The integral ∫x²sin(x²) dx cannot be easily solved using substitution or integration by parts. It can be approximated using a midpoint approximation or a Maclaurin series with three terms, and the Alternating Series Estimation Theorem can be used to estimate the error in the series approximation.
(a) Substitution is ineffective for this integral because there is no clear choice for a suitable substitution that simplifies the expression. Integration by parts also fails as it would require differentiating x² and integrating sin(x²), resulting in a similarly complex integral. Therefore, these standard integration techniques do not offer straightforward solutions.
(b) To approximate the integral using a midpoint approximation, we can divide the interval [0, x] into subintervals. With n = 4, the interval is divided into four equal subintervals: [0, 1], [1, 2], [2, 3], and [3, 4]. Within each subinterval, we evaluate the function at the midpoint and multiply it by the width of the subinterval. The sum of these products provides an approximation to the integral.
(c) Using a Maclaurin series, we expand sin(x²) as a power series centered at 0. Taking three terms of the series, we have sin(x²) ≈ x² - (x²)³/3! + (x²)⁵/5!. We substitute this approximation into the integral x² sin(x²) dx and integrate each term separately. This results in an estimate of the integral.
To predict the error in the Maclaurin series approximation, we can apply the Alternating Series Estimation Theorem. Since the alternating series converges for all x, the error is bounded by the absolute value of the next term in the series. By calculating the value of the fourth term, we can determine the maximum possible error in our estimation.
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The body temperatures in degrees Fahrenheit of a sample of adults in one small town are:
96.8 96.9 99.1 97.7 97.5 98.4 96.6
Assume body temperatures of adults are normally distributed. Based on this data, find the 99% confidence Interval of the mean body temperature of adults in the town. Enter your answer as an open-interval i.e., parentheses) accurate to 3 decimal places. Assume the data is from a normally distributed population.
99%C.I= _______________
Based on the given data and assuming a normal distribution of body temperatures, the 99% confidence interval for the mean body temperature of adults in the town is (96.169, 99.131) degrees Fahrenheit.
To calculate the 99% confidence interval for the mean body temperature, we need to estimate the population mean and the standard deviation. Given the sample data and assuming a normal distribution, we can use the formula for the confidence interval.
After performing the necessary calculations, we find that the lower limit of the confidence interval is 96.169 and the upper limit is 99.131. This means that we are 99% confident that the true mean body temperature of adults in the town falls within this range.
The confidence interval provides us with a range of values within which we can reasonably estimate the population mean. In this case, the interval suggests that the true mean body temperature of adults in the town is likely to be between 96.169 and 99.131 degrees Fahrenheit.
It's important to note that the confidence interval is an estimation and there is still some uncertainty involved. However, with a 99% confidence level, we can be quite confident in the accuracy of this interval.
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Find the volume of the Triangular Pyramid given below
The volume of the triangular prism is 12 in³
What is volume of a prism?Volume is defined as the space occupied within the boundaries of an object in three-dimensional space.
Prism is a three-dimensional solid object in which the two ends are identical.
The volume of the prism is expressed as;
V = base area × height
where v is the volume.
Base area = 1/2bh
= 1/2 × 3 × 2
= 3 in²
The height of the prism = 4in
Therefore the volume of the prism
= 3 × 4
= 12in³
Therefore the volume of the triangular prism is 12in³
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Find an equation of the line of intersection of planes below, and the acute angle between these two planes. P. : x + 2y – z = 1 and P2 : x + y + z = 1.
The equation of the line of intersection between planes P1 and P2 is x = 1 + 5z, y = -2z, z = z. The acute angle between the two planes is given by θ = arccos(2 / (√6 * √3)).
To determine the equation of the line of intersection between the two planes P1 and P2, we can set the equations of the planes equal to each other and solve for the variables.
First, let's set the equations equal to each other:
x + 2y - z = x + y + z
By rearranging the equation, we have:
y + 2z = 0
Now, we can express the equation in terms of a parameter. Let's choose z as the parameter:
y = -2z
Substituting this value back into the equation of P1, we have:
x + 2(-2z) - z = 1
x - 5z = 1
Therefore, the equation of the line of intersection between the two planes P1 and P2 is given by:
x = 1 + 5z
y = -2z
z = z
To determine the acute angle between the two planes, we can calculate the dot product of their normal vectors and use the formula:
cosθ = dot product of normal vectors / (magnitude of normal vector of P1 * magnitude of normal vector of P2)
The normal vector of P1 is [1, 2, -1] and the normal vector of P2 is [1, 1, 1]. Taking the dot product:
[1, 2, -1] ⋅ [1, 1, 1] = 1 + 2 - 1 = 2
The magnitude of the normal vector of P1 is √(1^2 + 2^2 + (-1)^2) = √6
The magnitude of the normal vector of P2 is √(1^2 + 1^2 + 1^2) = √3
Using the formula for the cosine of the angle:
cosθ = 2 / (√6 * √3)
θ = arccos(2 / (√6 * √3))
Thus, the acute angle between the two planes P1 and P2 is given by θ = arccos(2 / (√6 * √3)).
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A doctor's Order requests 500 mg of ampicillin IV in a 50-mL MiniBag of 0.9% sodium chloride injection. You have a 4-g vial of sterile powder, which, when reconstituted, will provide 100 mg/mL of ampicillin. How many milliliters of reconstituted solution will be needed to provide the 500-mg dose?
A. 4 ml
B. 5 ml
C. 2.5 ml
D. 25 ml
The correct answer is option B) 5 mL. This amount of the reconstituted solution will provide the necessary 500 mg dosage of ampicillin as specified in the doctor's order.
To provide a 5 mL dose of ampicillin, we need to calculate the required volume of the reconstituted solution.
First, we need to determine how much ampicillin is in each milliliter of the reconstituted solution. We know that the 4-g vial of sterile powder will provide 100 mg/mL of ampicillin when reconstituted.
To calculate how much ampicillin is in each milliliter, we divide the total amount of ampicillin in the vial (4 g or 4000 mg) by the total volume of the reconstituted solution:
4000 mg / X mL = 100 mg/mL
Solving for X, we get:
X = 4000 mg / 100 mg/mL = 40 mL
So, when the powder is reconstituted, we will have a total volume of 40 mL.
Next, we need to calculate how much of this solution we need to provide a 500-mg dose. We know that we want to deliver 500 mg of ampicillin and that the concentration of ampicillin in the reconstituted solution is 100 mg/mL. We can use this information to set up a proportion:
100 mg / 1 mL = 500 mg / X mL
Solving for X, we get:
X = (500 mg)(1 mL) / (100 mg) = 5 mL
Therefore, the correct option is B) 5 mL.
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(25 points) Find two linearly independent solutions of 2x²y" − xy' + (−3x + 1)y = 0, x > 0 of the form Y₁ = x¹(1+ a₁x + a²x² + A3x³ + ...) Y₂ = x¹² (1+b₁x + b₂x² + b3x³ + …..) where r₁ r₂. Enter r1 = a1 a2 = az = r2 = b₁ = b₂ = b3 =
Therefore the solutions are: y₁ = x¹(1+ a₁x + a²x² + A3x³ + ...) and y₂ = x¹²(1+b₁x + b₂x² + b3x³ + …..).
Two linearly independent solutions of 2x²y" − xy' + (−3x + 1)y = 0, x > 0 of the form Y₁ = x¹(1+ a₁x + a²x² + A3x³ + ...) Y₂ = x¹² (1+b₁x + b₂x² + b3x³ + …..) where r₁ r₂ are to be found. Let us try solution of the form Y₁ = x¹(1+ a₁x + a²x² + A3x³ + ...) y₁' = (1+a₁x +2a²x²+3a³x³+...) + x(a₁+4a²x+9a³x²+...), y₁" = (2a²+6a³x+...) + x(2a³x+...)+x(a₁+4a²x+9a³x²+...)On substituting the above expressions in the given differential equation, we get the value of r₁ as 1/2. Hence one of the solutions is y₁ = x¹(1+ a₁x + a²x² + A3x³ + ...)For second solution, we assume Y₂ = Y₁ ln x + x¹²(1+b₁x + b₂x² + b3x³ + …..)On differentiating once and twice we get:y₂' = (1+a₁x+2a²x²+...)+x(a₁+4a²x+9a³x²+...)+x¹¹(1+b₁x+b₂x²+...)y₂" = (2a²+6a³x+...)+x(2a³x+...)+x(a₁+4a²x+9a³x²+...)+x¹¹(b₁+2b₂x+...)On substituting the value in the given differential equation, we get the value of r₂ as 3/2. Hence the second solution is y₂ = x¹²(1+b₁x + b₂x² + b3x³ + …..).
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The length of time to failure (in hundreds of hours) for a transistor is a random variable X with the CDF given below: Fx $1 - e-x2 for x 20 0 elsewhere Ex(x) = {t-e = a. b. Compute the P(X 50.4) Compute the probability that a randomly selected transistor operates for at least 200 hours. Derive the pdf of this random variable. C.
(a) The probability that X is less than 50.4 is equal to 1.
To solve the given problem, we are given that the length of time to failure for a transistor is a random variable X with a CDF of Fx = [tex]1 - e^(-x^2)[/tex] for x >= 0 and 0 elsewhere. We are also given that Ex(x) = ∫(from 0 to infinity) t * f(x) dx = a.
a) To compute P(X < 50.4), we can use the CDF as follows:
P(X < 50.4) = Fx(50.4)
= 1 - [tex]e^(-(50.4)^2)[/tex]
= 1 - [tex]e^(-2540.16)[/tex]
= 1 - 0
= 1
(b) The probability that a randomly selected transistor operates for at least 200 hours is approximately equal to [tex]e^(-40000)[/tex].
To compute the probability that a randomly selected transistor operates for at least 200 hours, we need to find P(X >= 200). We can use the complement rule and the CDF as follows:
P(X >= 200) = 1 - P(X < 200)
= 1 - Fx(200)
= 1 - (1 - [tex]e^(-200^2)[/tex])
= [tex]e^(-40000)[/tex]
(c) The PDF of this random variable is f(x) = [tex]2xe^(-x^2)[/tex].
To derive the PDF of this random variable, we can differentiate the CDF with respect to x as follows:
f(x) = d/dx Fx(x)
= d/dx (1 - [tex]e^(-x^2)[/tex])
= [tex]2xe^(-x^2)[/tex]
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Suppose that G is a plane graph that has 15 edges in the boundary of its exterior region and all the other regions of G contain 4, 6, or 8 regions in their boundary. Use Grinberg's Theorem to show that G cannot contain a Hamilton circuit.
Based on Grinberg's Theorem, a plane graph with 15 edges in the exterior region and other regions containing 4, 6, or 8 edges in their boundaries cannot have a Hamilton circuit.
Grinberg's Theorem states that in a plane graph with n vertices, m edges, and r regions, the following inequality holds:
2m ≥ 3n + r - 6
Let's apply this theorem to the given situation:
Assume that G contains a Hamilton circuit. A Hamilton circuit is a closed path in a graph that visits each vertex exactly once. Since a Hamilton circuit visits each vertex once, the number of edges in the Hamilton circuit is equal to the number of vertices in G.
Let n be the number of vertices in G. Since G contains a Hamilton circuit, we have n edges.
The total number of regions in G can be determined by Euler's formula for planar graphs, which states:
n - m + r = 2
where r is the number of regions.
From the given information, we know that G has 15 edges in the boundary of its exterior region, which means there are 15 regions with a boundary of size 1.
Using the given information about the other regions, we can determine the number of regions with boundaries of size 4, 6, and 8, denoted as r4, r6, and r8, respectively.
Now, applying Grinberg's Theorem, we have:
2m ≥ 3n + r - 6
2n ≥ 3n + (15 + 4r4 + 6r6 + 8r8) - 6
2n - 3n ≥ 15 + 4r4 + 6r6 + 8r8 - 6
-n ≥ 9 + 4r4 + 6r6 + 8r8
Since the left-hand side of the inequality is negative and the right-hand side is positive (as the number of regions and boundaries are positive), the inequality is not satisfied.
Therefore, based on Grinberg's Theorem, G cannot contain a Hamilton circuit.
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Provide an appropriate response. The probability that an individual is left-handed is 0.1. In a class of 60 students, what is the probability of finding exactly five left-handers? O 0.083 O 0.1 O 0.000 O 0.166
The probability of finding exactly five left-handers in a class of 60 students, given that the probability of an individual being left-handed is 0.1, is approximately 0.166.
To calculate the probability of finding exactly five left-handers in a class of 60 students, we can use the binomial probability formula. In this case, the probability of success (finding a left-hander) is 0.1, the probability of failure (finding a right-hander) is 0.9 (1 - 0.1), and we want to find the probability of exactly five successes in a sample size of 60.
The formula for the binomial probability is P(X = k) = C(n, k) * p^k * (1 - p)^(n - k), where P(X = k) is the probability of getting exactly k successes, n is the sample size, p is the probability of success, and C(n, k) is the combination of n things taken k at a time.
Applying the formula, we have P(X = 5) = C(60, 5) * (0.1)^5 * (0.9)^(60 - 5). Evaluating this expression yields approximately 0.166.
Therefore, the probability of finding exactly five left-handers in a class of 60 students is approximately 0.166.
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Yerald weighed 92 kg. Then lost 4 kg 750 g. What is his weight now?
2. Jason is serving a 10 kg turkey to 28 people. How many grams of meat is he allowing for each person? round to nearest whole gram.
3. Young-Mi bought 2 kg 20 g of onion. The price was $1.49 per kilogram. How much did she pay, to nearest cent?
4. The standard dose of an antibiotic is 4 cc(cubic centimeters for every 25 pounds(lb) of body weight. At this rate,find the standard dose for a 140-lb woman.
5. A label printer prints 9 pages of labels in 1.7 seconds. How long will it take to print 72 pages of labels?
1. Using subtraction, Yerald's current weight after losing 4 kg 750 g is 87 kg and 250 g.
2. Using division operation, the allowed grams of meat for each person, Jason is serving the 10 kg turkey, are approximately 357 grams.
3. Using mathematical operations, the total cost that Young-Mi paid to buy 2 kg 20 g of onion priced at $1.49 per kilogram is $3.01.
4. Based on the standard rate of an antibiotic, the standard dose for a 140-pound woman is 22.4 cc.
5. Using the printing rate per second, the time to print 72 pages of labels is 13.6 seconds.
What are mathematical operations?The four basic mathematical operations include addition, subtraction, multiplication, and division.
These mathematical operations help us to determine the required values of algebraic expressions using mathematical operands and the equal symbol (=).
1. Yerald's Weight:
Past weight = 92 kg
The weight she lost = 4 kg and 750 g.
Current weight = 87.25 kg (92 - 4.75) = 87 kg and 250 g.
2. Jason:
The total weight of turkey = 10 kg
The total number of people being served = 28
1 kg = 1,000 grams
10 kg = 10,000 grams
The serving per person = 357.14 grams (10,000 ÷ 28)
= 357 grams
3. Young-Mi:
The weight of onions bought = 2 kg 20 g
1 kg = 1,000 grams
20 g = 0.02 kg (20 ÷ 1,000)
2 kg 20 g = 2.02 kg
The price per kilogram = $1.49
The total cost = $3.01 ($1.49 x 2.02)
4. Antibiotics:
Standard dose = 4 cc (cubic centimeters) for every 25 pounds of body weight
The standard dose for a 140-lb woman = 22.4 cc (4 x 140 ÷ 25)
5. Printing Rate:
The number of pages printed in 1.7 seconds = 9 pages
The total number of pages of labels = 72 pages
The time to print 72 pages = 13.6 seconds (72 ÷ 9 x 1.7)
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Let X1, X2, ..., Xm be a random sample from a population with mean mu1 and variance of sigma1^2=, and let Y1, Y2, ... , Yn be a random sample from a population with mean mu2 and variance sigma2^2, and that X and Y samples are independent of one another. Which of the following statements are true? Xbar is normally distributed with expected value mu1 and variance sigma1^2/m Ybar is normally distributed with expected value mu2 and variance sigma2^2/m Xbar-Ybar is normally distributed with expected value mul-mu2 and variance (sigma1^2/m+sigma2^2/n). Xbar-Ybar is an unbiased estimator of mul-mu2. All of the above statements are true.
Let X₁, X₂, ..., Xₙ be a random sample from a population with mean mu₁ and variance of sigma₁² then All of the above statements are true.
a) Xbar is normally distributed with expected value mu₁ and variance sigma₁²/m:
According to the Central Limit Theorem, when the sample size is large enough, the sampling distribution of the sample mean (X bar) approximates a normal distribution. The expected value of X bar is mu₁, the mean of the population, and the variance of X bar is sigma₁²/m, where sigma₁² is the variance of the population and m is the sample size.
b) Ybar is normally distributed with expected value mu₂ and variance sigma₂²/n:
Similar to Xbar, Ybar follows a normal distribution by the Central Limit Theorem. The expected value of Ybar is mu², the mean of the population, and the variance of Ybar is sigma₂²/n, where sigma₂² is the variance of the population and n is the sample size.
c) Xbar - Ybar is normally distributed with expected value mu₁ - mu₂ and variance (sigma₁²/m + sigma₂²/n):
Since Xbar and Ybar are independent, the difference Xbar - Ybar follows a normal distribution. The expected value of Xbar - Ybar is mu₁ - mu₂, and the variance of Xbar - Ybar is (sigma₁²/m + sigma₁²/n), where sigma₁² is the variance of the X population, sigma₂² is the variance of the Y population, m is the sample size for X, and n is the sample size for Y.
d) Xbar - Ybar is an unbiased estimator of mu₁ - mu₂:
An estimator is unbiased if its expected value equals the true value being estimated. Since the expected value of Xbar - Ybar is mu₁ - mu₂, it is an unbiased estimator of the difference in means, mu₁ - mu₂.
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Fulton is employed at an annual salary of S22,532 paid semi monthly. The regular workwerk in 36 hours (a) What is the regular salary per pay period? (b) What is the hourly rate of pay? c) What is the gross pay for a pay period in which the employee worked 9 hours overtime at time and one half regular pay?
a) The regular salary per pay period for Fulton is $938.83.
b) The hourly rate of pay is $13.04.
c) The gross pay for a pay period in which Fulton worked 9 hours overtime at time and one half is $645.48.
What is the gross pay?The gross pay is the total earning for a period before deductions are subtracted.
In this situation, the gross pay results from the addition of the regular pay and the overtime pay, which is computed at one and one half.
Annual salary = $22,532
The regular workweek = 36 hours
The number of pay periods per year = 24 (12 months x 2)
The regular salary per pay period = $938.83 ($22,532 ÷ 24)
The salary per week = $469.42 ($22,532 ÷ 48)
Hourly pay rate = $13.04 ($469.42 ÷ 36)
Gross pay with 9 hours overtime = $645.48 ($13.04 x 36 + $19.56 x 9)
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Consider the following matrices. (To make your job easier, an equivalent echelon form is given for the matrix.)
A = [1 0 −4 −6, −2 1 13, 5 0 1 5 −7] ~ [1 0 −4 −6, 0 1 5 −7, 0 0 0 0]
Find a basis for the column space of A. (If a basis does not exist, enter DNE into any cell.)
Find a basis for the row space of A. (If a basis does not exist, enter DNE into any cell.)
Find a basis for the null space of A. (If a basis does not exist, enter DNE into any cell.)
The basis for the null space of A is {(-5,0,-1,1)}.
Given matrix A = [1 0 -4 -6, -2 1 13, 5 0 1 5 -7] ~ [1 0 -4 -6, 0 1 5 -7, 0 0 0 0]The basis for the column space of matrix A is {(1,-2,5),(0,1,0),(-4,13,1),(-6,5,5)}. We can obtain the basis for the column space of matrix A by selecting the pivot columns. In this case, the pivot columns are columns 1 and 2. The non-zero columns in the row echelon form are columns 1, 2 and 3. To obtain the basis, we take columns 1 and 2 from the original matrix A, then write them in order followed by columns 3 and 4 of the original matrix A. So the basis for the column space of A is as shown below{(1,-2,5),(0,1,0),(-4,13,1),(-6,5,5)}.The basis for the row space of matrix A is {(1,0,-4,-6),(0,1,5,-7)}.
In order to find the basis for the row space, we take the nonzero rows from the row echelon form of A, which are rows 1 and 2. Then we select the corresponding rows of the original matrix A. The result is {(1,0,-4,-6),(0,1,5,-7)}.The basis for the null space of matrix A is {(-5,0,-1,1)}. We can obtain the basis for the null space of matrix A by solving the system Ax = 0. By writing this system in the form Rx = 0 where R is the row echelon form of A,
we get$$\begin{bmatrix}1&0&-4&-6\\0&1&5&-7\\0&0&0&0\end{bmatrix} \begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix}.$$ Solving this system, we get the general solution as x = (-5t, 0, -t, t) where t is a scalar. Therefore, the basis for the null space of A is {(-5,0,-1,1)}.
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What are the 3 important POST-implementation activities?
The three important post-implementation activities are evaluation and review, training and support, and maintenance and optimization.
After the implementation of a project or system, several important activities need to be carried out to ensure its success and ongoing effectiveness.
Evaluation and Review: This activity involves assessing the outcomes and performance of the implemented project or system. It includes gathering feedback from users, stakeholders, and other relevant parties to evaluate its functionality, efficiency, and user satisfaction. The evaluation helps identify any issues, shortcomings, or areas for improvement.
Training and Support: Post-implementation, providing adequate training and support to users is crucial. This involves conducting training sessions to familiarize users with the system's features, functionalities, and best practices. Ongoing technical support and assistance should be available to address user queries, troubleshoot issues, and ensure smooth operations.
Maintenance and Optimization: Regular maintenance is necessary to ensure the continued functioning and stability of the implemented project or system. This includes activities such as bug fixing, software updates, security patches, and performance optimizations. Monitoring system performance, collecting user feedback, and implementing necessary changes or enhancements contribute to the system's long-term success and efficiency.
By engaging in these post-implementation activities, organizations can effectively evaluate, support, and maintain the implemented project or system, maximizing its benefits and addressing any challenges that may arise.
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To estimate the proportion of smoker a sample of 100 mon was selected. In the selected sample, 80 mee were smoker. Determina a 95% confidence interval of proportion smoker OA (0.72 0.88) O B (0.72 0.85) OC (0.75 0.85) OD (0.75 0.88)
The 95% confidence interval for the proportion of smokers is (0.72, 0.88).
The proportion of smokers in a sample of 100 men has been calculated to be 80. We will utilize this information to create a 95% confidence interval of the proportion of smokers between two limits.
Here's how to do it:Solution:We have selected a sample of 100 men to estimate the proportion of smokers. In that sample, 80 of them were smokers.
Therefore, we may assume that the proportion of smokers in the sample is 0.8. Using this sample proportion, we can calculate the 95% confidence interval of the proportion of smokers.
Confidence interval of proportion of smokers is given by;[tex]CI = \left( {{\hat p} - {z_{\frac{\alpha }{2}}}\sqrt {\frac{{{\hat p}\left( {1 - {\hat p}} \right)}}{n}}} \right),\left( {{\hat p} + {z_{\frac{\alpha }{2}}}\sqrt {\frac{{{\hat p}\left( {1 - {\hat p}} \right)}}{n}}} \right)[/tex]where,[tex]n = 100[/tex][tex]\hat p = \frac{x}{n} = \frac{80}{100} = 0.8[/tex][tex]\alpha = 1 - 0.95 = 0.05[/tex][tex]\frac{\alpha }{2} = \frac{0.05}{2} = 0.025[/tex]
Since we know the values of [tex]\hat p[/tex], [tex]\alpha[/tex], [tex]\frac{\alpha}{2}[/tex] and [tex]n[/tex], we may calculate the confidence interval as follows:[tex]CI = \left( {0.8 - {z_{0.025}}\sqrt {\frac{{0.8\left( {1 - 0.8} \right)}}{{100}}}} \right),\left( {0.8 + {z_{0.025}}\sqrt {\frac{{0.8\left( {1 - 0.8} \right)}}{{100}}}} \right)[/tex][tex]CI = \left( {0.72,0.88} \right)[/tex]
Hence, the 95% confidence interval for the proportion of smokers is (0.72, 0.88).
Therefore, the answer is option OA (0.72 0.88).
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If the alternate hypothesis is justifiably directional (rather than non-directional), what should the researcher do when conducting a t test? O a one-tailed test O a two-tailed test O set the power to equal B O set ß to be less than the significance level
If the alternate hypothesis is justifiably directional (rather than non-directional), the researcher should conduct a one-tailed test.
A one-tailed test is a type of hypothesis test in which the alternative hypothesis is stated as a range of only one side of the probability distribution. The null hypothesis is rejected in a one-tailed test only if the test statistic is in the critical region of rejection for the upper or lower tail of the sampling distribution.What should be done when conducting a t test if the alternate hypothesis is justifiably directional?When conducting a t test, if the alternate hypothesis is justifiably directional, a one-tailed test should be used. It is because the direction of the difference is already stated in the alternative hypothesis. It is not necessary to test for the possibility of differences in both directions. A one-tailed test increases the power of the test to detect the difference in the direction specified by the alternative hypothesis. Thus, it is the most appropriate way to test the hypothesis when the direction is specified.
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Consider the set S = {(o,p,q,r): op-qr =0 }
Provide a counterexample to show that this set is not a subspace of R4
S is not a subspace of R^4. This shows the set is not a subspace of R4
Is the set S = {(o,p,q,r): op-qr = 0} a subspace of R^4?To determine if S is a subspace of R^4, we have to check if it satisfies the three properties of a subspace: closure under addition, closure under scalar multiplication, and contains the zero vector.
Closure under addition:
Let (o,p,q,r) and (o',p',q',r') be two vectors in S.
(op - qr) + (o'p' - q'r') = op + o'p' - qr - q'r'
= (o + o')p - (q + q')r
If (o + o')p - (q + q')r = 0, then (o + o', p + p', q + q', r + r') is also in S.
However, this is not always true.
Consider the vectors (1,0,1,0) and (-1,0,-1,0) in S:
= (1,0,1,0) + (-1,0,-1,0)
= (0,0,0,0)
But (0,0,0,0) does not satisfy the condition op - qr = 0, so it is not in S. Therefore, S does not satisfy closure under addition.
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You are evaluating the possibility that your company bids $150,000 for a particular construction job. (a) If a bid of $150,000 corresponds to a relative bid of 1.20, what is the dollar profit that your company would make from winning the job with this bid? Show your work. (b) Calculate an estimate of the expected profit of the bid of $150,000 for this job. Assume that, historically, 55 percent of the bids of an average bidder for this type of job would exceed the bid ratio of 1.20. Assume also that you are bidding against three other construction companies. Show your work.
a) The company will make a profit of $120,000 from winning the job with this bid.
b) The expected profit of the bid of $150,000 for this job is $13,500.
a)Given, Bid amount = $150,000 Relative Bid = 1.20
As per the question, Relative Bid = (Total cost of construction ÷ Bid amount) + 1i.e, (Total cost of construction ÷ Bid amount) = Relative Bid - 1Total cost of construction = (Relative Bid - 1) × Bid amount
Total cost of construction = (1.20 - 1) × $150,000 = $30,000Profit = Bid amount - Total cost of construction= $150,000 - $30,000 = $120,000
Therefore, the company will make a profit of $120,000 from winning the job with this bid.
b) Given, Bid amount = $150,000Relative Bid = 1.20As per the question,
Probability of Winning the bid = 1 - 55/100 = 0.45
Probability of winning among 4 construction companies = 0.45/4 = 0.1125
Expected profit = Probability of Winning the bid × Profit
Expected profit = 0.1125 × $120,000 = $13,500
Therefore, the expected profit of the bid of $150,000 for this job is $13,500.
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Which of the following story problems can be solved by calculating 3/4-1/2? (Select all that apply.) Select one: O a. A bird feeder was filled with 3/4 of a full bag of bird seed. The birds ate 1/2 of what was in the bird feeder. What fraction of a full bag of bird seed did the birds eat? O b. A bird feeder was filled with 3/4 of a full bag of bird seed. The birds ate 1/2 of what was in the bird feeder. What fraction of a full bag of bird seed is left in the bird feeder? Oc. Charlie has 3/4 of a full bag of bird seed. He has a bird feeder that holds half a full bag. What fraction of a full bag of bird seed will be left in the bag after he fills the bird feeder? O d. None of the story problems can be solved by calculating 3/4-1/2.
The story problem that can be solved by calculating 3/4 - 1/2 is option (b). It asks for the fraction of a full bag of bird seed that is left in the bird feeder after the birds ate half of it.
Option (b) presents a scenario where a bird feeder is initially filled with 3/4 of a full bag of bird seed. The birds eat 1/2 of what was in the bird feeder. To find out what fraction of a full bag of bird seed is left in the bird feeder, we need to subtract the amount the birds ate from the initial amount. Calculating 3/4 - 1/2 gives us 1/4, which represents the fraction of a full bag of bird seed that is left in the bird feeder.
The other options do not involve calculating 3/4 - 1/2. Option (a) asks for the fraction of a full bag of bird seed that the birds ate, not what is left. Option (c) involves a different scenario where Charlie has 3/4 of a full bag of bird seed and a bird feeder that holds half a full bag. It asks for the fraction of a full bag of bird seed that will be left in the bag after filling the bird feeder, which requires a different calculation. Therefore, option (d) is incorrect as option (b) can be solved by calculating 3/4 - 1/2.
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Let {N(t), t>0} be a Poisson process with rate 3 per minute. Let S_n be the time of the nth event. Find
a) E[S_10]
b) E[S_4 | N(1)=3)]
c) Var[S_10]
d) E[N(4)-N(2) | |N(1)=3]
e) P[T_20 > 3]
Should be visible now
The Poisson process is characterized by the rate at which the
Poisson
process has a rate of 3 events per minute. E[S_10] = 10/3 minutes. E[S_4 | N(1) = 3] = 1/3 minutes.
a) The
expected
value of the time of the 10th event, E[S_10], in a Poisson process with rate λ is given by E[S_10] = 10/λ. Therefore, E[S_10] = 10/3 minutes.
b) Given that there are 3 events in the first minute, the conditional expected value E[S_4 | N(1) = 3] is the expected time of the 4th event, given that 3 events occurred in the first minute. Since the time between events in a Poisson process is
exponentially
distributed with rate λ, we can use the memoryless property. The expected time of the 4th event is the same as the expected time of the 1st event in a Poisson process with rate 3. Hence, E[S_4 | N(1) = 3] = 1/3 minutes.
c) The variance of S_10, Var[S_10], in a Poisson process with rate λ is given by Var[S_10] = 10/λ^2. Therefore, Var[S_10] = 10/(3^2) = 10/9 minutes^2.
d) Given that there are 3 events in the first minute, the conditional expected value E[N(4)-N(2) | N(1) = 3] is the expected number of events
occurring
between time 2 and time 4, given that 3 events occurred in the first minute. The number of events in a Poisson process with rate λ is distributed as Poisson(λt), where t is the time duration. In this case, we have t = 2 minutes. So, E[N(4)-N(2) | N(1) = 3] = λt = 3*2 = 6 events.
e) To find the
probability
P[T_20 > 3], where T_20 represents the time of the 20th event, we can use the exponential distribution. The time until the 20th event follows an exponential distribution with
rate
λ. Therefore, P[T_20 > 3] = e^(-λt) = e^(-3*3) = e^(-9) = 0.00012341.
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14. If y = f(x) is a solution to the differential equation =et with the initial condition f(0) = 2, which of the dx
following is true?
(A) f(x)=1+e+²
(B) f(x) = 2xe¹²
(C) f(x) = [*e¹² dt
(D) f(x) = 2+ [*e²² dt
(E) f(x)=2+ fedt
The correct option is (A) f(x)=1+e+² since the value of y is obtained as et + 1, which is equal to 1+e^x 2. The other options do not satisfy the initial condition.
Given that, y = f(x) is a solution to the differential equation y' = et with the initial condition f(0) = 2. To find the correct option among the given options.
Therefore, let's solve this using the integration method. Let's integrate both sides with respect to x,y'=etdy/dx =etdy = etdx Integrating both sides, we get∫dy = ∫et dxy = ∫et dx + c ....(1) where c is the constant of integration. To find the constant c, we need to use the initial condition f(0) = 2.
Substituting x = 0 and y = f(0) = 2 in equation (1),2 = ∫e0 dx + c2 = 1 + c => c = 1. Therefore, the solution is y = ∫et dx + 1= et + 1
Therefore, the correct option is (A) f(x)=1+e+² since the value of y is obtained as et + 1, which is equal to 1+e^x 2. The other options do not satisfy the initial condition.
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Find the non-parametric equation of the plane with normal (−5,6,6)-5,6,6 which passes through point (5,−6,0)5,-6,0.
Write your answer in the form Ax+By+Cz+d=0Ax+By+Cz+d=0 using lower case x,y,zx,y,z and * for multiplication. Please Do Not rescale (simplify) the equation.
Sothe non-parametric equation of the plane with the given normal vector and passing through the point (5, -6, 0) is: -5x + 6y + 6z + 61 = 0
How to explain the equationIn order to find the non-parametric equation of the plane, we need the normal vector and a point on the plane. The normal vector is given as (-5, 6, 6), and a point on the plane is (5, -6, 0).
The non-parametric equation of a plane is given by:
Ax + By + Cz = D
where (A, B, C) is the normal vector and (x, y, z) is a point on the plane. We can substitute the values into the equation to find the values of A, B, C, and D.
(-5)(x - 5) + (6)(y + 6) + (6)(z - 0) = 0
Expanding this equation:
-5x + 25 + 6y + 36 + 6z = 0
-5x + 6y + 6z + 61 = 0
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The volume of a rectangular prism is represented by the function x^3 - 9x^2 + 6x - 16. The length of the box is x^2, while the height is x + 8. Find the expression representing the width of the box.
a. x - 8
b. x + 8
c. x^2 - 8
d. x^2 + 8
The volume of a rectangular prism is represented by the function x³ − 9x² + 6x − 16, while the length of the box is x² and the height is x + 8. We are required to find the expression representing the width of the box.
Solution: The formula for calculating the volume of a rectangular prism is: V = Iwhere V = Volume, l = Length, w = Width and h = Height Given that the length of the box is x² and the height is x + 8.The volume of the box is represented by x³ − 9x² + 6x − 16. Therefore, we can equate them as follows;x³ − 9x² + 6x − 16 = lwhx³ − 9x² + 6x − 16 = (x²)(w)(x + 8)Since we know that the length of the box is x² and the height is x + 8. We can write the expression for the width of the box as: w = (x³ − 9x² + 6x − 16)/((x²)(x + 8))w = (x − 8)Therefore, the expression representing the width of the box is x − 8. Answer: a. x − 8
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The expression representing the width of the box is (x - 16 - 10/x) / (x + 8).
Option A (x - 8) is incorrect.
Option B (x + 8) is incorrect.
Option C (x² - 8) is incorrect.
Option D (x² + 8) is incorrect.
The volume of a rectangular prism is represented by the function x^3 - 9x^2 + 6x - 16.
The length of the box is x^2, while the height is x + 8.
To find: The expression representing the width of the box.
Solution:
Volume of a rectangular prism = length x width x height
Given, the length of the box is x², while the height is x + 8.
And, the volume of the rectangular prism is given by the function x³ - 9x² + 6x - 16.
Therefore, Volume of the rectangular prism = x² × width × (x + 8)
Or, x³ - 9x² + 6x - 16 = x² × width × (x + 8)
Or, (x³ - 9x² + 6x - 16) / (x² × (x + 8)) = width
Or, [x³ - 16x² + 7x² - 16x + 6x - 16] / [x²(x + 8)] = width
Or, [(x³ - 16x² + 7x²) + (-16x + 6x - 16)] / [x²(x + 8)] = width
Or, [x²(x - 16) + x(-10)] / [x²(x + 8)] = width
Or, [x²(x - 16 - 10/x)] / [x²(x + 8)] = width
Or, (x - 16 - 10/x) / (x + 8) = width
Therefore, the expression representing the width of the box is (x - 16 - 10/x) / (x + 8).
Hence, option A (x - 8) is incorrect.
Option B (x + 8) is incorrect.
Option C (x² - 8) is incorrect.
Option D (x² + 8) is incorrect.
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which of the following illustrates the product rule for logarithmic equations?
a. log2(4x)- log24+log2x
b. log2(4x)- log24xlog2x
c. log2(4x)-log24-log2x
d. log2(4x)-log24+ log2x
Option d. log2(4x) - log24 + log2x illustrates the product rule for logarithmic equations.
The product rule for logarithmic equations states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.
Looking at the given options, the expression that illustrates the product rule is:
d. log2(4x) - log24 + log2x
In this expression, we have the logarithm of a product, log2(4x), which is being subtracted from the logarithm of another term, log24, and then added to the logarithm of the term x, log2x.
According to the product rule, we can rewrite this expression as the sum of the logarithms of the individual factors:
log2(4x) - log24 + log2x = log2(4x) + log2(x) - log2(4)
By applying the product rule, we can combine the logarithms and simplify further if necessary.
Therefore, option d. log2(4x) - log24 + log2x illustrates the product rule for logarithmic equations.
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Find the general solution to the differential equation y′ = x − x sin²
The general solution to the differential equation y' = x - x × sin²(x) is y = x + C, where C is a constant.
To find the general solution to the given differential equation, we'll separate the variables and integrate both sides.
The differential equation is: y' = x - x×sin²(x)
Step 1: Separate the variables
We can rewrite the equation as:
dy = (x - x×sin²(x)) dx
Step 2: Integrate both sides
Integrating the left side with respect to y gives us just y:
∫dy = ∫dx
On the right side, we need to integrate the expression (x - x×sin²(x)) with respect to x. This requires a bit more work.
Step 3: Expand the integrand
x - xsin²(x) can be expanded as follows:
x - xsin²(x) = x - x×(1 - cos²(x))
= x - x + xcos²(x)
= xcos²(x)
Step 4: Integrate xcos²(x) with respect to x
To integrate xcos²(x), we'll use integration by parts. Let's choose u = x and dv = cos²(x) dx.
Differentiating u, we get du = dx, and integrating dv, we have:
∫cos²(x) dx = ∫dv = v = (1/2)(x + sin(2x)/2)
Using the formula for integration by parts:
∫u dv = uv - ∫v du
We have:
∫x×cos²(x) dx = (1/2)(x + sin(2x)/2) - ∫(1/2)(x + sin(2x)/2) dx
Simplifying:
∫x×cos²(x) dx = (1/2)(x + sin(2x)/2) - (1/2)∫(x + sin(2x)/2) dx
We can integrate the remaining term on the right side.
Step 5: Integrate (x + sin(2x)/2) with respect to x
∫(x + sin(2x)/2) dx = (1/2)x² + (1/4)sin(2x) + C
Where C is the constant of integration.
Now, let's substitute this result back into our original equation:
y = ∫dx + C
= x + C
The general solution to the given differential equation is y = x + C, where C is a constant.
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A cohort study examined the effect of anti-smoking advertisements on smoking cessation among a group of smokers. For the purposes of this exercise, we are focusing on two groups in the study: 1) an unexposed control group that consists of 18,842 individuals contributing 351,551 person-years to the study, and 2) an exposed group of 798 individuals contributing 14,245 person-years These exposed smokers saw anti-smoking advertisements 1 a month for several years. Nine cases of smoking cessation were identified in the unexposed group. One case was identified in the exposed group. Follow-up occurred for 21 years. For risk calculations assume all individuals were followed for 21 years. Calculate the risk in the unexposed group. Select one:
a. 0.048% over 21 years of follow-up
b. 0.051% over 21 years of follow-up
c. 0.125% over 21 years of follow-up
d. 0.250% over 21 years of follow-up
The risk in the unexposed group over 21 years of follow-up is approximately 0.2562%.None of the provided options match the calculated value exactly.
To calculate the risk in the unexposed group, we need to divide the number of cases by the total person-years of follow-up.
Given information:
Unexposed group:
Individuals (n1): 18,842
Person-years (PY1): 351,551
Cases of smoking cessation (C1): 9
Risk in the unexposed group is calculated as:
Risk1 = (C1 / PY1) * 100%
Risk1 = (9 / 351,551) * 100%
Calculating the risk:
Risk1 ≈ 0.0025621 * 100%
Risk1 ≈ 0.2562%
Therefore, the risk in the unexposed group over 21 years of follow-up is approximately 0.2562%.
None of the provided options match the calculated value exactly.
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The raw data comparing the sex ratios for two subgroups of US: Native American and Japanese Americans. Find the relevant z-score and discuss its significance.
Years Native American Japanese American
1976 1070 1081
1977 1022 1077
1978 1044 1073
1979 1036 1071
1980 1048 7072
1981 1023 1014
1982 1032 1031
1983 1038 1037
1984 1016 1038
1985 1032 1062
1986 1047 1066
1987 1041 1096
1988 1021 1048
1989 1028 1061
1990 1023 1063
1991 1016 1042
1992 1034 1049
1993 1036 1063
1994 1031 1048
1995 1040 1054
1996 1031 1053
1997 1036 1068
1998 1038 1030
1999 1029 1063
2000 1035 1084
2001 1024 1041
2002 1023 1089
A z-score of 2.04 for the year 1976.
In the same year, the proportion of men was 1.5% above the mean.
However, this data point is not considered significant because the z-score is less than 2.0.
A z-score greater than 2.0 would indicate that the data is significant.
Sex ratios and z-score: The sex ratio is a statistical method for comparing the proportion of men to women in a population.
The raw data has been given for comparing sex ratios for two subgroups of the US: Native American and Japanese Americans.
The relevant z-score is calculated to check the significance of the data. The formula for the z-score is given by the following equation:
[tex]$z = (x - \mu) / \sigma$[/tex]
In this equation, x is the given raw data point, [tex]$\mu$[/tex] is the mean, and [tex]$\sigma$[/tex] is the standard deviation.
If the z-score is positive, it means the value is above the mean, and if it is negative, it means the value is below the mean.
If the z-score is close to zero, it indicates that the data is close to the mean.
Let's first calculate the mean and standard deviation for Native American and Japanese Americans separately.
Using the raw data, we can get the following results:
Native American:
[tex]$\mu = (1070 + 1022 + 1044 +...+ 1023 + 1024 + 1023) / 27$[/tex]
= 1036.7
[tex]$\sigma = 16.34$[/tex]
Japanese American:
[tex]$\mu = (1081 + 1077 + 1073 +...+ 1041 + 1089) / 27$[/tex]
= 1061.74
[tex]$\sigma = 18.39$[/tex]
Now that we have calculated the mean and standard deviation for both subgroups, we can move on to calculate the z-score.
Let's take one example for calculation:
For Native American data in the year 1976,
[tex]$z = (1070 - 1036.7) / 16.34$[/tex]
= 2.04
Similarly, z-scores can be calculated for all the raw data points.
A z-score of 2.04 for the year 1976 indicates that the proportion of men is above the mean for Native Americans.
In the same year, the proportion of men was 1.5% above the mean. However, this data point is not considered significant because the z-score is less than 2.0.
A z-score greater than 2.0 would indicate that the data is significant.
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A supermarket CEO claims that 26% of customers that enter the store purchase milk or bread. A survey of 320 customer showed that 67 customers purchased bread or milk on their trip to the store. Assuming the CEO's claim is correct, determine (to 4 decimal places):
1. the standard error for the sampling distribution of the proportion.
2. the probability that the sample proportion is no more than that found in the survey.
1. The standard error for the sampling distribution of the proportion is 0.0184.
2.The probability that the sample proportion is no more than that found in the survey is 0.0029 or 0.29%.
To determine the standard error for the sampling distribution of the proportion, we can use the formula:
Standard Error = √((p × (1 - p)) / n)
Where:
p = the proportion claimed by the CEO (0.26)
n = the sample size (320)
Standard error for the sampling distribution of the proportion:
Standard Error = √(0.26 × (1 - 0.26)) / 320)
Standard Error= 0.0184
2. To find the probability that the sample proportion is no more than that found in the survey, we need to calculate the z-score and use the standard normal distribution.
The sample proportion is calculated by dividing the number of customers who purchased bread or milk by the sample size:
Sample Proportion (p) = Number of customers who purchased bread or milk / Sample size
p = 67 / 320 = 0.2094
Now we can calculate the z-score using the formula:
z = (0.2094 - 0.26) / 0.0184
z = -2.7554
Using a standard normal distribution table the probability associated with a z-score of -2.7554 is 0.0029.
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4. Suppose you have a population with population with mean of u and a standard deviation of a and you take all possible samples of size 25. Suppose you take another random sample all possible samples of size 100 from the same population. Which of the following statements are true?
(a) The distribution of the population is symmetric.
(b) The distribution of both sample data are symmetric.
(c) The standard deviation of the sampling distribution of sample size 25 is more than standard deviation of the sampling distribution of sample size 100.
(d) The standard deviation of the sampling distribution of sample size 25 is less than standard deviation of the sampling distribution of sample size 100.
(e) No important information related to this situation can be determined.
Option (d) is correct.Option (a) cannot be determined because the population's symmetry is not related to the size of the sample. It is possible for an asymmetric population to have a symmetric sample distribution.Option (b) cannot be true because the sample distribution depends on the size of the sample.Option (c) is incorrect because the standard deviation of the sampling distribution is inversely proportional to the sample size, as mentioned earlier.Option (e) is incorrect because important information can be determined from the given scenario.
Given that a population has a mean of µ and a standard deviation of σ, and two random samples, one of size 25 and the other of size 100 are taken from the same population.
To determine which of the given options are correct, let's see the properties of the standard deviation, sample, and distribution. Standard deviation: It is a measure of the amount of variation or dispersion of a set of values from their mean.Sample: It is a subset of the population and is used to obtain information about the whole population. It is used to estimate population parameters. Distribution: It is a function that describes the probability of occurrence of a random variable. It can be represented in different ways depending on the context and the type of random variable. With these properties, we can say that the correct options are:
(d) The standard deviation of the sampling distribution of sample size 25 is less than the standard deviation of the sampling distribution of sample size 100.
(e) No important information related to this situation can be determined.
Explanation:It is known that the sample distribution is more likely to be normal than the population distribution. The standard deviation of the sampling distribution is inversely proportional to the sample size. That is, as the sample size increases, the standard deviation of the sampling distribution decreases. Therefore, the standard deviation of the sample of size 100 is less than the standard deviation of the sample of size 25.
Thus option (d) is correct.Option (a) cannot be determined because the population's symmetry is not related to the size of the sample. It is possible for an asymmetric population to have a symmetric sample distribution.Option (b) cannot be true because the sample distribution depends on the size of the sample.Option (c) is incorrect because the standard deviation of the sampling distribution is inversely proportional to the sample size, as mentioned earlier.Option (e) is incorrect because important information can be determined from the given scenario.
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Given that the population has a mean of u and a standard deviation of a and we take all possible samples of size 25, the distribution of the sampling means is approximately normal with mean u and standard deviation a/√25 = a/5
.Suppose we take another random sample, all possible samples of size 100 from the same population. Then, the distribution of the sampling means is approximately normal with mean u and standard deviation a/√100 = a/10.(a) The distribution of the population is symmetric - This is not true in general.(b) The distribution of both sample data is symmetric - This is not true in general.(c) The standard deviation of the sampling distribution of sample size 25 is more than standard deviation of the sampling distribution of sample size 100 - This is not true. The standard deviation of the sampling distribution of sample size 25 is less than the standard deviation of the sampling distribution of sample size 100.(d) The standard deviation of the sampling distribution of sample size 25 is less than the standard deviation of the sampling distribution of sample size 100 - This is true.(e) No important information related to this situation can be determined - This is false. The statements (c) and (d) are true.Answer: (d) The standard deviation of the sampling distribution of sample size 25 is less than standard deviation of the sampling distribution of sample size 100.
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A company produces packets of soap powder labeled "Giant Size 32 Ounces. The actual weight of soap powder in such a box has a Normal distribution, with a mean of 33 or and a standard deviation of 0.3 or. To ad having dissatisfied customers, the company says a box of soap is considered underweight if it weighs less than 32 or. To avoid losing money, it labels the top 5% (the heaviest 5%) overweight. How heavy does a bes have to be for it to be labeled overweight? 33.82 or 634,15 or 34.6202 31.80 o QUESTION 7 Some researchers have noted that adolescents who spend a lot of time playing video or computer games are at greater risk for depression and violence. This is an example of a a single-blind experiment, because the subjects knew they were playing games ha valid conclusion, because more time yields more aggression is a positive association c. a paired data experiment, because we are studying both aggression and game playing dan observational study with larking variables that may explain the association. QUESTIONS A company produces packets of soap poeder held"Chante 12 mes. The actual weight of top powder in a box una distribution of 33 oranda de roso having dised customers, the company box fit considered underwright if it weh less 1902. To avoiding help the best way does best be for is to be weled overweight XXX b.34.15 34.620 0.31.80 OR QUESTION 4 The time needed for college students to complete a certainpaper and pencil maze follows a Normal distribution, with a mean of 80 seconds and a standard deviation of 6 seconds. You wish to see if the mean time je is changed by meditation, so you have a group of 8 college students meditate for 30 minutes and then complete the mare. It takes them an average of 1-74 seconds to complete the mare. Use this information to test the hypotheses Hg: -80, Ha80 at significance level a-0.02. You conclude that a. Ha should be accepted. bHo should be rejected. Hy should not be rejected. d. this is a borderline case and so decision should be made.
In order to be labeled overweight the weight of box should be of 34.15 ounces.
P(boxe is underweight) = P(X < 32) = 0.0764
Here, we have,
Given mean of 33 ounces, standard deviation=0.7 ounces. Production of packets of 32 ounces.
Because it is normally distributed samples are solved using the z score formula.
In this Z=X-meu/st
meu is sample mean and st is population standard deviation.
We have to find the z score and then p value from the z table.
meu =33 and st=0.7
Top 5% so X when Z has a p value of 1-0.05=0.95. So X when Z=1.645.
Z=(X-meu)/st
1.645=(X-33)/0.7
X-33=0.7*1.645
X-33=1.1515
X=1.1515+33
X=34.1515 ounces.
Hence to be labeled overweight the box must have 34.15 ounces weight.
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A survey conducted by Sallie Mae and Gallup of 1404 respondents found that 323 students paid for their education by student loans. Find the 90% confidence interval of the true proportion of students who paid for their education by student loans.
a. Find the sample proportion
b. Determine the critical value
c. Find the margin of error
d. Find the confidence interval for the true population proportion
a. To find the sample proportion, we divide the number of students who paid for their education by student loans (323) by the total number of respondents (1404):
[tex]Sample proportion (\(\hat{p}\)) = \(\frac{323}{1404}\)[/tex]
b. To determine the critical value, we need to use the z-table for a 90% confidence level. Since the sample size is large (n = 1404), we can use the standard normal distribution.
The critical value for a 90% confidence level is [tex]\(z = 1.645\)[/tex].
c. To find the margin of error, we use the formula:
[tex]Margin of error (E) = \(z \times \sqrt{\frac{\hat{p} \times (1 - \hat{p})}{n}}\)[/tex]
Substituting the values, we have:
[tex]Margin of error (E) = \(1.645 \times \sqrt{\frac{\frac{323}{1404} \times (1 - \frac{323}{1404})}{1404}}\)[/tex]
d. Finally, to find the confidence interval for the true population proportion, we use the formula:
[tex]Confidence interval = \(\left(\hat{p} - E, \hat{p} + E\right)\)[/tex]
Substituting the values, we have:
[tex]Confidence interval = \(\left(\frac{323}{1404} - E, \frac{323}{1404} + E\right)\)[/tex]
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