The solution to the initial value problem is y = 8e/t - 2, where t ≠ 0.
The given differential equation is:
t^2 di/dt - t = 1 + y + ty
We can rearrange the terms as:
di/(1+y) = (1+t)/(t^2) dt
Integrating both sides, we get:
ln|1+y| = -1/t + ln|t| + C1
where C1 is the constant of integration.
Taking the exponential of both sides, we get:
|1+y| = e^(-1/t) * |t| * e^(C1)
Using the initial condition y(1) = 7, we get:
|1+7| = e^(-1/1) * |1| * e^(C1)
8 = e^(-1) * e^(C1)
e^(C1) = 8e
C1 = ln(8e)
Therefore, the solution is:
1 + y = ± e^(-1/t) * t * e^(ln(8e))
y = -1 ± 8e/t - 1
y = 8e/t - 2
So, the solution to the initial value problem is y = 8e/t - 2, where t ≠ 0.
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Find the absolute minimum and absolute maximum values off on the given interval. f(x) = x - 6x2 + 9x + 7 (-1, 4] absolute minimum value absolute maximum value
The absolute minimum value of the function is -1 and the absolute maximum value is approximately 7.5417.
To find the absolute minimum and absolute maximum values of the function f(x) = x - 6x² + 9x + 7 on the interval (-1, 4], we need to follow these steps:
1. Find the critical points by setting the derivative equal to zero.
2. Evaluate the function at the critical points and the endpoints of the interval.
3. Compare the values to find the absolute minimum and maximum.
Step 1: Find the critical points.
f'(x) = d/dx (x - 6x² + 9x + 7)
f'(x) = 1 - 12x + 9
To find the critical points, set f'(x) = 0:
0 = 1 - 12x + 9
12x = 10
x = 5/6
Step 2: Evaluate the function at the critical points and the endpoints of the interval.
f(-1) = -1 - 6(-1)² + 9(-1) + 7 = -1
f(5/6) = (5/6) - 6(5/6)^2 + 9(5/6) + 7 ≈ 7.5417
f(4) = 4 - 6(4)² + 9(4) + 7 = -87
Step 3: Compare the values.
Absolute minimum value: f(-1) = -1 (since -1 is the lowest value)
Absolute maximum value: f(5/6) ≈ 7.5417 (since 7.5417 is the highest value)
So, the absolute minimum value of the function is -1 and the absolute maximum value is approximately 7.5417.
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Use the t-distribution table to find the critical value(s) for the indicated alternative hypotheses, level of significance α, and sample sizes n1 and n2. Assume that the samples are independent, normal, and random. Answer parts (a) and (b).Ha : μ1 ≠ μ2 , α = 0.10 , n1 = 14 , n2 = 13(a) Find the critical value(s) assuming that the population variances are equal.____(Type an integer or decimal rounded to three decimal places as needed. Use a comma to separate answers as needed.)(b) Find the critical value(s) assuming that the population variances are not equal.____(Type an integer or decimal rounded to three decimal places as needed. Use a comma to separate answers as needed.)
a) The t critical values are -1.708 and 1.708
b) The t critical values are -1.782 and 1.782
What is t-statistic ?
The t-statistic in statistics measures how far an estimated value of a parameter deviates from its hypothesised value in relation to its standard error.
T symbol in statistics mean Test statistic for t-test ( t-score )
a) At α = 0.10, df=14+13-2 = 25( two tailed)
t critical values are -1.708 and 1.708
b) At α = 0.10,
df= smaller(n1-1, n2-1)= smaller(14-1, 13-1)= smaller(13, 12) = 12
t critical values are -1.782 and 1.782
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r
10
O
For each radius length of a circle that is given, mark the correct area of the circle.
Use = 3.14
Radius of
Circle
5 cm
6 cm
9 cm
10 cm
Porfavorere helppp plis 15 points for it
Answer:
Below
Step-by-step explanation:
formula for area of circle: A = (pie)(r)^2
1. 5 cm radius:
A = (pie)(r)^2
A = (pie)(5)^2
A = 78.54 and 78.5 cm^2
2. 6 cm radius:
A = (pie)(r)^2
A = (pie)(6)^2
A = 113.04 cm^2
3. 9 cm radius:
A = (pie)(r)^2
A = (pie)(9)^2
A = 254.34 cm^2
10. 10 cm radius:
A = (pie)(r)^2
A = (pie)(10)^2
A = 314 cm^2
Hope this helps^^
i need the profit maximizing price and OUTPUT PLS14. Profit For a monopolist's product, the demand function is p=50/√q= and the average-cost function is c = 1/4 + 2500/q Find the profit-maximizing price and output. noduce at most 120 units of a
if the monopolist can produce at most 120 units, they will produce 120 units and charge approximately $4.56 per unit.
To find the profit-maximizing price and output for a monopolist, we need to find the point where marginal revenue (MR) equals marginal cost (MC).
Given the demand function, we can derive the total revenue (TR) as follows:
TR = p * q
TR = (50/√q) * q
TR = 50√q
We can then derive the marginal revenue (MR) function by taking the derivative of TR with respect to q:
MR = dTR/dq = 25/√q
To find the marginal cost (MC) function, we can derive the total cost (TC) function as follows:
TC = VC + FC
VC = q * (1/4 + 2500/q) = 1/4*q + 2500
FC = 0
We can then derive the marginal cost (MC) function by taking the derivative of TC with respect to q:
MC = dTC/dq = 1/4 - 2500/q^2
Now, we can set MR equal to MC and solve for q:
MR = MC
25/√q = 1/4 - 2500/q^2
100 = √q - 625000/q^2
100q^2 = q^2 - 625000
q^2 = 625000/99
q ≈ 251.3
Note that since the demand function is p=50/√q, we can plug in q=251.3 to find the corresponding price:
p = 50/√q
p ≈ $3.16
Therefore, the profit-maximizing output is approximately 251 units, and the profit-maximizing price is approximately $3.16 per unit.
However, we also need to check whether this output level is feasible given the production constraint of producing at most 120 units. In this case, the monopolist will produce 120 units since this is the maximum amount they can produce, and the price will be determined by the demand function:
q = 120
p = 50/√q
p = 50/√120
p ≈ $4.56
Therefore, if the monopolist can produce at most 120 units, they will produce 120 units and charge approximately $4.56 per unit.
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The garden area is 48 000 cm². What is the area in square meters?
Answer:
4.8 squared meters
Step-by-step explanation:
divide the area value by 10,000
is it bad for the dependent variable y to be correlated with the error term e if the independent variable x is not?
It is generally considered bad for the dependent variable y to be correlated with the error term e, even if the independent variable x is not.
This is because the presence of such correlation indicates that there may be omitted variables or measurement errors that are affecting both the dependent variable and the error term. In turn, this can lead to biased and inefficient estimates of the parameters in the regression model, as well as invalid hypothesis testing and confidence intervals.
To address this issue, it is recommended to carefully examine the data and model assumptions, consider alternative specifications or estimation methods, and possibly include additional variables or controls that may help explain the relationship between y and e.
This correlation between Y and E violates one of the key assumptions of the classical linear regression model, which states that the error term should be uncorrelated with both the dependent and independent variables. This violation can lead to biased and inconsistent estimators, ultimately affecting the reliability of the regression results.
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Help me look in this image below
We can write our linear equation as:
y + 1 = (4/3)*(x + 1)
or
y - 3 = (4/3)*(x - 2)
How to write the linear equation?If a linear equation passes through two points (x₁, y₁) and (x₂, y₂), then the slope is:
a = (y₂ - y₁)/(x₂ - x₁).
Here the line passes through (-1, -1) and (2, 3), so the slope is:
a = (3 + 1)/(2 + 1) = 4/3
Now, if a line has a slope a and passes through a point (x₁, y₁),then we can write that line as:
y - y₁ = a*(x - x₁)
So with our two points, we can write our line as:
y + 1 = (4/3)*(x + 1)
y - 3 = (4/3)*(x - 2)
The correct option is the first one.
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Use the Ratio Test to determine whether the series is convergent or divergent. Σ n=1 (-1)^n + 1 n^6 6^n/n!Identify an_____Evaluate the following limit. lim n--> [infinity] |an + 1|/|an|
The series is convergent according to Ratio test.
To determine whether the series Σ n=1 (-1)ⁿ + 1 n⁶ 6ⁿ/n! is convergent or divergent, we can use the Ratio Test. First, we need to find the limit of the ratio of consecutive terms as n approaches infinity.
|an+1|/|an| = (n+1)⁶ 6^(n+1)/(n+1)! * n!/n⁶ 6ⁿ
= (n+1)⁶/6n⁶ * 6/((n+1)(n)(n-1)(n-2)(n-3)(n-4))
= [(n+1)/n]⁶ * 6/[(n+1)(n)(n-1)(n-2)(n-3)(n-4)]
As n approaches infinity, the first term in the product approaches 1, and the second term approaches 0. Therefore, the limit of the ratio of consecutive terms is 0.
Since the limit of the ratio of consecutive terms is less than 1, the series is convergent by the Ratio Test.
The value of the limit lim n--> [infinity] |an + 1|/|an| is 0, as we found in the previous calculation. This limit represents the rate at which the terms of the series Σ n=1 (-1)ⁿ + 1 n⁶ 6ⁿ/n! approach zero as n approaches infinity.
Since the limit is 0, the terms of the series approach zero very quickly as n becomes large, which supports the conclusion that the series is convergent.
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There is a 15% increase in tuition at UT for next fall. If the current tuition is $3,500 per semester, which equation could be used to find x, the new tuition for the fall? A. 0.15 • 3500 = x B. 1.15 • 3500 = xC. 0.85 • 3500 = x D. (15/100) = (x/3500)
The new tuition for the fall will be $4,025 per semester. This is exactly what we get by using equation B, since: 1.15 • 3500 = 4025
The correct equation to find the new tuition for the fall is: B. 1.15 • 3500 = x
Here's why: The problem states that there is a 15% increase in tuition, which means that the new tuition will be the current tuition plus 15% of the current tuition. Mathematically, we can represent this as:
new tuition = current tuition + 15% of current tuition
Using x to represent the new tuition, and 3500 to represent the current tuition, we can write this equation as:
x = 3500 + 0.15(3500)
Simplifying the right side, we get:
x = 3500 + 525
x = 4025
Therefore, the new tuition for the fall will be $4,025 per semester. This is exactly what we get by using equation B, since: 1.15 • 3500 = 4025
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Solve for x.
15
5
Set up the proportion.
[?]
XI5
X
The calculated exact value of x in the right triangle is 5√5
Finding the exact value of x in the right triangleFrom the question, we have the following parameters that can be used in our computation:
Similar triangles
Using the theorem of corresponding sides in similar triangles, we have the following proportion
x/15 = 5/x
When both sides of the equation are cross multplied, we have
x^2 = 75
Take the square root of both sides
So, we have
x = 5√5
Hence, the exact value of x is 5√5
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part (b) would you prefer as an estimate of the effect of the law on women's wages? Why? 4. Least Squares Estimator and Measurement Errors Consider a simple bivariate regression model: Yi = Bo + 91 11 + Ui, (1) where {Yi, Ili} are I.I.D. draws from their joint distribution, and both have non-zero finite fourth moments. (a) Recall that the least squares estimator is given by (1-7)(y-7) (2) EL (XL-7) 2 what sense the OLS stimator linear? Given your definition, show that (2) indeed linear. (b) Using expression (2), derive conditions for the OLS estimator 2 to be unbiased. (c) Suppose you do not have access to X1i; and instead observe xii, which is measured with an error, i.e., zmi = Xii+Vli, where vli is a measurement error. Derive a bias of the OLS estimator when instead of the true model (1) you are running a model with xt. (d) Evaluate these statements: "Measurement error in the r's is a serious problem. Measurement error in y is not." 5. Paper: Acemoglu, Johnson and Robinson
The bias can be corrected by using instrumental variables, which are correlated with the true value of the independent variable but uncorrelated with the measurement error.
The OLS estimator is linear because it satisfies the superposition principle.
To show that equation (2) is linear, we can write it in the form of a linear equation:
β1 = ∑(Xi - x)(Yi - y) / ∑(Xi - x)²
where β1 is the estimated slope coefficient.
To derive conditions for the OLS estimator to be unbiased, we need to assume that the error term Ui has a zero mean, constant variance, and is uncorrelated with the independent variable X1i. Under these assumptions, the OLS estimator is unbiased if and only if the expected value of the error term is zero.
Suppose we do not have access to the true independent variable X1i and instead observe a measured variable xi, which is subject to measurement error.
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If n is a positive integer divisible by 7, and if n < 70, what is the greatest possible value of n?
Answer:
63
Step-by-step explanation:
We can find the other multiples of a number by adding or subtracting the number.
n is less than 70 and is divisible by 7, which means we need to subtract 7 from 70.
70 - 7 = 63
2. You are interested in the effect that government directed innovation activities has on patenting. After some time spent at the National Archives, you learn about a program during the Vietnam War that gave qualifying applicants the opportunity to work at the US Government Office of Research as their service during the war as opposed to being drafted for combat. Out of the qualifying candidates, the government randomly selected who would be offered to work at the office of research
Based on the information provided, it appears that the government directed innovation activities during the Vietnam War had a unique program that gave qualifying applicants the opportunity to work at the US Government Office of Research instead of being drafted for combat. This program had a specific goal, which may have been to support innovation efforts during the war or to provide an alternative service opportunity for those who did not want to fight in combat.
Overall, this program is an interesting example of how the government can direct innovation activities through specific programs and initiatives. It also demonstrates how random selection can be used to determine who participates in these programs, which may have implications for how successful they are in achieving their goals.
In this scenario, the government implemented a program during the Vietnam War that allowed qualifying applicants to work at the US Government Office of Research as an alternative to being drafted for combat. The program's objective was to promote innovation and increase patenting by utilizing the skills and expertise of these individuals. The government randomly selected the candidates who would be offered the opportunity to work in the Office of Research, ensuring a fair selection process. This program highlights the potential impact of government-directed innovation activities on patenting and showcases how the government can actively support and encourage advancements in technology and research.
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About 24 randomly selected people were asked how long they slept at night. The mean time was 6 hours, and the standard deviation was 1.3 hour. Calculate the 80% confidence interval of the mean time by assuming that the variable is normally distributed. Provide only the value required below. Express your answer in 3 decimal places.
The 80% confidence interval for the meantime is 5.660 to 6.340 hours.
To calculate the 80% confidence interval for the meantime, we will use the following formula:
CI = Mean ± (Z-score * (Standard deviation / √Sample size))
Here, Mean = 6 hours, Standard deviation = 1.3 hours, and Sample size = 24.
For an 80% confidence interval, the Z-score is 1.282 (from the standard normal distribution table).
Now, plug in the values:
CI = 6 ± (1.282 * (1.3 / √24))
CI = 6 ± (1.282 * (1.3 / 4.899))
CI = 6 ± (1.282 * 0.265)
CI = 6 ± 0.340
The 80% confidence interval for the meantime is 5.660 to 6.340 hours.
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Find the test statistic t0 for a sample with n = 10, = 7.9, s = 1.3, and ifH1:µ > 8.0. Round your answer to three decimal places.
The test statistic t0 is -0.243. This can be answered by the concept of Standard deviation.
To find the test statistic t0, you can use the following formula:
t0 = (x - µ) / (s / √n)
where x is the sample mean, µ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.
Plugging in the given values:
t0 = (7.9 - 8.0) / (1.3 / √10)
t0 = (-0.1) / (1.3 / 3.162)
t0 = -0.1 / 0.411
t0 = -0.243
After rounding to three decimal places, the test statistic t0 is -0.243.
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Why does the mean value theorem not apply to the function on the interval 0 6?
The mean value theorem states that if a function is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in (a,b) where the derivative of the function at c is equal to the average rate of change of the function over [a,b].
However, the mean value theorem does not apply to a function on the interval [0,6] if the function is not continuous on this interval or if it is not differentiable on the open interval (0,6). Therefore, it is possible that the function does not satisfy the conditions required for the mean value theorem to hold on the interval [0,6].
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2÷/9=3 therefore 27÷0.9=30, TRUE OR FALSE? 5TH GRADE Question
Answer:
True
Step-by-step explanation:
27 = 3 x 9
0.9 = 9 x 0.1
27 ÷ 0.9
= 27/0.9
= (3 x 9)/(9 x 0.1)
= 3/0.1
= 3/0.1 x 10/10
= (3 x 10)/(0.1 x 10)
= 30/1
= 30
(5 points) Find the sum of the following infinite series. If it is divergent, type "Diverges" or "D". 11 +2 + 4 11 + 8 121 + ... Sum: Preview My Answers Submit Answers
The sum of the given infinite series 11 +2 + 4/11 + 8/121 + ... is 121.
A geometric series is a sequence of numbers in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero constant called the common ratio.
The given series is a geometric series with the first term being 11 and the common ratio being 2/11. Thus, the sum of the infinite series is given by
S = a / (1 - r)
where a is the first term and r is the common ratio.
Substituting the values, we get
S = 11 / (1 - 2/11) = 11 / (9/11) = 121
Hence, the sum of the given infinite series is 121.
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The given question is incomplete, the complete question is:
Find the sum of the following infinite series. If it is divergent, type "Diverges" or "D". 11 +2 + 4/11 + 8/121 + ...
an investigator anticipates that the proportion of red blossoms in his hybrid plants is 0.15. a random sample of 50 of his plants indicated that 22% of the blossoms were red. assuming that the proportion of red blossoms is .15, the standard deviation of the sampling distribution of the sample proportion is approximately group of answer choices 0.116 0.051 0.059 0.07
The standard deviation of the sampling distribution of the sample proportion can be calculated using the formula: σp = sqrt[p(1-p)/n]
where p is the expected proportion of red blossoms in the hybrid plants, n is the sample size, and sqrt represents the square root.
Here, p = 0.15, n = 50, and the sample proportion of red blossoms is 0.22.
So, the standard deviation of the sampling distribution of the sample proportion is:
σp = sqrt[(0.15)(1-0.15)/50]
= sqrt[(0.1275)/50]
= 0.051
Therefore, the standard deviation of the sampling distribution of the sample proportion is approximately 0.051.
Answer options B, 0.051, is the correct answer.
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(Translations LU)
Use the graph to answer the question.
-6 -5
A'
-4
В'
-3
D'
-2 -1
A
0
C'
4
-5
-6
B
2
D
Determine the translation used to create the image.
3
4
C
5p
The translation used to create the image on the graph is (5, -4), which means that all points on the original figure have been moved 5 units to the right and 4 units down.
What is graph?A graph is a visual representation of data or information, often displayed on a coordinate system with points or lines indicating the values or relationships between variables. It can be used to show trends, patterns, and comparisons.
What is translation?Translation is a type of transformation in geometry that involves moving an object or shape from one position to another without changing its size, shape, or orientation. It is also known as a slide.
According to the given information:
Based on the graph, we can see that points A have been translated 5 units to the right and 4 units down to create point A'. This means that a translation of (5, -4) was used to create the image.
Similarly, point B has been translated 5 units to the right and 6 units up to create point B', so a translation of (5, 6) was used. Point C has been translated 5 units to the right and 3 units up to create point C', so a translation of (5, 3) was used. Finally, point D has been translated 5 units to the right and 4 units down to create point D', so a translation of (5, -4) was used again.
Therefore, the translation used to create the image is (5, -4).
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a triangle has sides of length 3 inches, 5 inches, and 6 inches. is the triangle a right triangle? explain how you know.
Answer: no
Step-by-step explanation: ITS not because 25 36 are not the same size.
How is discriminate validity estimated according to the MTMM Matrix?
By examining the diagonal of the matrix, we can compare the correlations between different constructs measured using the same method to the correlations between the same construct measured using different methods.
The MTMM matrix, or the Multi-Trait Multi-Method matrix, is a popular tool used in psychology and social sciences to assess the validity of measurements.
To understand how discriminate validity is estimated using the MTMM matrix, let's first explore what the matrix is. The MTMM matrix is a table that displays the correlations between multiple traits (constructs) and multiple methods of measuring these traits.
Now, to estimate discriminate validity, we need to examine the diagonal of the matrix. The diagonal represents the correlations between each construct and the same method of measurement. For example, the correlation between intelligence measured using self-report and intelligence measured using objective tests.
Discriminate validity can be assessed by comparing these correlations to the correlations between different constructs measured using the same method.
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A carnival performer claims to be able to guess a persons weight within 4 pounds of their actual weight or the person wins a prize. If the person weighs 142 pounds, which equations can be used to find the minimum and maximum weights the performer can guess without the person winning a prize.
a) |x-142|=4
b) |x-4|=142
c) |x+4|=142
d) |x+142|=4
Option (1) is correct. The minimum weight an interpreter can guess without winning a prize is 138 pounds and the maximum weight is 146 pounds.
What do you mean by linear equation?
Linear equations are first-order equations. Linear equations are defined for the lines of the coordinate system. If an equation has a homogeneous variable of degree 1 (that is only one variable), it is called a linear equation in one variable. A linear equation can have more than one variable.
The correct equation would be: a) |x - 142| ≤ 4
Explanation:
The carnival performer claims to be able to guess a person's weight within 4 kilograms of their actual weight, which means that the difference between the guessed weight and the actual weight must not be more than 4 kilograms.
Let x be the weight of the carnival performer. In this case the difference between x and 142 cannot be more than 4 pounds. It can be written as:
|x - 142| ≤ 4
This inequality means that x can be any weight within 4 pounds of 142, giving the range:
142 - 4 ≤ x ≤ 142 4
138 ≤ x ≤ 146
Therefore, the minimum weight an interpreter can guess without winning a prize is 138 pounds and the maximum weight is 146 pounds.
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Find the mean of the data summarized in the given frequency distribution. Daily Low Temperature (F) Frequency 35-39 1 40-44 3 45-49 5 50-54 11 55-59 7 60-64 7 65-69 1
The mean of the data summarized in the given frequency distribution is approximately 53.43°F.
To find the mean of the data summarized in the given frequency distribution, we'll first determine the midpoint of each interval and then multiply it by the respective frequency. Finally, we'll add these products together and divide by the total frequency.
1. Determine the midpoints of each interval:
35-39: 37
40-44: 42
45-49: 47
50-54: 52
55-59: 57
60-64: 62
65-69: 67
2. Multiply each midpoint by its frequency:
37 × 1 = 37
42 × 3 = 126
47 × 5 = 235
52 × 11 = 572
57 × 7 = 399
62 × 7 = 434
67 × 1 = 67
3. Add these products together:
37 + 126 + 235 + 572 + 399 + 434 + 67 = 1870
4. Divide the sum by the total frequency (1 + 3 + 5 + 11 + 7 + 7 + 1 = 35):
1870 ÷ 35 = 53.43
The mean of the data summarized in the given frequency distribution is approximately 53.43°F.
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We consider salaries of 45 college graduates who took a statistics course in college. Based on these data we have a sample variance of $25,150. Find 99% upper confidence bound for σ2. Let and
The 99% upper confidence bound for σ2 is $16,751.57.
To find the 99% upper confidence bound for σ2, we can use the chi-square distribution with n-1 degrees of freedom, where n is the sample size (in this case, n=45). The upper confidence bound can be found using the formula:
Upper Confidence Bound = (n-1) × sample variance / chi-square value
We need to find the chi-square value that corresponds to a 99% confidence level and n-1 degrees of freedom. From the chi-square distribution table, we can see that the value is 67.505.
Substituting the values, we get:
Upper Confidence Bound = (45-1) × 25,150 / 67.505
= 16,751.57
Therefore, the 99% upper confidence bound for σ2 is $16,751.57.
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5. (16 marks) It is given that the moment generating function of a negative binomial random variable is mx(t)= (1 - p)^r /(1 - pe^t)^r where p and r are the parameters. Find the expected value and variance using the moment generating function.
To find the expected value and variance of a negative binomial random variable with moment generating function mx(t) = (1 - p)^r / (1 - pe^t)^r, we need to use the following formulas:
The nth moment of a random variable is given by mx^(n)(0), the nth derivative of the moment generating function evaluated at t = 0.
The expected value of a random variable is given by mx^(1)(0).
The variance of a random variable is given by mx^(2)(0) - [mx^(1)(0)]^2.
Using these formulas, we can find the expected value and variance of the negative binomial random variable.
First, let's find the first two derivatives of the moment generating function:
mx'(t) = r(1-p)^rpe^t / (1-pe^t)^(r+1)
mx''(t) = r(1-p)^rpe^t(r+1-pe^t) / (1-pe^t)^(r+2)
Now we can evaluate the moment generating function and its derivatives at t = 0 to find the expected value and variance:
mx(0) = (1 - p)^r / (1 - p)^r = 1
mx'(0) = r(1-p)^r p / (1-p)^(r+1)
mx''(0) = r(r+1)(1-p)^r p / (1-p)^(r+2) + r(1-p)^r p / (1-p)^(r+1)
Using the formulas for the expected value and variance, we have:
E[X] = mx'(0) = r(1-p) / p
Var[X] = mx''(0) - [mx'(0)]^2 = r(1-p) / p^2
Therefore, the expected value of the negative binomial random variable is E[X] = r(1-p) / p, and the variance is Var[X] = r(1-p) / p^2, where p and r are the parameters of the distribution.
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Use the normal approximation to the binomial to find that probability for the specific value of X.
n = 30, p = 0.7, X = 22
The value for z is 0.45
What are binomial words?
binomial. noun. bi no mi al b-n-m-l.: an equation consisting of a pair of terms joined by a plus or minus sign.: a biological species description consisting of 2 terms pursuant to the binomial nomenclature system.
To use the typical approximation we must first determine the binomial distribution's mean and standard deviation. The mean of a distribution that is bin is = np, while the standard deviations is = sqrt(np(1-p)). With n equals thirty & a p value 0.7, we get:
= np = 30(0.7) = 21 = [tex]\sqrt{(np(1-p)}[/tex] =[tex]\sqrt{(30(0.7)(1-0.7)}[/tex] = 2.24 = sqrt(30(0.7)(1-0.7)) = 2.24
Following that, we standardise X = 22 utilising the following equation:
z = (X - μ) / σ
With X = 22, X = 21, and X = 2.24, we obtain:
z = (22 - 21) / 2.24 = 0.45
Finally, we employ a conventional normal distribution tables (or calculator) to
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Find the perimeter of the following polygon. Be sure to include the correct unit in your answer.
12ft
16ft
9ft
9ft
15ft
The perimeter of the polygon is 61 feet.
What is polygon?A polygon is a two-dimensional geometric shape that is defined as a closed plane figure with three or more straight sides and angles. It is a flat shape made up of line segments that are connected end to end to form a closed shape.
According to given information:The perimeter of a polygon is the total length of its sides. To find the perimeter of a polygon, we need to add up the lengths of all its sides.
In this problem, we are given the lengths of the sides of a polygon in feet: 12ft, 16ft, 9ft, 9ft, and 15ft.
To find the perimeter, we simply add up these lengths:
Perimeter = 12ft + 16ft + 9ft + 9ft + 15ft
Perimeter = 61ft
Therefore, the perimeter of the polygon is 61 feet. Note that the unit for the answer is feet, since all the lengths were given in feet.
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Evaluate the following limits :
x→0lim( xe 2+x −e 2)
The limit of the given function as x approaches 0 is e².
The given limit is:
lim(x→0) (xe²⁺ˣ - e²)
To evaluate this limit, we can use algebraic manipulation and basic limit rules. First, we can factor out e² from the expression:
lim(x→0) (xe²⁺ˣ - e²) = lim(x→0) e²(xeˣ - 1)
Next, we can use the fact that the limit of a product is the product of the limits, as long as both limits exist:
lim(x→0) e²(xeˣ - 1) = lim(x→0) e² x lim(x→0) (xeˣ - 1)
The limit of e² as x approaches 0 is simply e², so we can evaluate the second limit:
lim(x→0) (xeˣ - 1) = lim(x→0) [(eˣ - 1)/x] x x = lim(x→0) (eˣ - 1)/1 = lim(x→0) (eˣ - 1)
We can use L'Hôpital's rule to evaluate this limit:
lim(x→0) (eˣ - 1) / x = lim(x→0) eˣ / 1 = e⁰ = 1
Therefore, the original limit is:
lim(x→0) (xe²⁺ˣ - e²) = lim(x→0) e² x lim(x→0) (xeˣ - 1) = e² x 1 = e²
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a sack contains n unbiased coins. among them, n-1 coins are normal (i.e., head on one side andtail on the other side), and one coin is fake, having heads on both sides. you pick a coin,uniformly at random, from the sack and flip it twice. you get heads both times. what is theconditional probability that you picked the fake coin?
The conditional probability that you picked the fake coin given that you got heads twice is 4/(3n-1).
Let E be the case where you choose the bogus coin and F be the case where you got heads twice. We wish to calculate P(E|F), which is the likelihood that you chose the fake coin given that you received heads twice.
By Bayes' theorem, we have:
P(E|F) = P(F|E)P(E) / P(F)
We can calculate each term on the right-hand side as follows:
P(F|E) = 1, Because the fake coin contains heads on both sides and always results in two heads when flipped.
P(E) = 1/n, since there is only one fake coin among n coins.
P(F) = P(F|E)P(E) + P(F|not E)P(not E), where not E is the event that you picked a normal coin. We can calculate:
P(F|not E) = (n-1) * (1/2)^2 = (n-1)/4, Because each normal coin has a 50% chance of revealing heads on each given flip and there are n-1 normal coins
P(not E) = (n-1)/n, since there are n-1 normal coins among n coins.
Therefore, we have:
P(F) = 1 * (1/n) + (n-1)/4 * (n-1)/n = (3n-1)/(4n)
Substituting these values into Bayes' theorem, we get:
P(E|F) = 1 * (1/n) / ((3n-1)/(4n)) = 4/(3n-1)
Thus, the conditional probability that you picked the fake coin given that you got heads twice is 4/(3n-1).
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