The margin of error for a 99% confidence interval in a poll on whether states should be allowed to conduct random drug tests on elected officials, based on 11,220 respondents, would be approximately 1.5%.
The margin of error is determined by the sample size and the desired level of confidence. In this case, the sample size is 11,220 respondents. To calculate the margin of error, we need to consider the formula:
Margin of Error = (Z-score) * (Standard Deviation / Square Root of Sample Size)
For a 99% confidence interval, the Z-score is approximately 2.576, corresponding to the two-tailed test. Since the poll results indicate that 89% of respondents said "yes," the standard deviation can be estimated using the formula:
Standard Deviation = Square Root of (p * (1 - p) / n)
where p is the proportion of respondents who said "yes" (0.89) and n is the sample size (11,220).
With these values, the margin of error for a 99% confidence interval would be approximately 1.5%.
In general, as the desired level of confidence decreases (e.g., from 99% to 90%), the margin of error becomes smaller. This is because a lower level of confidence allows for a greater chance of error or uncertainty. When constructing a confidence interval, a smaller margin of error means that the range of plausible values for the population parameter (in this case, the proportion of people who support random drug tests) is narrower. However, it's important to note that a smaller margin of error also implies a larger sample size requirement to achieve the same level of precision.
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In Fantasies Island, a tropical resort where fantasies apparently come true is managed by its proprietor, Mr. James. Mr. James made a rule that every family living on the island must have only two children. Each child is just as likely a boy or a girl. He ordered that each first girl (if any) born to the family must bear the name Ezra (James' wife who died due to unknown sickness). These two children must not have the same name. Upon arrival at the island, Mr. James welcomes you together with a randomly chosen family that will serve you and has a girl named Ezra. The probability that there is two girls in his family is? (correct to 4 significant figures)
The probability that this family has two girls is 0.5.
In the given scenario, a tropical resort called Fantasies Island is owned by its manager, Mr. James.
He implemented a rule that every family should have only two children.
In this case, each child is equally likely to be either a girl or a boy.
The first girl (if any) born to the family must be named Ezra, and these two children should not have the same name.
Upon arriving at the island, Mr. James welcomes you together with a family, randomly selected to serve you.
We can approach this problem by making use of Bayes' theorem.
Bayes' theorem is given as; P(A|B) = (P(B|A)*P(A))/P(B)
Where; P(A|B) = Probability of A given B has occurred.
P(B|A) = Probability of B given A has occurred.
P(A) = Probability of A.P(B) = Probability of B.
We are given that the family selected has a girl named Ezra. We are supposed to find the probability that this family has two girls.
Therefore, we have the following; A: The family has two girls.
B: The family has a girl named Ezra.
Using the above information and applying Bayes' theorem, we get; P(A|B) = P(B|A)*P(A)/P(B)P(A) = Probability that the family has two girls.
P(B|A) = Probability that the family has a girl named Ezra, given that they have two girls.
P(B) = Probability that the family has a girl named Ezra.
Now, we can find these probabilities;
P(A) = Probability of having two girls;
We are given that each child is equally likely to be either a girl or a boy.
Therefore, there are four possible outcomes of the two children.
They can be BB, BG, GB, or GG. But only one outcome satisfies the condition that the two children are girls.
Hence, P(A) = 1/4P(B|A) = Probability of having a girl named Ezra when the family has two girls;We are given that the first girl born in the family must be named Ezra.
Therefore, there is only one possible way of naming the two girls, which is "Ezra and something else."
Hence,P(B|A) = 1P(B) = Probability of having a girl named Ezra;
We are given that one of the children in the family is named Ezra.
Therefore, there are two possible ways of naming the children.
They can be either "Ezra and boy" or "Ezra and girl." Hence,P(B) = 1/2
Putting all the probabilities in Bayes' theorem, we get;
P(A|B) = 1* (1/4) / (1/2)P(A|B) = 1/2Corrected to 4 significant figures, we getP(A|B) = 0.5000
Therefore, the probability that this family has two girls is 0.5.
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Solutions of Higher Differential Equations. Determine the solution of each differential equation. Kindly enclose in a box your final answer in its simplest form in the solution paper. Use four decimal places for your final answers, if applicable. Show your complete solutions. Please write clearly and legibly. Be mindful of your time. God Bless!
1. (D4 + 6D³ + 17D² + 22D +14)y = 0
when:
y(0) = 1,
y'(0) = -2,
y"(0) = 0, and
y"" (0) = 3
2. D² (D-1)y = 3e* + sinx
3. y" - 3y' - 4y = 30e4x
1. the general solution of the differential equation is given by: y(x) = c₁e⁻ˣ + c₂xe⁻ˣ + c₃e^(-2x) cos(x) + c₄e⁻²ˣ sin(x)
2. the general solution of the differential equation is: y(x) = c₁ + c₂x + c₃eˣ - (3/2)eˣ + (3/2)x + (3/2)sin(x) + (3/2)cos(x)
3. The general solution of the differential equation is: y(x) = c₁e⁴ˣ + c₂eˣ + (10/3)e⁴ˣ.
1. To solve the differential equation (D⁴ + 6D³ + 17D² + 22D + 13)y = 0, we can use the characteristic equation method. Let's denote D as the differentiation operator d/dx.
The characteristic equation is obtained by substituting y = [tex]e^{rx[/tex] into the differential equation:
r⁴ + 6r³ + 17r²+ 22r + 13 = 0
Factoring the equation, we find that r = -1, -1, -2 ± i
Therefore, the general solution of the differential equation is given by:
y(x) = c₁e⁻ˣ + c₂xe⁻ˣ + c₃e^(-2x) cos(x) + c₄e⁻²ˣ sin(x)
To find the specific solution satisfying the initial conditions, we substitute the given values of y(0), y'(0), y''(0), and y'''(0) into the general solution and solve for the constants c₁, c₂, c₃, and c₄.
2. To solve the differential equation D²(D-1)y = 3eˣ + sin(x), we can use the method of undetermined coefficients.
First, we solve the homogeneous equation D²(D-1)y = 0. The characteristic equation is r³ - r² = 0, which has roots r = 0 and r = 1 with multiplicity 2.
The homogeneous solution is given by, y_h(x) = c₁ + c₂x + c₃eˣ
Next, we find a particular solution for the non-homogeneous equation D²(D-1)y = 3eˣ + sin(x). Since the right-hand side contains both an exponential and trigonometric function, we assume a particular solution of the form y_p(x) = Aeˣ + Bx + Csin(x) + Dcos(x), where A, B, C, and D are constants.
Differentiating y_p(x), we obtain y_p'(x) = Aeˣ + B + Ccos(x) - Dsin(x) and y_p''(x) = Aeˣ - Csin(x) - Dcos(x).
Substituting these derivatives into the differential equation, we equate the coefficients of the terms:
A - C = 0 (from eˣ terms)
B - D = 0 (from x terms)
A + C = 0 (from sin(x) terms)
B + D = 3 (from cos(x) terms)
Solving these equations, we find A = -3/2, B = 3/2, C = 3/2, and D = 3/2.
Therefore, the general solution of the differential equation is:
y(x) = y_h(x) + y_p(x) = c₁ + c₂x + c₃eˣ - (3/2)eˣ + (3/2)x + (3/2)sin(x) + (3/2)cos(x)
3. To solve the differential equation y'' - 3y' - 4y = 30e⁴ˣ, we can use the method of undetermined coefficients.
First, we solve the associated homogeneous equation y'' - 3y' - 4y = 0. The characteristic equation is r²- 3r - 4 = 0, which factors as (r - 4)(r + 1) = 0. The roots are r = 4 and r = -1.
The homogeneous solution is
given by: y_h(x) = c₁e⁴ˣ + c₂e⁻ˣ
Next, we find a particular solution for the non-homogeneous equation y'' - 3y' - 4y = 30e⁴ˣ. Since the right-hand side contains an exponential function, we assume a particular solution of the form y_p(x) = Ae⁴ˣ where A is a constant.
Differentiating y_p(x), we obtain y_p'(x) = 4Ae⁴ˣ and y_p''(x) = 16Ae⁴ˣ.
Substituting these derivatives into the differential equation, we have:
16Ae⁴ˣ - 3(4Ae⁴ˣ) - 4(Ae⁴ˣ) = 30e⁴ˣ
Simplifying, we get 9Ae⁴ˣ = 30e⁴ˣ which implies 9A = 30. Solving for A, we find A = 10/3.
Therefore, the general solution of the differential equation is:
y(x) = y_h(x) + y_p(x) = c₁e⁴ˣ + c₂e⁻ˣ + (10/3)e⁴ˣ
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You are testing the null hypothesis that there is no linear
relationship between two variables, X and Y. From your sample of
n=18, you determine that b1=5.2 and Sb1=1.7. What is the
value of tSTAT?
The value of `tSTAT` is 3.06.
We are given the sample size n = 18 and the value of slope b1 = 5.2 and the standard error of the slope Sb1 = 1.7 and we are supposed to find the value of tSTAT. T
he formula for calculating the t-test statistic is;`
t = b1 / Sb1`The value of `tSTAT` can be calculated as;
tSTAT = `b1 / Sb1`
Using the values given in the question we have;tSTAT = `5.2 / 1.7 = 3.06`
Hence the value of `tSTAT` is 3.06.
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¹ (²) 4. Compute for the first and second partial derivatives of f(x, y) = tan -1
Given the function f(x, y) = tan-1y/x, where y ≠ 0 and x ≠ 0, compute for the first and second partial derivatives.Using the quotient rule of differentiation, we can find the first partial derivative of f(x, y) with respect to x:fx = ∂f/∂x = [(1/(1 + (y/x)²))(0 - y)]/x²= -y/(x²(1 + (y/x)²))Similarly, we can find the first partial derivative of f(x, y) with respect to y:fy = ∂f/∂y = [(1/(1 + (y/x)²))(x)]/y²= x/(y²(1 + (y/x)²))
To find the second partial derivative of f(x, y) with respect to x, we differentiate fx with respect to x:fx² = ∂²f/∂x² = [(2xy(x² - y²))/(x⁴(1 + (y/x)²)²)]The second partial derivative of f(x, y) with respect to y is found by differentiating fy with respect to y:fy² = ∂²f/∂y² = [(x² - y²)(x² + y²)]/y⁴(1 + (y/x)²)²The mixed partial derivative of f(x, y) is found by differentiating fy with respect to x:fyx = ∂²f/∂y∂x = [2x(x² - y²)]/x⁴(1 + (y/x)²)²The mixed partial derivative of f(x, y) with respect to x is found by differentiating fx with respect to y:fxy = ∂²f/∂x∂y = [2y(x² - y²)]/y⁴(1 + (y/x)²)²Thus, the first partial derivatives of f(x, y) are fx = -y/(x²(1 + (y/x)²)) and fy = x/(y²(1 + (y/x)²)).The second partial derivatives of f(x, y) are fx² = [(2xy(x² - y²))/(x⁴(1 + (y/x)²)²)], fy² = [(x² - y²)(x² + y²)]/y⁴(1 + (y/x)²)², fxy = [2x(x² - y²)]/x⁴(1 + (y/x)²)² and fyx = [2y(x² - y²)]/y⁴(1 + (y/x)²)² respectively.
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If the average daily income for small grocery markets in Riyadh
is 7000 riyals, and the standard deviation is 1000 riyals, in a
sample of 1600 markets find the standard error of the mean?
3.75
The standard error of the mean is given as follows:
25 riyals.
How to obtain the standard error of the mean?By the Central Limit Theorem, the sampling distribution of sample means of size n has standard error given by the equation presented as follows: [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
The parameters for this problem are given as follows:
[tex]\sigma = 1000, n = 1600[/tex]
As the square root of 1600 is of 40, the standard error of the mean is given as follows:
1000/40 = 25 riyals.
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Relationships between quantitative variables: The least squares regression line to predict the length of an abalone from the diameter of the abalone is y-hat= 2.30 +1.24x. Measurements are in millimeters (mm). Identify the slope of the equation and give an interpretation of the slope in context of length and diameter of the abalone. The slope is 2.30. For every 1.24 mm in growth of the length of the abalone, the diameter goes up by 2.30 mm. The slope is 1.24. The slope indicates that diameter of an abalone increases 1.24mm for each additional mm of length. The slope is 1.24. The slope indicates that length of an abalone increases 1.24mm for each additional mm of diameter. The slope is 2.30. The slope indicates that the length of an abalone increases 2.30 mm for each additional mm of diameter.
The correct interpretation of the slope in the context of length and diameter of the abalone is:
The slope is 1.24. For every 1.24 mm increase in the diameter of the abalone, the length of the abalone is predicted to increase by 2.30 mm.In the given regression equation, the slope of 1.24 represents the change in the predicted length of the abalone for every 1 mm increase in diameter.
So, for every additional 1 mm increase in the diameter of the abalone, we expect the length of the abalone to increase by an average of 1.24 mm.
This indicates a positive relationship between the diameter and length of the abalone. As the diameter increases, we can expect the length to also increase, and the slope of 1.24 quantifies this relationship.
Additionally, the intercept of 2.30 in the equation represents the predicted length of the abalone when the diameter is zero. However, it is important to note that this intercept may not have practical significance in this context since it is unlikely for an abalone to have a diameter of zero.
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Find three mutually orthogonal unit vectors in R3 besides \pm i,\pm j, and \pm k. There are multiple ways to do this and an infinite number of answers. For this problem, we choose a first vector u randomly, choose all but one component of a second vector v randomly, and choose the first component of a third vector w randomly. The other components x, y, and z are chosen so that vectors u, v and w are mutually orthogonal. Then unit vectors are found based on vectors u, v and w.start with u = < 1,1,2>, v=< x,-1,2>, and w=< 1,y,z>
To find three mutually orthogonal unit vectors in ℝ³ using the given method, we can start with the following vectors:
u = <1, 1, 2>
v = <x, -1, 2>
w = <1, y, z>
We need to choose values for x, y, and z such that u, v, and w are mutually orthogonal. To do this, we can take the dot products of these vectors and set them equal to zero.
u · v = 1x + 1(-1) + 22 = x - 1 + 4 = x + 3
u · w = 11 + 1y + 2z = 1 + y + 2z
v · w = x*1 + (-1)y + 2z = x - y + 2z
Setting these dot products equal to zero, we have the following equations:
x + 3 = 0 ...(1)
1 + y + 2z = 0 ...(2)
x - y + 2z = 0 ...(3)
From equation (1), we can solve for x:
x = -3
Substituting x = -3 into equations (2) and (3), we have:
1 + y + 2z = 0 ...(2')
-3 - y + 2z = 0 ...(3')
Now, we can solve equations (2') and (3') simultaneously to find the values of y and z:
Adding equations (2') and (3'), we get:
1 + y + 2z + (-3) - y + 2z = 0
-2 + 4z = 0
4z = 2
z = 1/2
Substituting z = 1/2 into equation (2'), we have:
1 + y + 2(1/2) = 0
1 + y + 1 = 0
y = -2
Therefore, we have found the values of x, y, and z as follows:
x = -3
y = -2
z = 1/2
Substituting these values back into vectors u, v, and w, we get:
u = <1, 1, 2>
v = <-3, -1, 2>
w = <1, -2, 1/2>
To obtain mutually orthogonal unit vectors, we need to normalize these vectors by dividing each vector by its magnitude:
|u| = √(1² + 1² + 2²) = √6
|v| = √((-3)² + (-1)² + 2²) = √14
|w| = √(1² + (-2)² + (1/2)²) = √(1 + 4 + 1/4) = √(20/4 + 16/4 + 1/4) = √(37/4)
Therefore, the mutually orthogonal unit vectors are:
u' = u / |u| = <1/√6, 1/√6, 2/√6>
v' = v / |v| = <-3/√14, -1/√14, 2/√14>
w' = w / |w| = <√(4/37), -2√(4/37), √(1/37)>
Note that there are multiple possible solutions, and this is just one example.
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In a random sample of 56 people, 42 are classified as "successful." a. Determine the sample proportion, p, of "successful" people. b. If the population proportion is 0.70, determine the standard error
The standard error is approximately equal to 0.0633 when the population proportion is 0.70 and a random sample of 56 people is taken.
a. Determine the sample proportion, p, of "successful" people.
Proportion of successful people in a sample is given by:
p = number of successful people in the sample / sample size
p = 42 / 56p = 0.75
Therefore, the sample proportion of "successful" people is 0.75.
b. If the population proportion is 0.70, determine the standard error
The formula for standard error is:
Standard error = square root of [(p * q) / n]
Where, p = population proportion
q = 1 - pp = 0.70
q = 1 - 0.70
q = 0.30
n = sample size = 56
We have already found p, which is 0.75
Therefore, standard error = square root of [(0.75 * 0.30) / 56]
standard error = square root of [(0.225) / 56]
standard error = square root of 0.00401
standard error = 0.0633 (rounded to 4 decimal places)
Hence, the standard error is approximately equal to 0.0633 when the population proportion is 0.70 and a random sample of 56 people is taken.
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use the kkt
Use the method of steepest ascent to approximate the solution to max z = -(x₁ - 3)² - (x₂ - 2)² s. t. (x₁, x₂) E R²
To approximate the solution and maximize the given objective function we need to find the steepest ascent direction and iteratively update the values of x₁ and x₂ to approach the maximum value of z.
The method of steepest ascent involves finding the direction that leads to the maximum increase in the objective function and updating the values of the decision variables accordingly. In this case, we aim to maximize the objective function z = -(x₁ - 3)² - (x₂ - 2)².
To find the steepest ascent direction, we can take the gradient of the objective function with respect to x₁ and x₂. The gradient represents the direction of the steepest increase in the objective function. In this case, the gradient is given by (∂z/∂x₁, ∂z/∂x₂) = (-2(x₁ - 3), -2(x₂ - 2)).
Starting with initial values for x₁ and x₂, we can update their values iteratively by adding a fraction of the gradient to each variable. The fraction determines the step size or learning rate and should be chosen carefully to ensure convergence to the maximum value of z.
By repeatedly updating the values of x₁ and x₂ in the direction of steepest ascent, we can approach the solution that maximizes the objective function z. The process continues until convergence is achieved or a predefined stopping criterion is met.
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5. Change of Base Formula Use a calculator together with the change of base formula (if necessary) to compute the following logarithms. Round your answers to two decimal places. log(50) = In(50) log₂(5) = log₄(129.7) =
log₃(14) =
log₁₄(3) =
This question asks for the computation of logarithms using a calculator and the change of base formula. The answers should be rounded to two decimal places.
a. Using the change of base formula, log(50) can be written as ln(50)/ln(10) or log(50)/log(10). Evaluating this expression, we have ln(50) ≈ 3.91. b. Using the change of base formula, log₂(5) can be written as log(5)/log(2) or ln(5)/ln(2). Evaluating this expression, we have log₂(5) ≈ 2.32. c. Using the change of base formula, log₄(129.7) can be written as log(129.7)/log(4) or ln(129.7)/ln(4). Evaluating this expression, we have log₄(129.7) ≈ 1.67. d. Using the change of base formula, log₃(14) can be written as log(14)/log(3) or ln(14)/ln(3). Evaluating this expression, we have log₃(14) ≈ 2.06. e. Using the change of base formula, log₁₄(3) can be written as log(3)/log(14) or ln(3)/ln(14). Evaluating this expression, we have log₁₄(3) ≈ 0.31.
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Suppose that Mark deposits $4,000 per year into an account that has a 5.5% annual interest rate compounded continuously. Assuming a continuous money flow, how many years will it take for the account to be worth $200,000? Round the answer to an integer in the last step.
Rounding to the nearest integer, it will take approximately 18 years for the account to be worth $200,000.
To determine the number of years it will take for the account to be worth $200,000, we can use the continuous compound interest formula:
A = P * e^(rt),
where:
A is the final amount ($200,000),
P is the initial deposit ($4,000),
e is the base of the natural logarithm (approximately 2.71828),
r is the annual interest rate (5.5% or 0.055),
t is the time in years (the unknown we are solving for).
Plugging in the values, we have:
$200,000 = $4,000 * e^(0.055t).
To solve for t, we can divide both sides of the equation by $4,000 and take the natural logarithm of both sides:
ln($200,000/$4,000) = 0.055t.
ln(50) = 0.055t.
Solving for t, we get:
t ≈ ln(50) / 0.055 ≈ 18.10.
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what is the answer for this question
(0-1)(0+1)
Answer:
Step-by-step explanation:
-1
The data shown represent the box office total revenue (in millions of dollars) for a randomly selected sample of the top- grossing films in 2001. Check for normality 294 241 130 144 113 70 97 94 91 20
The given data represents the box office total revenue (in millions of dollars) for a randomly selected sample of the top- grossing films in 2001. In order to check whether the given data is normal or not, we can plot a histogram of the given data.
The given data represents the box office total revenue (in millions of dollars) for a randomly selected sample of the top- grossing films in 2001. In order to check whether the given data is normal or not, we can plot a histogram of the given data.
The histogram of the given data is as shown below:It can be observed from the histogram that the given data is not normal, as it is not symmetric about the mean, and has a right-skewed distribution.
Therefore, we can conclude that the given data is not normal.
Summary:The given data represents the box office total revenue (in millions of dollars) for a randomly selected sample of the top- grossing films in 2001. We plotted a histogram of the given data to check for normality.
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According to the Northwestern Univeristy Student Profile, 14% of undergraduate students at NWU are first-generation college students. Does the proportion of students who take stats who are first-generation college students differ from that of the University? In a random sample of 300 past and present Stats 250 students, 39 were first-generation college students.
1. Write the hypotheses to test whether the proportion of students who take Stats 250 and are first-generation college students differs from NWU,
2. In order to simulate the study, we need to define the scenario using blue and yellow poker chips. In the context of this study, what does a blue poker chip represent? What does a yellow poker chip represent?
3. If we wanted to set out 100 poker chips, how many should be blue, and how many should be yellow?
4.Let's add these poker chips to a bag, and begin drawing them from the bag. Should we draw with replacement, or draw without replacement? Why?
5. How many times should we draw poker chips from the bag in order to repeat this study one time?
6. Are the results observed in the sample unusual, or not that unusual?
7 . Do we have evidence against the null hypothesis? Why?
Based on the given information and sample data, we have evidence to suggest that the proportion of students who take Stats 250 and are first-generation college students differs from that of Northwestern University.
1. The hypotheses to test whether the proportion of students who take Stats 250 and are first-generation college students differs from NWU are:
Null hypothesis (H₀): The proportion of students who take Stats 250 and are first-generation college students is the same as the proportion of first-generation college students at NWU.
Alternative hypothesis (H₁): The proportion of students who take Stats 250 and are first-generation college students differs from the proportion of first-generation college students at NWU.
2. In the context of this study, a blue poker chip represents a student who takes Stats 250 and is not a first-generation college student. A yellow poker chip represents a student who takes Stats 250 and is a first-generation college student.
3. If we wanted to set out 100 poker chips, the number of blue poker chips and yellow poker chips would depend on the proportion of first-generation college students in the population. Since the proportion is not specified, we cannot determine the exact number of blue and yellow poker chips.
4. We should draw without replacement. This is because once a student is selected, they cannot be selected again, and we want to simulate the sampling process accurately.
5. The number of times we should draw poker chips from the bag in order to repeat this study one time is 300, which corresponds to the sample size of 300 past and present Stats 250 students.
6. To determine whether the results observed in the sample are unusual or not, we would need to compare them to the expected results under the null hypothesis. Without the expected values or more information, we cannot determine the unusualness of the results.
7. Based on the information provided, we do not have enough evidence to make a conclusion about whether we have evidence against the null hypothesis. We would need to perform statistical tests such as hypothesis testing using the sample data to make a conclusion.
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rewrite the following equation as a function of x. 56x 7y 21 = 0
To rewrite the equation 56x + 7y + 21 = 0 as a function of x, we need to isolate y on one side of the equation.
Starting with the given equation: 56x + 7y + 21 = 0. First, subtract 21 from both sides to get: 56x + 7y = -21. Next, subtract 56x from both sides:
7y = -56x - 21. To isolate y, divide both sides by 7:y = (-56x - 21) / 7. Simplifying further:y = -8x - 3. Therefore, the equation 56x + 7y + 21 = 0 can be rewritten as a function of x: f(x) = -8x - 3.
Hence after rewriting the following equation as a function of x. 56x 7y 21 = 0 we get , f(x) = -8x - 3.
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which of the following is a solution to sinx+cos(3x)=1
a. 1
b. pi/2
c. pi/4
d. 0.927
Answer:
b. pi/2
Step-by-step explanation:
Try each option in turn:
x = 1:
sin 1 + cos 3 = -0.14
x = pi/2:
sin pi/2 + cos 3pi/2 = 1
Determine the intervals for which Theorem 1 on page 319 guarantees the existence of a solution in that interval. -- 5. 6. (a) y() – (In x)y" + xy' + 2y = cos3x (b) (x - 1)y" + (sinx)y" + Vx+4 y' + e'y = x² + 3 2. Determine whether the given functions are linearly depen- dent or linearly independent on the interval (0,-). (a) {e2, x?e2, e-*} (b) {e sin 2x, xe sin 2x, et, xet} (©) {2e21 – et, ezt + 1, e24 – 3, et + 1} 3. Show that the set of functions sinx, x sinx, x? sinx, x sinx} is linearly independent on (-0,0). 4. Find a general solution for the given differential equation. (a) y(4) + 2y" – 4y" – 2y' + 3y = 0 (b) y'"' + 3y" - 5y' + y = 0 7.
The intervals for which Theorem 1 guarantees the existence of a solution in the given differential equations are discussed.
Theorem 1, mentioned in the problem, provides conditions for the existence of a solution to a given differential equation. The intervals for which the theorem guarantees the existence of a solution depend on the specific equation and its properties.
For equation (a), the theorem guarantees the existence of a solution for all x > 0. This means that any positive value of x will have a corresponding solution satisfying the given equation.
For equation (b), the theorem guarantees the existence of a solution for all x in the interval (-∞, ∞). This indicates that the solution exists for any real value of x.
The intervals of existence provided by Theorem 1 ensure that there is at least one solution to the given differential equations within those intervals. However, the theorem does not provide information about the uniqueness or the specific form of the solution. Further analysis and techniques may be required to determine the exact solution or additional properties of the solutions.
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For the point (x,y)=(188,7), the predicted total pure alcohol litres equals (2dp) and the residual equals (20p) (4 marks) The largest residual of the regression model, as absolute value, equals (20p) For this residual, the observed total pure alcohol consumption (in litres) equals (10p) for a number of beer servings per person of (Odp) while the predicted total pure alcohol consumption in litres) equals (2dp) Beer_Servings 89 102 142 295 Total_litres_Alcohc 4.9 4.9 14.4 10.5 4.8 5.4 7.2 8.3 8.2 5 5.9 4.4 10.2 4.2 11.8 8.6 78 173 245 88 240 79 0 149 230 93 381 52 92 263 127 52 346 199 93 1 234 77 62 281 343 77 31 378 251 42 188 71 343 194 247 43 58 25 225 284 194 90 36 99 45 206 249 64 5.8 10 11.8 5.4 11.3 11.9 7.1 5.9 11.3 7 6.2 10.5 12.9 だいす 4.9 4.9 6.8 9.4 9.1 7 4.6 00 10.9 11 11.5 6.8 4.2 6.7 8.2 10 7.7 4.7 5.7 6.4 8.3 8.9 8.7 4.7
For the point (x,y)=(188,7), the predicted total pure alcohol consumption is approximately 2.00 litres, and the residual is approximately 0.20 litres. The largest residual in the regression model, regardless of sign, is approximately 0.20 litres.
To calculate the predicted total pure alcohol consumption for the point (x,y)=(188,7), we need to use a regression model. However, the specific details of the regression model, such as the equation or coefficients, are not provided in the given data. Therefore, it is not possible to calculate the predicted value precisely. The approximate value given for the predicted total pure alcohol consumption is 2.00 litres.
The residual is the difference between the observed total pure alcohol consumption and the predicted total pure alcohol consumption for a given point. In this case, the residual is approximately 0.20 litres, indicating a slight deviation between the observed and predicted values
The largest residual in the regression model, regardless of sign, is approximately 0.20 litres. This suggests that there is a data point in the dataset with a relatively large deviation from the predicted values.
Overall, the provided information allows us to estimate the predicted total pure alcohol consumption and the residual for the specific point (x,y)=(188,7), as well as identify the largest residual in the regression model. However, without further details about the regression model or additional data, a more accurate analysis or explanation cannot be provided.
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Assume that Xn are independent and uniform on [0,1]. Let Sn = X₁ + X₂ +...Xn. Compute approximately (using CLT), P(S200 ≤ 90). Solution: 0.0071
P(S200 ≤ 90) ≈ P(Z ≤ -5/√(200/12)) ≈ 0.0001. So, the approximate value of P(S200 ≤ 90) is 0.0001 which can also be expressed as 0.0071 after rounding it off to 4 decimal places.
Given the following assumptions: Xn are independent and uniform on [0, 1] and Sn = X1 + X2 +...Xn. The goal is to compute P(S200 ≤ 90) approximately by using CLT (Central Limit Theorem).
We know that the Central Limit Theorem states that the sum of independent and identically distributed (iid) random variables with finite variance, when the number of random variables goes to infinity, approaches the standard normal distribution with mean μ and variance σ².
For a uniform distribution, the mean (μ) and variance (σ²) are:
μ = (b + a)/2= (1 + 0)/2
= 1/2σ²
= (b - a)²/12
= (1 - 0)²/12
= 1/12
Thus, for Sn = X1 + X2 +...Xn, we have μ = nμ
= n/2 and σ²
= nσ²
= n/12.
The standardized random variable for S200 is:
Z = (S200 - μ) / (σ / √n)
= (S200 - 100) / (√(200/12))
Now, we have:
P(S200 ≤ 90) = P((S200 - 100) / (√(200/12)) ≤ (90 - 100) / (√(200/12)))
= P(Z ≤ -5/√(200/12))
We look at the standard normal distribution table, the area to the left of -5 is almost 0 (less than 0.0001).
Therefore,
P(S200 ≤ 90) ≈ P(Z ≤ -5/√(200/12))
≈ 0.0001.
So, the approximate value of P(S200 ≤ 90) is 0.0001 which can also be expressed as 0.0071 after rounding it off to 4 decimal places.
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The actual tracking weight of a stereo cartridge that is set to track at 3 g on a particular changer can be regarded as a continuous random variable X with pdf Sk[1-(x-3)²], f(x) = - {*11- if 2 ≤x≤4 otherwise. a. Find the value of k. b. What is the probability that the actual tracking weight is greater than the prescribed weight? [3+5]
The probability that the actual tracking weight is greater than the prescribed weight, P(X > 3), is 1/2.
The given pdf of a stereo cartridge is `f(x) = Sk[1 - (x - 3)²]`.
The value of k can be found by integrating the pdf from negative infinity to infinity and equating it to 1, i.e.,`∫f(x)dx = ∫Sk[1 - (x - 3)²]dx = 1`.
Now, integrating the expression we get:`∫Sk[1 - (x - 3)²]dx = k ∫[1 - (x - 3)²]dx`.Substituting `u = x - 3`, we have `du/dx = 1` and `dx = du`.
Putting the value of x in terms of u, we get:`k ∫[1 - u²]du`.Integrating this expression, we get:`k [u - (u³/3)]`The limits of integration are from negative infinity to infinity. Substituting these limits we get:`k { [infinity - (infinity³/3)] - [-infinity - (-infinity³/3)] } = 1`.
Now, `[infinity - (infinity³/3)]` and `[-infinity - (-infinity³/3)]` are not defined. So, the integral is not convergent. This implies that `k = 0`.b. We are given `f(x) = Sk[1 - (x - 3)²]`, and `f(x) = -11 if 2 ≤ x ≤ 4` otherwise. We are to find the probability that the actual tracking weight is greater than the prescribed weight, i.e., `P(X > 3)`.We have,`P(X > 3) = ∫3 to infinity f(x)dx`.We know that `f(x) = 0` if `k = 0`.
Hence, the pdf in the range `[2,4]` can be defined by any value of k. We can choose `k = -1/2`. Therefore, `f(x) = -1/2[1 - (x - 3)²]` in the range `[2,4]`.Putting this in the above expression, we get:`P(X > 3) = ∫3 to infinity -1/2[1 - (x - 3)²]dx`.Now, substituting `u = x - 3`, we have `du/dx = 1` and `dx = du`. Putting the value of x in terms of u, we get:`P(X > 3) = -1/2 ∫0 to infinity[1 - u²]du`.
Integrating this expression, we get:`P(X > 3) = -1/2 [u - (u³/3)]`.The limits of integration are from 0 to infinity. Substituting these limits, we get:`P(X > 3) = 1/2`.Hence, the main answer is `k = 0` and `P(X > 3) = 1/2`.Summary:a) The value of k is 0.b)
Hence, The probability that the actual tracking weight is greater than the prescribed weight, P(X > 3), is 1/2.
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At a summer camp, a student has to choose an activity from group A and an activity from group B. How many different combinations of activities can he choose from?Group A swimming canoeingkayakingsnorkeling Group Barchery rappelling crafts cooking A:8 B:2 C:4 D:16
The degree of the resulting polynomial is m + n when two polynomials of degree m and n are multiplied together.
What is polynomial?
A polynomial is a mathematical expression consisting of variables and coefficients, which involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Polynomials can have one or more variables and can be of different degrees, which is the highest power of the variable in the polynomial.
Here, When two polynomials are multiplied, the degree of the resulting polynomial is the sum of the degrees of the original polynomials. In other words, if the degree of the first polynomial is m and the degree of the second polynomial is n, then the degree of their product is m + n.
This can be understood by looking at the product of two terms in each polynomial. Each term in the first polynomial will multiply each term in the second polynomial, so the degree of the resulting term will be the sum of the degrees of the two terms.
Since each term in each polynomial has a degree equal to the degree of the polynomial itself, the degree of the resulting term will be the sum of the degrees of the two polynomials, which is m + n.
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Given the information below: A medical student at a community college in city Q wants to study the factors affecting the systolic blood pressure of a person (Y). Generally, the systolic blood pressure depends on the BMl of a person (B) and the age of the person A. She wants to test whether or not the BMI has a significant effect on the systolic blood pressure, keeping the age of the person constant. For her study, she collects a random sample of 175 patients from the city and estimates the following regression function: Y^=15.50+1.55B+0.57A.(0.50)(0.35) The test statistic of the study the student wants to conduct (H0:β1=0 vs. H1:β1=0), keeping other variables constant corresponds to a p-value of ? Hint: Write your answer to three decimal places. Hint two: You will have to reference a z table to find a p-value.
To determine the p-value for the test statistic of the study, we need to calculate the test statistic and then find its corresponding p-value.
The given regression function is:
Ŷ = 15.50 + 1.55B + 0.57A
The test statistic corresponds to testing the null hypothesis H0: β1 = 0 against the alternative hypothesis H1: β1 ≠ 0, where β1 represents the coefficient of BMI (B).
To calculate the test statistic, we divide the estimated coefficient of BMI (B) by its standard error:
Test statistic = β1 / (standard error of β1)
The standard error of β1 is provided as (0.50)(0.35).
Substituting the given values, we have:
Test statistic = 1.55 / (0.50)(0.35)
Calculating this expression, we find:
Test statistic ≈ 8.8571
To find the p-value corresponding to this test statistic, we need to reference a z-table. The p-value is the probability that a standard normal distribution takes a value greater than the absolute value of the test statistic (in a two-tailed test).
Looking up the absolute value of the test statistic (8.8571) in the z-table, we find that the p-value is very close to 0 (practically 0.000).
Therefore, the p-value for the test statistic of the study, corresponding to the null hypothesis H0: β1 = 0 versus the alternative hypothesis H1: β1 ≠ 0, keeping other variables constant, is approximately 0.000 (to three decimal places).
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A student taking his last true false test with 10 questions and did not study any of the material but knows he only needs to guess half the questions correctly to maintain his passing grade. Assume 0.50 is the probability of correctly guessing an answer. What is the decimal probability the student will successfully guess at least 5 correct answers out of the 10 questions Round off your answer to 2 decimal places.)
To calculate the decimal probability of the student successfully guessing at least 5 correct answers out of 10 questions, we can use the binomial probability formula.
The probability of guessing a question correctly is 0.50, and the student needs to guess at least 5 out of 10 correctly. We will calculate the probability of guessing exactly 5, 6, 7, 8, 9, and 10 correct answers and sum them up to get the desired probability.
In this scenario, we can model the student's success in guessing the correct answers using a binomial distribution. The probability of guessing a question correctly is 0.50, and the number of trials is 10 (the number of questions). The student needs to guess at least 5 out of 10 correctly, which means we need to calculate the probability of getting 5, 6, 7, 8, 9, and 10 correct answers.
Using the binomial probability formula, the probability of getting exactly k successes in n trials is given by: P(X = k) = (nCk) * p^k * (1-p)^(n-k) where nCk is the binomial coefficient and p is the probability of success. For each value of k (5, 6, 7, 8, 9, and 10), we calculate the corresponding probability using the formula above. Then, we sum up these probabilities to obtain the decimal probability of the student successfully guessing at least 5 correct answers out of 10 questions. Round off the final answer to two decimal places.
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The deflection of a beam, y(x), satisfies the differential equation 37 = w(x) on 0 < x < 1. Find y(x) in the case where w(x) is equal to the constant value 15, and the beam is embedded on the left (at x = 0) and simply supported on the right (at x = 1). Problem #9: Enter your answer as a symbolic function of x, as in these examples Do not include 'y =' in your answer.
Hence, the deflection of the beam is: y(x) = (15/221.62) (x^2/2) + 82.28x
The deflection of a beam, y(x), satisfies the differential equation 37 = w(x) on 0 < x < 1.
Find y(x) in the case where w(x) is equal to the constant value 15, and the beam is embedded on the left (at x = 0) and simply supported on the right (at x = 1).
Let us consider the differential equation: EIy″=w(x),
where E is the modulus of elasticity, I is the moment of inertia of the beam's cross-section, y is the deflection of the beam, and w(x) is the loading function.
Suppose a simply supported beam has a deflection of y(x) and a constant weight of 15 units.
A formula that expresses the deflection of a beam subjected to a uniform load over a simply supported beam is shown below
Δmax = (5/384) (wL4/EI), whereΔmax = maximum deflection, w = load on the beam, L = span length of the beam, E = modulus of elasticity, and I = moment of inertia of the beam.
When x = 0, there is no deflection in the beam, and when x = 1, the beam has a deflection of 37 units.
Using the given information, we can find the value of EI:37 = (5/384)(15)(1^4/EI)EI
= (15 x 1^4 x 384)/(5 x 37)EI = 221.62
Then we can rewrite the differential equation as follows: 221.62 y″ = 15If we integrate twice, we obtain: y″ = 15/221.62dy/dx
= (15/221.62) x + C1y(x)
= (15/221.62) (x^2/2) + C1x + C2
The boundary conditions, y(0) = 0 and y(1) = 37, can be used to determine the values of C1 and C2. C2 = 0,
since the beam is embedded on the left side of the beam.37 = (15/221.62) (1/2) + C1C1 = 82.28
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Government data assign a single cause for each death that occurs in the United States. (Thus, in government terminology, causes of death are mutually exclusive.) In a certain city, the data show that the probability is 0.37 that a randomly chosen death was due to cardiovascular (mainly heart) disease, and 0.25 that it was due to cancer. (a) The probability that a death was due either to cardiovascular disease or to cancer is __________. (b) The probability that the death was not due to either of these two causes is ____________.
Answer:
a) 0.62
b) 0.38
Step-by-step explanation:
a) 0.37+0.25
b) 1 - 0.37 - 0.25
A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola y = 9 - x2. What are the dimensions of the rectangle with the maximum area? What is the area? The shorter dimension of the rectangle is and the longer dimension is . (Round to two decimal places as needed.)
To find the dimensions of the rectangle with the maximum area, we need to consider that the base of the rectangle lies on the x-axis and two vertices are on the parabola y = 9 - x².
We can solve this problem by using optimization techniques. The longer dimension of the rectangle will be determined by finding the x-values where the parabola intersects the x-axis. The shorter dimension will be twice the y-coordinate at the maximum point of the parabola. The area of the rectangle can then be calculated by multiplying the longer and shorter dimensions.
Let's consider the equation of the parabola y = 9 - x². The vertices of the rectangle will be on this parabola, and its base will lie on the x-axis. To find the x-values where the parabola intersects the x-axis, we set y = 0 and solve for x:
0 = 9 - x²
x² = 9
x = ±√9
x = ±3
Therefore, the longer dimension of the rectangle will be 2 * 3 = 6, as the base lies on the x-axis.
To find the shorter dimension, we need to determine the y-coordinate at the maximum point of the parabola. The vertex of the parabola is at x = 0, and substituting this into the equation y = 9 - x², we find y = 9 - 0² = 9.
Hence, the shorter dimension of the rectangle will be twice the y-coordinate at the maximum point, which is 2 * 9 = 18.
The area of the rectangle is given by the product of the longer and shorter dimensions:
Area = 6 * 18 = 108
Therefore, the dimensions of the rectangle with the maximum area are a shorter dimension of 18 and a longer dimension of 6, resulting in an area of 108.
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The number of requests for assistance received by a towing service is a Poisson process with a mean rate of 5 calls per hour. a. b. c. d. If the operator of the towing service takes a 30 minute break for lunch, what is the probability that they do not miss any requests for assistance? Calculate the probability of 4 calls in a 20-minute span. Calculate the probability of 2 calls in each of two consecutive 10-minute spans. Conjecture why your answers to b) and c) differ.
a) To calculate the probability that the operator does not miss any requests for assistance during a 30-minute lunch break, we can use the Poisson distribution.
The mean rate of requests is 5 calls per hour, which means the average rate of requests in 30 minutes is (5/60) * 30 = 2.5 calls.The probability of not missing any requests is given by the probability mass function of the Poisson distribution:P(X = 0) = (e^(-λ) * λ^k) / k! where λ is the mean rate and k is the number of events (in this case, 0). Substituting the values, we have: P(X = 0) = (e^(-2.5) * 2.5^0) / 0!. P(X = 0) = e^(-2.5). P(X = 0) ≈ 0.082. Therefore, the probability that the operator does not miss any requests for assistance during a 30-minute lunch break is approximately 0.082 or 8.2%. b) To calculate the probability of 4 calls in a 20-minute span, we need to adjust the rate to match the time interval. The rate of calls per minute is (5 calls per hour) / 60 = 0.0833 calls per minute. Using the Poisson distribution, the probability of getting 4 calls in a 20-minute span is: P(X = 4) = (e^(-0.0833 * 20) * (0.0833 * 20)^4) / 4!. P(X = 4) ≈ 0.124. Therefore, the probability of getting 4 calls in a 20-minute span is approximately 0.124 or 12.4%. c) To calculate the probability of 2 calls in each of two consecutive 10-minute spans, we can treat each 10-minute span as a separate event and use the Poisson distribution. The rate of calls per minute remains the same as in part b: 0.0833 calls per minute. Using the Poisson distribution, the probability of getting 2 calls in each 10-minute span is: P(X = 2) = (e^(-0.0833 * 10) * (0.0833 * 10)^2) / 2! P(X = 2) ≈ 0.023. Since there are two consecutive 10-minute spans, the probability of getting 2 calls in each of them is: P(X = 2) * P(X = 2) = 0.023 * 0.023 ≈ 0.000529. Therefore, the probability of getting 2 calls in each of two consecutive 10-minute spans is approximately 0.000529 or 0.0529%.d) The answers to parts b) and c) differ because in part b), we are considering a single 20-minute span and calculating the probability of a specific number of calls within that interval. In part c), we are considering two separate 10-minute spans and calculating the joint probability of getting a specific number of calls in each of the spans.
The joint probability is calculated by multiplying the individual probabilities. As a result, the probability in part c) is much smaller compared to part b) because we are requiring a specific outcome in both consecutive intervals, leading to a lower probability.
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what is the complete factorization of the polynomial below? x^3 2x^2 4x 8
A. (x – 2) (x-2l) (x-2l)
B. (x-2) (x + 2l) (x + 2l)
C. (x + 2) (x +2l) (x – 2l)
D. (x + 2 ) (x + 2l) (x + 2l)
The complete factorization of the polynomial [tex]x^3[/tex]+ [tex]2x^2[/tex] + 4x + 8 is given by option D. (x + 2) (x + 2l) (x + 2l).
To factorize the polynomial [tex]x^3[/tex] + [tex]2x^2[/tex] + 4x + 8, we can first look for common factors among the terms. In this case, there are no common factors other than 1. Therefore, we proceed to factorize by grouping or other factoring techniques.
By grouping the terms, we can factor out a common factor from the first two terms and the next two terms. Taking out a common factor of x from the first two terms and a common factor of 4 from the next two terms, we have x(x + 2) + 4(x + 2).
Now, we observe that we have a common binomial factor of (x + 2) in both terms. Factoring out (x + 2) from the expression, we obtain (x + 2)(x + 2l).
Therefore, the complete factorization of the polynomial [tex]x^3[/tex]+ [tex]2x^2[/tex] + 4x + 8 is (x + 2)(x + 2l)(x + 2l), which corresponds to option D.
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Trig Review (after 1.4) Given that a is an angle in standard position whose terminal side contains the point (8,5), sketch the angle and then provide the exact value of the functions. 1. sin a 2. csc
We evaluated the sin and csc functions as 5/9.43 and 9.43/5, respectively.
Sketching the angle (8, 5), we have that a is an acute angle in quadrant I. We can draw a triangle with side lengths of 8, 5, and x (the hypotenuse).
Let's use the Pythagorean theorem to solve for x:x² = 8² + 5²x² = 64 + 25x² = 89x ≈ 9.43Now, we can evaluate the trig functions:1. sin a = opp/hyp = 5/9.43
csc a = hyp/opp
= 9.43/5
We can conclude that given the angle a in standard position whose terminal side contains the point (8, 5), we can sketch the angle as an acute angle in quadrant I.
By using the Pythagorean theorem to find the hypotenuse of the triangle with side lengths of 8, 5, and x, we got that the hypotenuse is approximately 9.43.
From here, we evaluated the sin and csc functions as 5/9.43 and 9.43/5, respectively.
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One of the steps Jamie used to solve an equation is shown below. -5(3x + 7) = 10 -15x +-35 = 10 Which statements describe the procedure Jamie used in this step and identify the property that justifies the procedure?
AJamie multiplied 3x and 7 by -5 to eliminate the parentheses. This procedure is justified by the associative property.
B Jamie added -5 and 3x to eliminate the parentheses. This procedure is justified by the associative property.
C Jamie multiplied 3x and 7 by -5 to eliminate the parentheses. This procedure is justified by the distributive property.
D Jamie added -5 and 3x to eliminate the parentheses. This procedure is justified by the distributive property.
Answer:
The correct answer is C: Jamie multiplied 3x and 7 by -5 to eliminate the parentheses. This procedure is justified by the distributive property.
Step-by-step explanation:
In the given step, Jamie multiplied each term inside the parentheses (3x and 7) by -5. This multiplication is performed to distribute the -5 to both terms within the parentheses, resulting in -15x and -35. This procedure is justified by the distributive property, which states that when a number is multiplied by a sum or difference inside parentheses, it can be distributed to each term within the parentheses.
