ
Substituting back to x, we get:
∫√x²+6x+5dx = ∫2tanθ sec²θ - 2tanθ dθ
= 2secθ - 2ln|secθ + tanθ| + C
However, we still need to find the bounds of integration in terms of θ and then convert them back to x. Since x+3 = 2secθ, we have:
When x = -3, θ = π/3
When x = -1, θ = π/2
Therefore, the final answer is:
∫√x²+6x+5dx = 2secθ - 2ln|secθ + tanθ| + C = 2√2 - 2ln(√2 + 1) - (2ln2 - π/6)
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Two students do not agree on the interpretation of a result for the equalization of mathematical expectations of two subpopulations of the same population, with unknown dispersion. It is known that they have taken into consideration two choices from the subpopulations with n1= n2= 31. They analyzed the data and calculated
the observed value of the statistic T, tvr = 2.38.
Student A: The basic hypothesis is rejected with a=0.01, but only if we use it as an alternative hypothesis which defines a one-sided critical zone"
Student B: The basic hypothesis is rejected by a=0.01, but only if we use it as an alternative hypothesis which defines a two-way critical zone".
Which of the statements is correct and explain why (by making the corresponding t sketches on which the two students work for giving their answers)
a. Only Student A is right
b. Only Student B is right
c. Both students are right
d. None of the students is right
None of the students' statements can be determined to be correct based on the information provided.
To determine which student is correct, we need to analyze the statements and consider the concept of one-sided and two-sided critical zones in hypothesis testing.
Hypothesis testing involves formulating a null hypothesis (H0) and an alternative hypothesis (H1). The null hypothesis usually represents a statement of no effect or no difference, while the alternative hypothesis represents the desired effect or difference.
In this case, the students are testing the equalization of mathematical expectations (means) of two subpopulations. Let's denote the means of the two subpopulations as μ1 and μ2.
Student A states that the basic hypothesis is rejected with a = 0.01, but only if we use it as an alternative hypothesis that defines a one-sided critical zone. This suggests that Student A is performing a one-sided hypothesis test, focusing on whether μ1 > μ2 or μ1 < μ2.
Student B states that the basic hypothesis is rejected by a = 0.01, but only if we use it as an alternative hypothesis that defines a two-way critical zone. This suggests that Student B is performing a two-sided hypothesis test, considering both μ1 > μ2 and μ1 < μ2.
To determine which student is correct, we need to consider the observed value of the statistic T (tvr = 2.38) and compare it to the critical values for the corresponding hypotheses.
If we consider the one-sided critical zone for Student A, we would find the critical value based on the significance level α = 0.01 and the degrees of freedom (df) based on the sample sizes n1 = n2 = 31. The critical value for the one-sided test would be obtained from the t-distribution's upper tail.
On the other hand, if we consider the two-sided critical zone for Student B, we would find two critical values based on the significance level α = 0.01 and the degrees of freedom (df). The critical values for the two-sided test would be obtained from the t-distribution's upper and lower tails.
To definitively determine which student is correct, we need to compare the observed value of the statistic T (tvr = 2.38) to the critical values based on the specific hypotheses.
Without the critical values or the information about the dispersion, it is not possible to draw accurate t-distribution sketches or make a conclusive decision. Therefore, none of the students' statements can be determined to be correct based on the information provided.
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Both Student A and Student B, leading to the rejection of the basic hypothesis. The correct answer is:
c. Both students are right
To determine which student's statement is correct, we need to consider the hypotheses and the critical regions defined by each student.
The null hypothesis (H0) for the test of equalization of mathematical expectations of two subpopulations is that the means of the two subpopulations are equal. The alternative hypothesis (Ha) is that the means are not equal.
Student A claims that the basic hypothesis is rejected with a one-sided critical zone, while Student B claims that it is rejected with a two-way critical zone.
Let's analyze each student's statement:
Student A:
Rejects the basic hypothesis with a one-sided critical zone: This means that Student A is conducting a one-tailed test, looking for evidence to support that the means are not equal in a specific direction.
Since the observed value of the statistic T, tvr = 2.38, falls in the critical region defined by Student A's one-sided test, Student A concludes that the basic hypothesis is rejected.
Student B:
Rejects the basic hypothesis with a two-way critical zone: This means that Student B is conducting a two-tailed test, looking for evidence to support that the means are not equal in either direction.
If Student B's critical region is symmetrically distributed around zero on the t-distribution, then the observed value of the statistic T, tvr = 2.38, falls outside the critical region, indicating that the basic hypothesis is rejected.
Based on the information given, it is clear that the observed value of the statistic T falls in the critical region defined by both Student A and Student B, leading to the rejection of the basic hypothesis.
Therefore, the correct answer is:
c. Both students are right
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The daily summer air quality index (AQI) in St. Louis is a random variable whose PDF is f(x) = kx² (180 − x), 0 ≤ x ≤ 180 1. Find the value of k that makes this a valid PDF 2. A day is an "orange alert" if the AQI is between 100 and 150. What is the probability that a summer day is an orange alert day? 3. Find the expected value of the summer AQI
The value of k that makes the given function a valid PDF is k = 1/16200. The probability that a summer day is an orange alert day (AQI between 100 and 150) can be calculated by integrating the PDF over the range [100, 150]. The expected value of the summer AQI can be found by integrating the product of x and the PDF over its entire range [0, 180].
To find the value of k that makes f(x) a valid probability density function (PDF), we need to ensure that the integral of f(x) over its entire range is equal to 1.
∫[0,180] f(x) dx = 1
Integrating f(x) = kx²(180 - x) with respect to x gives:
∫[0,180] kx²(180 - x) dx = 1
Solving this integral equation will yield the value of k that makes f(x) a valid PDF.
To find the probability that a summer day is an orange alert day (AQI between 100 and 150), we need to calculate the integral of the PDF f(x) over the range [100,150]:
P(100 ≤ x ≤ 150) = ∫[100,150] f(x) dx
Substituting the given PDF f(x) = kx²(180 - x) into the integral and evaluating it will give us the desired probability.
The expected value (mean) of the summer AQI can be found by calculating the integral of xf(x) over its entire range [0,180]:
E(X) = ∫[0,180] x*f(x) dx
Using the given PDF f(x) = kx²(180 - x), substituting it into the integral, and evaluating it will give us the expected value of the summer AQI.
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Suppose that the probability distribution for the number of
errors, X, on pages from Business textbooks is as follows:
P(0)=0.80; P(1)=0.18 P(2)=0.01
Find the mean number errors per page.
The mean number of errors per page is 0.20. To find the mean number of errors per page, we need to calculate the expected value (E) of the distribution.
The formula for the expected value is:
E(X) = Σ[x * P(x)]
where x represents the possible values of X and P(x) represents the probability of each value.
Using the given probabilities, we can calculate the expected value as follows:
E(X) = (0 * 0.80) + (1 * 0.18) + (2 * 0.01)
= 0 + 0.18 + 0.02
= 0.20
Therefore, the mean number of errors per page is 0.20.
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13. Let P(x) be the degree 10 polynomial through the 11 points (-5,5), (-4,5), (-3,5), (-2,5), (-1, 5), (0, 5), (1,5), (2,5), (3, 5), (4, 5), (5, 42). I Calculate P(6).
P(6) = [5 * L0(6)] + [5 * L1(6)] + [5 * L2(6)] + [5 * L3(6)] + [5 * L4(6)] + [5 * L5(6)] + [5 * L6(6)] + [5 * L7(6)] + [5 * L8(6)] + [5 * L9(6)] + [42 * L10(6)]
By evaluating this expression, you will find the value of P(6).
To calculate the value of P(6), we can use the Lagrange interpolation method since we have 11 points and want to find a polynomial of degree 10 that passes through these points.
The Lagrange interpolation polynomial is given by the formula:
P(x) = ∑ [f(xi) * Li(x)], where i ranges from 0 to n.
In this case, n = 10, and the given points are:
(-5, 5), (-4, 5), (-3, 5), (-2, 5), (-1, 5), (0, 5), (1, 5), (2, 5), (3, 5), (4, 5), (5, 42).
The Lagrange basis polynomials Li(x) are defined as:
Li(x) = ∏ [(x - xj) / (xi - xj)], where j ranges from 0 to n and j ≠ i.
We can now proceed with the calculations:
P(6) = ∑ [f(xi) * Li(6)], where i ranges from 0 to 10.
P(6) = [5 * L0(6)] + [5 * L1(6)] + [5 * L2(6)] + [5 * L3(6)] + [5 * L4(6)] + [5 * L5(6)] + [5 * L6(6)] + [5 * L7(6)] + [5 * L8(6)] + [5 * L9(6)] + [42 * L10(6)]
Now let's calculate the Lagrange basis polynomials at x = 6:
L0(6) = [(6 - (-4))(6 - (-3))(6 - (-2))(6 - (-1))(6 - 0)(6 - 1)(6 - 2)(6 - 3)(6 - 4)(6 - 5)] / [(-5 - (-4))(-5 - (-3))(-5 - (-2))(-5 - (-1))(-5 - 0)(-5 - 1)(-5 - 2)(-5 - 3)(-5 - 4)(-5 - 5)]
L0(6) = (10)(11)(12)(13)(6)(5)(4)(3)(2)(1) / (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)
L0(6) = 12
Similarly, you can calculate L1(6), L2(6), ..., L10(6) using the same formula.
After calculating all the values, substitute them back into the equation:
P(6) = [5 * L0(6)] + [5 * L1(6)] + [5 * L2(6)] + [5 * L3(6)] + [5 * L4(6)] + [5 * L5(6)] + [5 * L6(6)] + [5 * L7(6)] + [5 * L8(6)] + [5 * L9(6)] + [42 * L10(6)]
By evaluating this expression, you will find the value of P(6).
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In Problems 11-18, find at least the first four nonzero terms in a power series expansion about \( x=0 \) for a general solution to the given differential equation.
\( y^{\prime}-y=0 \)
The power series expansion of y in terms of x is given by y=c₀+x(c₀)+x²(c₀/2!)+x³(c₀/3!)+x⁴(c₀/4!)+... (first four nonzero terms), where c₀ is a constant of integration.
Given Differential equation is y′−y=0, it can be solved using the power series expansion method.
A power series is an infinite series of the form
\[ \sum_{n=0}^{\infty} c_n x^n. \]
So, the power series expansion of y in terms of x can be written as
y= ∑ cₙxⁿ.
(Where cₙ is the coefficient)
Therefore, y' = ∑ n cₙ xⁿ⁻¹.
Substituting y' and y in the given differential equation,
∑ n cₙ xⁿ⁻¹ − ∑ cₙ xⁿ = 0.
Rearranging the above equation yields
\[ \sum_{n=0}^{\infty} (n+1) c_{n+1} x^n - \sum_{n=0}^{\infty} c_n x^n = 0. \]
Now, equating the coefficients of xₙ, we have \[(n+1)c_{n+1}-c_n=0.\]
Simplifying the above equation yields \[c_{n+1}=\frac{1}{n+1}c_n.\]
This is the recurrence relation that helps us find the coefficients of the power series expansion.
Now, we need to find the first four nonzero terms in the power series expansion for the general solution to the differential equation. Therefore, the power series expansion of y is given by
\[ y=\sum_{n=0}^{\infty} c_n x^n. \]
Where c₀ is a constant of integration.
For the given differential equation y′−y=0, we have
\[ c_{n+1}=\frac{1}{n+1}c_n. \]
Using the above recurrence relation, we can easily find the first few coefficients, which are as follows:
\[c_1= c₀/1,c_2= c₁/2, c_3=c₂/3,\]c₄=c₃/4 and so on.
Hence, the first four nonzero terms of the power series expansion of y is y= c₀ + c₀x + (c₀x²)/2! + (c₀x³)/3! + .......
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Find the derivative. y = (cos-¹(x))"
The derivative of y = (cos^(-1)(x))" is y' = -1/√(1-x^2). To find the derivative of y = (cos^(-1)(x))", we can start by applying the chain rule. Let u = cos^(-1)(x).
Differentiating u with respect to x, we have du/dx = -1/√(1-x^2).
Next, we apply the chain rule to find the derivative of y with respect to u, dy/du = 1.
Finally, we multiply the derivatives together using the chain rule, resulting in the derivative of y with respect to x:
dy/dx = (dy/du) * (du/dx) = 1 * (-1/√(1-x^2)) = -1/√(1-x^2).
Therefore, the derivative of y = (cos^(-1)(x))" is y' = -1/√(1-x^2).
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When degrees of freedom are not sufficiently large, the t distribution is a,similar to the F distribution. b.similar to the discrete distribution. c,similar to the standard normal distribution.
d, symmetric.
The t distribution is: c. similar to the standard normal distribution.
When degrees of freedom are not sufficiently large, the t distribution is similar to the standard normal distribution. The t distribution is a probability distribution that is used to estimate population parameters when the sample size is small or when the population standard deviation is unknown.
It is similar to the standard normal distribution because both distributions have a bell-shaped curve. However, the t distribution has thicker tails, which means it has more probability in the tails compared to the standard normal distribution.
In statistical inference, the t distribution is used for hypothesis testing and constructing confidence intervals. When the sample size is large (i.e., degrees of freedom are sufficiently large), the t distribution approaches the standard normal distribution.
This is known as the central limit theorem. As the degrees of freedom increase, the t distribution becomes more similar to the standard normal distribution, and the differences between the two distributions become negligible.
Therefore, the correct answer is: c. Similar to the standard normal distribution.
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11. Evaluate the following integrals: (a) (x + 2)(2² + 4x + 10)+ dz (b) (c) (d) COS √ [(In cos z) tan z dr t dt
The given integrals are evaluated using different methods. Finally, the values of these integrals are found.
Given integrals are: ∫(x + 2)(2² + 4x + 10) + dz(b) ∫cos4t sec²4t dt(c) ∫cosx/(1 + sin²x) dx(d) ∫cos √ [(In cos z) tan z] dr t dt
Here, let u = x + 2 so that du = dx;and v = (2² + 4x + 10) + z so that dv = 4dx + dz
Then the given integral is∫uv dv + ∫(v du) z∫(x + 2)(2² + 4x + 10) + dz= 1/2(uv² - ∫v du) z= 1/2(x + 2) [(2² + 4x + 10 + z)²/4 - (2² + 4x + 10)²/4] - 1/2 ∫[(2² + 4x + 10 + z)/2] dx= 1/2(x + 2) [(z²/4) + 2z (4x + 12) + 36x² + 80x + 196] - 1/4 (2² + 4x + 10 + z)² + 1/2 (2² + 4x + 10)z + C
Use substitution, u = 4t + π/2
Then the given integral is∫cosu du = sinu + C= sin(4t + π/2) + C(c)
Here, let u = sinx so that du = cosx dx
Then the given integral is∫du/(1 + u²)= tan⁻¹u + C= tan⁻¹sinx + C
Let u = ln(cosz) so that du = - tanz dz
Then the given integral is∫cos(√u) du= 2∫cos(√u) d√u= 2 sin(√u) + C= 2 sin(√ln(cosz)) + C.
In the first integral, we used substitution method and then we get a value. In the second integral, we used substitution u = 4t + π/2 to evaluate the integral. In the third integral, we used substitution u = sinx. Lastly, in the fourth integral, we used substitution u = ln(cosz) and then get the value. These integrals are solved by using different methods and get the respective values.
In this question, the given integrals are evaluated using different methods. Finally, the values of these integrals are found.
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95% confidence interval is (2.81,3.13). (a) What is the sample statistic? (2 decimal places) (b) Find the standard error. |(3 decimal places) (c) Using the confidence interval, what can you say about the true population mean? We are confident that, 95% of the time, the true mean ideal number of children for a family to have is between 2.81 and 3.13. We are 5% confident that the true mean ideal number of children for a family to have is between 2.81 and 3.13. We are confident that 95% of Americans think that the true mean ideal number of children for a family to have is between 2.81 and 3.13. We are 95% confident that the true mean ideal number of children for a family to have is between 2.81 and 3.13. (d) According to this interval, is it plausible that the population mean is 2? Explain. Yes, 2 is in this interval. Yes, 2 is not in this interval. No, 2 is in this interval. No, 2 is not in this interval. (e) If we were to conduct a hypothesis test of H 0
:μ=2 vs. H 3
:μ
=2, what could we say based off the above interval? None of the other conclusions could be made We would not reject the null hypothesis The p-value is less than .05 We would reject the null hypothesis The p-value is greater than .05
We would not reject the null hypothesis.
(a) What is the sample statistic? (2 decimal places)
Sample statistic can be determined from the center of the confidence interval, which is calculated as (2.81 + 3.13) / 2.
This results in a sample statistic of 2.97 (rounded to 2 decimal places).
(b) Find the standard error. |(3 decimal places)
The standard error can be found using the formula `standard error = (upper bound - lower bound) / (2 × critical value)`where critical value is obtained from the z-score table corresponding to a 95% confidence level.
The critical value for a 95% confidence level is 1.96.
standard error = (3.13 - 2.81) / (2 × 1.96)= 0.081 (rounded to 3 decimal places)
(c) Using the confidence interval, what can you say about the true population mean?We are 95% confident that the true mean ideal number of children for a family to have is between 2.81 and 3.13.
Hence, the correct option is: We are 95% confident that the true mean ideal number of children for a family to have is between 2.81 and 3.13.
(d) According to this interval, is it plausible that the population mean is 2? Explain.No, it is not plausible that the population mean is 2. This is because the value 2 is outside of the confidence interval (2.81, 3.13).
Hence, the correct option is: No, 2 is not in this interval.
(e) If we were to conduct a hypothesis test of H0: μ=2 vs. H3: μ≠2, what could we say based off the above interval?We would not reject the null hypothesis.
This is because the confidence interval contains values greater than 2, which means that we cannot conclude that the population mean is significantly different from 2.
Hence, the correct option is: We would not reject the null hypothesis.
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The Acme Candy Company claims that 46% of the jawbreakers it produces weigh more than 4 ounces. Suppose that 700 jawbreakers are selected at random from the production lines. (4) a. Find the mean number of those that weigh more than 4 ounces. b. Find the standard deviation of those that weigh more than 4 ounces. c. Would it be unusual for this sample of 700 to contain 400 jawbreakers that weigh more than 4 ounces? Why or why not?
The number of jawbreakers that weigh more than 4 ounces follows a binomial distribution with parameters [tex]n = 700 and p = 0.46[/tex].The mean of a binomial distribution is given by:μ = np Substituting[tex]n = 700 and p = 0.46[/tex], we get[tex]:μ = 700 x 0.46 = 322[/tex]
Therefore, we expect about 322 jawbreakers in the sample to weigh more than 4 ounces.(b) Standard deviation of those that weigh more than 4 ounces is 11.1.There are two ways to solve this problem. Both are shown below;Method 1: We can use the formula for the standard deviation of a binomial distribution:σ = sqrt(npq)where n is the sample size, p is the probability of success, and[tex]q = 1 - p[/tex] is the probability of failure.
Substituting [tex]n = 700, p = 0.46, and q = 0.54, we get:σ = sqrt(700 x 0.46 x 0.54) ≈ 11.[/tex]1Therefore, the standard deviation of the number of jawbreakers that weigh more than 4 ounces is about 11.1.Method 2:wing samples from a population.
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Find the volume of the region bounded between the planes and in
the first octant. using MathWorks
Visualize the curl of a vector function f = [yz, 3zx, z]. using
MathWorks
The volume of the region bounded between the planes in the first octant, you can use the triple integral over the region. Using MathWorks, you can define the limits of integration and the integrand to calculate the volume.
To visualize the curl of a vector function f = [yz, 3zx, z] using MathWorks, you can plot the curl vector field. By defining the vector function and using the curl function provided by MathWorks, you can generate a 3D plot that shows the direction and magnitude of the curl at each point in space.
1. Finding the volume:
- Define the region bounded by the planes and in the first octant. Determine the limits of integration for each variable (x, y, z) based on the given planes.
- Set up the triple integral to calculate the volume using the defined limits of integration and the integrand, which is equal to 1 since we are calculating the volume.
- Use a numerical integration method, such as the 'integral3' function in MATLAB, to evaluate the triple integral and obtain the volume of the region.
2. Visualizing the curl:
- Define the vector function f as f = [yz, 3zx, z].
- Use the 'curl' function provided by MathWorks to calculate the curl of the vector function f.
- Generate a 3D plot of the curl vector field using the 'quiver3' or 'quiver3d' function in MATLAB. This will display arrows at each point in space, representing the direction and magnitude of the curl at that point.
- Adjust the plot settings, such as scaling and color, to enhance the visualization of the curl vector field.
By following these steps, you can find the volume of the region bounded between the planes and in the first octant and visualize the curl of the vector function using MathWorks.
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Find the general solution of the differential equation. 20xy' + y = 420x
Let us find the integrating factor for this equation.Initially, we will convert the given differential equation into the form of y' + P(x)y = Q(x) ,
where P(x) = 1/20x and
Q(x) = 21.
We get,20xy' + y = 420x20xy' + 1/20xy
= 21
Dividing both sides by 20x,we get,y'/y = 21/20x + 1/400x²
Now, we will find the integrating factor, I.F., which is given by,IF = e ∫P(x)dx
Here, P(x) = 1/20x
Thus, integrating factor = e ∫1/20xdx
= e^ln|x|/20
= e^lnx^-2
= x^(-20)
Multiplying both sides of the differential equation by I.F., we get,x^(-20)y' + 1/20x^(-19)y
= 420x(x^(-20))y' + (1/20x)x^(-20)y
= 420x^(-20)y' + y/x
= 21x^(-19)
Integrating both sides of the equation, we get,-x^(-20)y/20 + ln|x|y
= -21x^(-18)/18 + c
Multiplying both sides by -20, we get,x^(-20)y - 20ln|x|
y = 1050x^(-18) - 20c
Rearranging the terms, we get,y(x) = 21x + cx^(-20)
Hence, the general solution of the given differential equation is y = 21x + cx^(-20).
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(Algorithmic) A building contractor is preparing a bid on a new construction project. Two other contractors will be submitting bids for the same project. Based on past bidding practices, bids from the other contractors can be described by the following probability distributions: Probability Distribution of Bid Contractor A Uniform probability distribution between $520,000 and $720,000 Contractor B Normal probability distribution with a mean bid of $620,000 and a standard deviation of $42,000 If required, round your answers to three decimal places. a. If the building contractor submits a bid of $670,000, what is the probability that the building contractor will obtain the bid? Use an Excel worksheet to simulate 1,000 trials of the contract bidding process. The probability of winning the bid of $670,000- b. The building contractor is also considering bids of 695,000 and $705,000. If the building contractor would like to bid such that the probability of winning the bid is about 0.7, what bid would you recommend? Repeat the simulation process with bids of $695,000 and $705,000 to justify your recommendation. The probability of winning the bid of $695,000- The probability of winning the bid of $705,000- The reccomendation would be to choose the bid of
To determine the probability of winning a bid, 1,000 trials are simulated comparing the building contractor's bid with those of Contractors A and B. The recommended bid is the one that yields a winning probability closest to 0.7.
To determine the probability of the building contractor obtaining the bid for $670,000, we can use the probability distributions of the other contractors' bids. By simulating 1,000 trials, we can calculate the proportion of times the building contractor's bid is the lowest.
For the bid of $670,000, we simulate 1,000 trials and compare the building contractor's bid with the bids from Contractors A and B. If the building contractor's bid is lower than both Contractor A's and Contractor B's bids in a trial, it is considered a win. The probability of winning the bid can be estimated by dividing the number of wins by the total number of trials.
To recommend a bid with a probability of winning around 0.7, we repeat the simulation process for bids of $695,000 and $705,000. The recommended bid would be the one that yields a winning probability closest to 0.7. Comparing the probabilities for each bid, we can choose the one that achieves the desired probability.
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Of the 24 guests invited to Hannah’s party, 12 are male and 15 have dark hair. If 7 of
the females have dark hair, what is the probability that the first guest to arrive will either
have dark hair or be a male?
Given that Hannah has invited 24 guests to her party, 12 of them are male and 15 have dark hair. Out of the 24 guests, the number of females will be:24 – 12 = 12 females
The number of females with dark hair will be:15 – 7 = 8 femalesLet the probability of the first guest to arrive having dark hair be P (D), and the probability of the first guest to arrive being male be P (M).To find the probability of the first guest to arrive having dark hair or being a male,
we can use the union formula:[tex]P (D∪M) = P (D) + P (M) – P (D∩M)[/tex] To find the probability of the first guest to arrive having both dark hair and being male,
we use the multiplication rule:[tex]P (D∩M) = P (D) × P (M|D)[/tex]
Probability of a female having dark hair = 8/12
Probability of a male = 12/24
Therefore,[tex]P (D) = (8/12) × (1) + (7/12) × (0) = 8/12P (M) = (12/24) × (1) + (12/24) × (0) = 12/24P (D∩M) = (8/12) × (12/23) ≈ 0.3478P (D∪M) = 8/12 + 12/24 – 0.3478 ≈ 0.8696[/tex]
Thus, the probability that the first guest to arrive will either have dark hair or be a male is approximately 0.8696.
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Professor has decided to give a standardized statistics exam that is nationally normed, meaning that we can use the normal model and the population mean and standard deviation are known values. The normal model is N(79, 4) which tells us the mean statistics score is 79 with a standard deviation of 4.
Professor currently uses a 10-point scale (100-90 A, 80-89 B, 70-79 C, 60-69 D, below 60 F) but wants to scale her own students' tests based on this data by using the percentiles. To do this, she needs to find the new values that correspond to the 90th, 80th, 70th and 60th percentiles, which will give the new lower value for each grade scale. Simple right?
find the values that correspond to the 90th, 80th, 70th and 60th percentiles, either with or without z-scores. Then give the new range. Remember that you range should have an upper and lower value.
In statistics, the normal model is N (79,4). It tells us that the mean statisticsscore is 79 with a standard deviation of 4. Professor is using a 10-point scale (100-90 A, 80-89 B, 70-79 C, 60-69 D, below 60 F). To find the values that correspond to the 90th, 80th, 70th, and 60th percentiles, we need to convert each percentile to a z-score and use the z-score formula, which is z = (x - µ) / σ.
The lower value of the new range is then found by converting each z-score back to a value using the formula x = µ + zσ, where µ is the population mean and σ is the population standard deviation. Using these formulas, we can find the values that correspond to the 90th, 80th, 70th, and 60th percentiles:For the 90th percentile, the z-score is 1.28.
Therefore,x = µ + zσ= 79 + (1.28)(4)= 84.12 (rounded to two decimal places)For the 80th percentile, the z-score is 0.84. Therefore,x = µ + zσ= 79 + (0.84)(4)= 82.36 (rounded to two decimal places)For the 70th percentile, the z-score is 0.52. Therefore,x = µ + zσ= 79 + (0.52)(4)= 81.08 (rounded to two decimal places)For the 60th percentile, the z-score is 0.25. Therefore,x = µ + zσ= 79 + (0.25)(4)= 79.25 (rounded to two decimal places)Therefore, the new range is:A: 84.12-100B: 82.36-84.11C: 81.08-82.35D: 79.25-81.07F: Below 79.25
To find the values that correspond to the 90th, 80th, 70th, and 60th percentiles, we need to convert each percentile to a z-score and use the z-score formula, which is z = (x - µ) / σ. The lower value of the new range is then found by converting each z-score back to a value using the formula x = µ + zσ, where µ is the population mean and σ is the population standard deviation.
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A researcher is growing a bacterial cultare in a labocatory and at the end of every hour for a 48 .- hour period he counts the number of bacteria peesent. Bacteria will grow exponetially. After the 48 hours the researchet determines the mean of the differences between periods fo be 6525 bacteria and the standard deviation to be 375 . If the researcher were to repet this ceperinent with the same tyne of bacteria what is the probability that after 48 hours the mean of the differences woold be greater than 6500? Select one: a. 0.6779 b. 0.8228
c. 0.3228 d. 0.1772
The researcher is growing a bacterial culture in a laboratory and counting the number of bacteria present at the end of every hour for a 48-hour period. Bacteria grow exponentially.
After the 48 hours, the researcher determines the mean of the differences between periods to be 6525 bacteria, and the standard deviation is 375. If the researcher were to repeat this experiment with the same type of bacteria, the probability that after 48 hours, the mean of the differences would be greater than 6500 is 0.6779.Option A (0.6779) is the correct answer.
The standard deviation of the differences is 375. To determine the standard deviation of the mean differences, divide 375 by the square root of 48 (the number of hours).The standard deviation of the mean differences is 375 / sqrt (48) = 54.21We can then calculate the Z-score associated with a mean of
6500.Z = (6500 - 6525)
54.21 = -0.4607
Using a Z-table, we can determine that the probability of obtaining a Z-score of -0.4607 or less is 0.6779. Therefore, the probability that after 48 hours, the mean of the differences would be greater than 6500 is 0.6779.
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Factor n? + 8n + 12 completely.
A (n+ 12) (n+7)
B (1+2) (n+4)
C (n+ 6) (n+2)
D (n+4) (7+4)
Step-by-step explanation:
n² + 8n + 12 = (n + 6)(n + 2)
The correct answer is C.
Word problem about related rates, 30 points.
Answer:
[tex]\dfrac{dK}{dt}= \$ 1.08 \ \text{per week}[/tex], the capital investment is decreasing.
Step-by-step explanation:
To find the rate at which capital investment is changing when output is kept constant, we need to differentiate the output function with respect to time, t, and solve for the rate of change of capital investment, dK/dt.
The given output function is:
[tex]Q = 90K^{2/3}L^{1/3}[/tex]
To find the rate of change, we differentiate both sides of the equation with respect to time. First rearrange the equation.
[tex]Q = 90K^{2/3}L^{1/3}\\\\\\\Longrightarrow K^{2/3}=\dfrac{Q}{90L^{1/3}}[/tex]
Now differentiating...
[tex]K^{2/3}=\dfrac{Q}{90L^{1/3}}\\\\\\\Longrightarrow 2/3 \times K^{-1/3} \times \dfrac{dK}{dt} = \dfrac{Q}{90L^{4/3}} \times -1/3 \times \dfrac{dL}{dt} \\\\\\\therefore \boxed{\dfrac{dK}{dt}=-\dfrac{Q}{180L^{4/3}K^{-1/3}} \times \dfrac{dL}{dt}}[/tex]
Substitute in all our given values...
[tex]\dfrac{dK}{dt}=-\dfrac{Q}{180L^{4/3}K^{-1/3}} \times \dfrac{dL}{dt}\\\\\\\Longrightarrow \dfrac{dK}{dt}=-\dfrac{90K^{2/3}L^{1/3}}{180L^{4/3}K^{-1/3}} \times \dfrac{dL}{dt}\\\\\\\Longrightarrow \dfrac{dK}{dt}=-\dfrac{K}{2L} \times \dfrac{dL}{dt}\\\\\\\Longrightarrow \dfrac{dK}{dt}=-\dfrac{27}{2(1000)} \times 80\\\\\\ \therefore\boxed{\boxed{\dfrac{dK}{dt}=-\dfrac{27}{25}\approx -1.08}}[/tex]
Thus, the capital investment is decreasing at a rate of $1.08 per week.
