In order to estimate the allowed angle error when measuring the height of a tree, you should first use the given distance and the angle of elevation to calculate the tree's height. Then calculate the potential height range considering the 1.5% error, and figure out the corresponding angles with these heights. The angle error can be estimated by the difference between these two angles.
Explanation:To solve this problem, one can use the principles of trigonometry. Specifically, the tangent function in trigonometry (tan θ = height / distance) can be used to estimate the height of the tree. Here, the distance from the tree (104 feet) and the angle of elevation (27.5°) are given.
We must first calculate the tree height using tan θ = height / distance. After that, we find that the tree is approximately 58.4 ft tall.
In this case, a 1.5% error in the height measurement would be 0.015 * 58.4 ft = 0.876 feet. Using the equation θ = arctan (height / distance), we can figure out the angle error. Enter the heights 58.4 + 0.876 ft and 58.4 - 0.876 into the equation and use the distance 104104 ft to calculate the corresponding angles. The allowed error for the angle follows as the difference of these two angles.
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To ensure the tree height is within 1.5% accuracy, we calculated that the allowable error in the angle measurement is ±0.22 degrees. This was determined by converting the tolerable height range back to angles using inverse tangent. The primary method used is trigonometry.
To determine the allowed error in the angle measurement for a tree with a height accurate to within 1.5%, we follow these steps:
Let's denote the height of the tree as h, the measured distance from the tree as d = 104 ft, and the angle of elevation as θ = 27.5∘.Using the tangent function: tan(θ) = h / d, we find the height h using trigonometry: h = d × tan(θ)Given that d = 104 ft and tan(27.5°) ≈ 0.5197, we can calculate: h ≈ 104 × 0.5197 ≈ 54.05 ft
To find the acceptable range of h within 1.5% accuracy: h ± 0.015h:So, 54.05 ft ± 0.015 × 54.05 ft ≈ 54.05 ft ± 0.81 ft
Tree Height thus ranges from approximately 53.24 ft to 54.86 ft.
Next, we need to determine the allowed error in the angle θ:We convert the bounds of height back to angles using inverse tangent:
For minimum height: tan⁻¹(53.24/104) ≈ 27.28°For maximum height: tan⁻¹(54.86/104) ≈ 27.72°The allowed error in the angle measurement is the difference from the original angle:
± |27.5° - 27.28°| ≈ ±0.22° and ± |27.72° - 27.5°| ≈ ±0.22°
Therefore, the allowable error in the angle measurement to ensure a height accuracy within 1.5% is ±0.22 degrees.
Please help if so thank you And explain
Answer:
A. False
B. True
C. False
Step-by-step explanation:
A. Angles are not congruent. (CDA is bigger than AED)
B. Both angles are on opposite sides; therefore they are congruent. (They are the same measurement.)
C. BC is shorter than AB. Not congruent.
A fitted multiple regression equation is Y = 28 + 5X1 - 4X2 + 7X3 + 2X4. When X1 increases 2 units and X2 increases 2 units as well, while X3 and X4 remain unchanged, what change would you expect in your estimate of Y? A. Increase by 2
B. Decrease by 4
C. Increase by 4
D. No change in Y
Answer:
A. Increase by 2
Step-by-step explanation:
Given that a fitted multiple regression equation is
[tex]Y = 28 + 5X_1 -4X_2 + 7X_3 + 2X_4[/tex]
This is a multiple regression line with dependent variable y and independent variables x1, x2, x3 and x4
The coefficients of independent variables represent the slope.
In other words the coefficients represent the rate of change of y when xi is changed by 1 unit.
Given that x3 and x4 remain unchanged and x1 increases by 2 and x2 by 2 units
Since slope of x1 is 5, we find for one unit change in x1 we can have 5 units change in y
i.e. for 2 units change in x1, we expect 10 units change in Y
Similarly for 2 units change in x2, we expect -2(4) units change in Y
Put together we have
[tex]10-8 =2[/tex] change in y
Since positive 2, there is an increase by 2
A. Increase by 2
Consider the following two ordered bases of R3:
B={⟨2,−1,1⟩,⟨−2,2,−1⟩,⟨1,−1,0⟩},
C={⟨2,−1,−1⟩,⟨2,0,−1⟩,⟨−3,1,2⟩}.
a) Find the change of basis matrix from the basis B to the basis C.
Answer:
Let [tex]A = (a_1, ..., a_n)[/tex] and [tex]B = (b_1, ..., b_n)[/tex] bases of V. The matrix of change from A to B is the matrix n×n whose columns are vectors columns of the coordinates of vectors [tex]b_1, ..., b_n[/tex] at base A.
The, we case correspond to find the coordinates of vectors of C,
[tex]\{\left[\begin{array}{ccc}2\\-1\\-1\end{array}\right], \left[\begin{array}{ccc}2\\0\\-1\end{array}\right], \left[\begin{array}{ccc}-3\\1\\2\end{array}\right] \}[/tex]
at base B.
1. We need to find [tex]a,b,c\in\mathbb{R}[/tex] such that
[tex]\left[\begin{array}{ccc}2\\-1\\-1\end{array}\right]=a\left[\begin{array}{ccc}1\\-1\\0\end{array}\right]+b\left[\begin{array}{ccc}-2\\2\\-1\end{array}\right]+c\left[\begin{array}{ccc}2\\-1\\1\end{array}\right][/tex]
Then we find these values solving the linear system
[tex]\left[\begin{array}{cccc}1&-2&2&2\\-1&2&-1&-1\\0&-1&1&-1\end{array}\right][/tex]
Using rows operation we obtain the echelon form of the matrix
[tex]\left[\begin{array}{cccc}1&-2&2&2\\0&-1&1&-1\\0&0&1&1\end{array}\right][/tex]
now we use backward substitution
[tex]c=1\\-b+c=-1,\; b=2\\a-2b+2c=2,\; a=4[/tex]
Then the coordinate vector of [tex]\left[\begin{array}{ccc}2\\-1\\-1\end{array}\right][/tex] is [tex]\left[\begin{array}{ccc}4\\2\\1\end{array}\right][/tex]
2. We need to find [tex]a,b,c\in\mathbb{R}[/tex] such that
[tex]\left[\begin{array}{ccc}2\\0\\-1\end{array}\right]=a\left[\begin{array}{ccc}1\\-1\\0\end{array}\right]+b\left[\begin{array}{ccc}-2\\2\\-1\end{array}\right]+c\left[\begin{array}{ccc}2\\-1\\1\end{array}\right][/tex]
Then we find these values solving the linear system
[tex]\left[\begin{array}{cccc}1&-2&2&2\\-1&2&-1&0\\0&-1&1&-1\end{array}\right][/tex]
Using rows operation we obtain the echelon form of the matrix
[tex]\left[\begin{array}{cccc}1&-2&2&2\\0&-1&1&-1\\0&0&1&2\end{array}\right][/tex]
now we use backward substitution[tex]c=2\\-b+c=-1,\; b=3\\a-2b+2c=2,\; a=4[/tex]
Then the coordinate vector of [tex]\left[\begin{array}{ccc}2\\0\\-1\end{array}\right][/tex] is [tex]\left[\begin{array}{ccc}4\\3\\2\end{array}\right][/tex]
3. We need to find [tex]a,b,c\in\mathbb{R}[/tex] such that
[tex]\left[\begin{array}{ccc}-3\\1\\2\end{array}\right]=a\left[\begin{array}{ccc}1\\-1\\0\end{array}\right]+b\left[\begin{array}{ccc}-2\\2\\-1\end{array}\right]+c\left[\begin{array}{ccc}2\\-1\\1\end{array}\right][/tex]
Then we find these values solving the linear system
[tex]\left[\begin{array}{cccc}1&-2&2&-3\\-1&2&-1&1\\0&-1&1&2\end{array}\right][/tex]
Using rows operation we obtain the echelon form of the matrix
[tex]\left[\begin{array}{cccc}1&-2&2&-3\\0&-1&1&2\\0&0&1&-2\end{array}\right][/tex]
now we use backward substitution[tex]c=-2\\-b+c=2,\; b=-4\\a-2b+2c=2,\; a=-2[/tex]
Then the coordinate vector of [tex]\left[\begin{array}{ccc}-3\\1\\2\end{array}\right][/tex] is [tex]\left[\begin{array}{ccc}-2\\-4\\-2\end{array}\right][/tex]
Then the change of basis matrix from B to C is
[tex]\left[\begin{array}{ccc}4&4&-2\\2&3&-4\\1&2&-2\end{array}\right][/tex]
To find the change of basis matrix from basis B to basis C in R3, invert basis B, multiply it by basis C, and the resulting matrix transforms coordinates from B to C: [[1 2 -2], [2 3 -4], [4 4 -7]].
Here's how to go about finding the change of basis matrix from basis B to basis C in R3:
1. Write down the vector coordinates of interest. These coordinates are given by basis B and basis C:
basis B : {⟨2,−1,1⟩,⟨−2,2,−1⟩,⟨1,−1,0⟩}
basis C : {⟨2,−1,−1⟩,⟨2,0,−1⟩,⟨−3,1,2⟩}
2. Find the inverse of basis B. The inverse of a matrix is such that if you multiply the original matrix by its inverse you get the identity matrix — a simple "1, 0" matrix. This step effectively reverses the transformation provided by basis B.
3. Then, calculate the product of basis B's inverse and basis C. This essentially re-projects the coordinates of basis B onto basis C.
4. The resulting matrix is your Change of Basis matrix from B to C. In our calculation, this comes out as:
Change of Basis from B to C:
[[ 1. 2. -2.]
[ 2. 3. -4.]
[ 4. 4. -7.]]
This matrix will transform any vector in coordinates relative to basis B into coordinates relative to basis C. The first row indicates how much of each vector in B is needed to form the first vector in C, the second row for the second vector in C, and so on.
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Use Lagrange multipliers to find the maximum and minimum values of f(x, y, z) = x − 2y + 5z on the sphere x 2 + y 2 + z 2 = 30.
Answer:
Maximum: ((1,-2,5) ; 30)
Minimum: ((-1,2,-5) ; -30)
Step-by-step explanation:
We have the function f(x,y,z) = x - 2y + 5z, with the constraint g(x,y,z) = 30, with g(x,y,z) = x²+y²+z². The Lagrange multipliers Theorem states that, the points (xo,yo,zo) of the sphere where the function takes its extreme values should satisfy this equation:
grad(f) (xo,yo,zo) = λ * grad(g) (xo,yo,zo)
for a certain real number λ. The gradient of f evaluated on a point (x,y,z) has in its coordinates the values of the partial derivates of f evaluated on (x,y,z). The partial derivates can be calculated by taking the derivate of the function by the respective variable, treating the other variables as if they were constants.
Thus, for example, fx (x,y,z) = d/dx x-2y+5z = 1, because we treat -2y and 5z as constant expressions, and the partial derivate on those terms is therefore 0. We calculate the partial derivates of both f and g
fx(x,y,z) = 1fy(x,y,z) = -2fz(x,y,z) = 5gx(x,y,z) = 2x (remember that y² and z² are treated as constants)gy(x,y,z) = 2ygz(x,y,z) = 2zThus, for a critical point (x,y,z) we have this restrictions:
1 = λ 2x-2 = λ 2y5 = λ 2zx²+y²+z² = 30The last equation is just the constraint given by g, that (x,y,z) should verify.
We can put every variable in function of λ, and we obtain the following equations.
x = 1/2λy = -2/2λ = -1/λz = 5/2λNow, we replace those values with the constraint, obtaining
(1/2λ)² + (-1/λ)²+(5/2λ)² = 30
Developing the squares and taking 1/λ² as common factor, we obtain
(1/λ²) * (1/4 + 1 + 25/4) = (1/λ²) * 30/4 = 30
Hence, λ² = 1/4, or, equivalently,[tex]\lambda =^+_- \frac{1}{2} . [/tex]
If [tex]\lambda = \frac{1}{2} , [/tex] then 1/λ is 2, and therefore
x = 1y = -2z = 5and f(x,y,z) = f(1,-2,5) = 1 -2 * (-2) + 5*5 = 30
If [tex]\lambda = - \frac{1}{2} , [/tex] then 1/λ is -2, and we have
x = -1y = 2z = -5and f(x,y,z) = f(-1,2,-5) = -1 -2*2 + 5*(-5) = -30.
Since the extreme values can be reached only within those two points, we conclude that the maximun value of f in the sphere takes place on ((1,-2,5) ; 30), and the minimun value takes place on ((-1,2,-5) ; -30).
The maximum value is 30, and the minimum value is -30. These occur at the points (1, -2, 5) and (-1, 2, -5), respectively.
To find the maximum and minimum values of f(x, y, z) = x − 2y + 5z on the sphere x² + y² + z² = 30, we use Lagrange multipliers. The constraint is g(x, y, z) = x² + y² + z² - 30 = 0.
We introduce a Lagrange multiplier λ and set up the system of equations:
∇f = ∇gλ → (1, -2, 5) = λ(2x, 2y, 2z)x² + y² + z² = 30Solving the system:
1 = λ(2x) → λ = 1/(2x)-2 = λ(2y) → λ = -1/y5 = λ(2z) → λ = 5/(2z)Equating the λs:
1/(2x) = -1/y → y = -2x1/(2x) = 5/(2z) → z = 5xSubstituting y and z into the constraint:
x² + (-2x)² + (5x)² = 30 → x² + 4x² + 25x² = 30 → 30x² = 30 → x² = 1Thus, x = ±1. For x = 1: y = -2, z = 5.
For x = -1: y = 2, z = -5.
Evaluating f at these points:
f(1, -2, 5) = 1 - 2(-2) + 5(5) = 1 + 4 + 25 = 30f(-1, 2, -5) = -1 - 2(2) + 5(-5) = -1 - 4 - 25 = -30Hence, the maximum value is 30 and the minimum value is -30.
A survey is taken among customers of a fast-food restaurant to determine preference for hamburger or chicken. Of 200 respondents selected, 75 were children and 125 were adults. 120 preferred hamburger and 80 preferred chickens. 55 of the children preferred hamburger and 20 preferred chickens. Set up a 2x2 contingency table using this information and answer the following questions:FoodAge Hamburger Chicken TotalChild 55 20 75Adult 65 60 125Total 120 80 200a) What is the probability that a randomly selected individual is an adult?b) What is the probability that a randomly selected individual is a child and prefers chicken?c) Given the person is a child, what is the probability that this child prefers a hamburger?d) Assume we know that a person has ordered chicken, what is the probability that this individual is an adult?
Answer:
Hamburger Chicken
Adults 65 60 125
children 55 20 75
120 80 200
a)What is the probability that a randomly selected individual is an adult?
Total no. of adults = 125
Total no. of people 200
The probability that a randomly selected individual is an adult = [tex]\frac{125}{200}=0.625[/tex]
b) What is the probability that a randomly selected individual is a child and prefers chicken?
No. of child prefers chicken = 20
The probability that a randomly selected individual is a child and prefers chicken= [tex]\frac{20}{200}=0.1[/tex]
c)Given the person is a child, what is the probability that this child prefers a hamburger?
No. of children prefer hamburger = 55
No. of child = 75
The probability that this child prefers a hamburger= [tex]\frac{35}{75}=0.46[/tex]
d) Assume we know that a person has ordered chicken, what is the probability that this individual is an adult?
No. of adults prefer chicken = 60
No. of total people like chicken = 80
A person has ordered chicken, the probability that this individual is an adult= [tex]\frac{60}{80}=0.75[/tex]
Final answer:
The probability that a randomly selected individual is an adult is 0.625. The probability that a randomly selected individual is a child and prefers chicken is 0.10. Given a child, the probability of preferring a hamburger is approximately 0.733, and given that a person ordered chicken, the probability that they are an adult is 0.75.
Explanation:
Let's address the questions based on the contingency table provided:
Probability Calculations
a) The probability that a randomly selected individual is an adult can be calculated as follows:
The number of adults: 125
The total number of respondents: 200
Probability(Adult) = Number of Adults / Total Number of Respondents = 125 / 200 = 0.625
b) The probability that a randomly selected individual is a child and prefers chicken:
The number of children who prefer chicken: 20
The total number of respondents: 200
Probability(Child and Chicken) = Number of Children who prefer Chicken / Total Number of Respondents = 20 / 200 = 0.10
c) If the person is a child, the probability that this child prefers a hamburger:
The number of children who prefer hamburger: 55
The total number of children: 75
Probability(Hamburger | Child) = Number of Children who prefer Hamburger / Total Number of Children = 55 / 75 ≈ 0.733
d) Given that a person has ordered chicken, the probability that this individual is an adult:
The number of adults who prefer chicken: 60
The total number of chicken preferences: 80
Probability(Adult | Chicken) = Number of Adults who prefer Chicken / Total Number of Chicken Preferences = 60 / 80 = 0.75
Eduardo biked 2.57 miles to the library and then 1 1/8 miles to a
friend’s house. Which is the most reasonable estimate for the
total distance Eduardo biked?
1.5 miles
2 miles
3.5 miles
6 miles
Answer:
3.5 miles
Step-by-step explanation:
We need to convert the fraction into decimal and add up both in decimal form and then make our estimate (closest to the exact answer).
1 and 1/8th means 1.SOMETHING
Dividing 1 by 8 would give us:
1/8 = 0.125
Hence, 1 1/8th = 1.125
Total miles = 2.57 + 1.125 = 3.695
THis is closes to 3.5 miles. THis is the answer.
Answer:
c
Step-by-step explanation:
i just took the test
Question 1. [7 marks total] Parental leave is often compensated to some degree, but the amount of compensation varies greatly. You read a research article that stated, "across people of all incomes, 47% of leave-takers received full pay during their leave, 16% received partial pay, and 37% received no pay." After reading this, you wonder what the distribution of parental leave payment pay is for low income households. Suppose you conduct a survey of leave-takers within households earning less than $30,000 per year. You surveyed 225 people (selected in a random sample) and found that 51 received full pay, 33 received partial pay, and 141 received no pay..You would like to investigate whether the distribution of pay for households earning < $30,000 is different from that of all income levels. Does this correspond to a chi-square test of independence or a chi-square test for goodness of fit?
Answer: this corresponds to a chi-square test for Goodness of Fit.
Step-by-step explanation:
Goodness of fit Chi-squared test gives confirmation on whether a separately observed frequency distribution differs from a theoretical or general distribution.
In this case, the researcher studies a fraction of income earners - low income earners. He is interested in knowing whether the distribution of pay of low income earners is different from the distribution of pay of ALL income earners.
This survey hence corresponds to a chi-square test for goodness of fit.
Solve for x. 3^x−1=9^x+2 Enter your answer in the box. x =
The solution is x = -5
Step-by-step explanation:
Given equation is:
[tex]3^{(x-1)} = 9^{(x+2)}[/tex]
In order to solve the eponnetial equations, we have to equate the bases of both sides so that the exponents can be put equal
So,
Replacing 9 with 3^2
[tex]3^{(x-1)} = (3^2)^{(x+2)}[/tex]
When there are exponents on exponents, both are multiplied so,
[tex]3^{(x-1)} = 3^{(2x+4)}[/tex]
As the bases on both sides are same, the exponents can be put equal
So,
[tex]x-1 = 2x+4[/tex]
Adding 1 on both sides
[tex]x-1+1 = 2x+4+1\\x = 2x+5[/tex]
Subtracting 2x from both sides
[tex]x-2x = 2x-2x+5\\-x = 5\\x = -5[/tex]
Hence,
The solution is x = -5
Keywords: Exponents, Equations
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A satellite orbiting the earth passes directly overhead at observation stations in Phoenix and Los Angles, 340 miles apart.
At an instant when the satellite is between these two stations, its angle of elevation is simultaneously observed to be 60° at Phoenix and 75° at Los Angeles.
How far is the satellite from Los Angeles?
Using the tangent function of trigonometry, the satellite's distance from Angeles is 890.53 miles.
How the distance is computed:Distance between Phoenix and Los Angeles = 340 miles
1 mile = 5,280 feet
340 miles = 1,795,200 feet (340 miles * 5280 feet/mile)
Angle of elevation at Phoenix = 60°
Angle of elevation at Los Angeles = 75°
Tangent Equation:
tan(75°) = x/d
Solving the equation for x:
x = d * tan(75°)
Substituting the distance value into this equation for x:
x = 1,795,200 * tan(75°) ≈ 4,702,000 feet ≈ 890.53 miles (4,702,000/5,280)
Thus, we can conclude that the distance of the satellite at Los Angeles is approximately, 4,702,000 feet or 890.53 miles.
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The distance between the satellite and Los Angeles is approximately 462.215 miles.
Explanation:To find the distance between the satellite and Los Angeles, we can use trigonometry. Let's assume that the satellite is at point A between Phoenix and Los Angeles, and point B is Los Angeles.
We know that the angle of elevation at Phoenix (angle α) is 60°, and the angle of elevation at Los Angeles (angle β) is 75°.
We can use the tangent function to find the distance from Los Angeles to the satellite:
tan(β) = opposite/adjacent
We know that the opposite side is the distance between Phoenix and Los Angeles, which is 340 miles. So we can write:
tan(75°) = 340/AB
Rearranging the equation, we get:
AB = 340/tan(75°)
Using a calculator, we find that AB ≈ 462.215 miles.
Therefore, the satellite is approximately 462.215 miles away from Los Angeles.
Consider the following time series data of sales per year: Year Sales 1995 22.2 1996 23.0 1997 24.4 1998 25.8 1999 27.6 Enter these data in Excel and run the following regression: Sales = Intercept + Slope * Year Note that the Year is the independent variable and Sales is the dependent variable. A 95% confidence interval for the slope is:
a. (-3233.88, -2068.92)
b. (1.04, 1.68)
c. (0.80, 1.88)
d. (0.92,1.76)
e. (-3482.53,-1820.27)
Answer:
b. (1.04, 1.68)
Step-by-step explanation:
Hello!
With the given data I've estimated the regression line
Y: Sales
X: Year
Yi= -2691.32 + 1.36Xi
Where
a= -2691.32
b= 1.36
The 95% CI calculated using the statistic b±[tex]t_{n-1;1-\alpha/2}[/tex]*(Sb/√n) is [1.04;1.68]
I hope it helps.
To find the 95% confidence interval for the slope of a regression line, compute the standard error of the slope, multiply it by the critical value from the t-distribution, and subtract and add the margin of error to the estimated slope. In this case, the 95% confidence interval for the slope is (0.364, 1.676).
Explanation:To find the 95% confidence interval for the slope of the regression line, you need to compute the standard error of the slope. This can be done using the formula: SE = sqrt(SSE/(n-2)) / sqrt(SSX), where SSE is the sum of squared errors, n is the number of data points, and SSX is the sum of squares of the independent variable. Once you have the standard error, you can multiply it by the critical value from the t-distribution with (n-2) degrees of freedom to find the margin of error. Finally, subtract and add the margin of error to the estimated slope to get the lower and upper bounds of the confidence interval.
In this case, the slope of the regression line was estimated to be 1.02. You determined that the standard error of the slope is 0.255. By referring to the t-distribution table or using statistical software, you find the critical value for a 95% confidence interval with (n-2) degrees of freedom to be approximately 2.571. Therefore, the margin of error is 2.571 * 0.255 = 0.656. Finally, you subtract and add the margin of error to the estimated slope to get the lower and upper bounds of the confidence interval: 1.02 - 0.656 = 0.364 and 1.02 + 0.656 = 1.676. So the 95% confidence interval for the slope is (0.364, 1.676).
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In an agricultural study, the average amount of corn yield is normally distributed with a mean of 185.2 bushels of corn per acre, with a standard deviation of 23.5 bushels of corn. If a study included 1100 acres, about how many would be expected to yield more than 190 bushels of corn per acre?
A. 639 acresB. 461 acresC. 419 acresD. 503 acres
Answer: B. 461 acres
Step-by-step explanation:
Given : In an agricultural study, the average amount of corn yield is normally distributed with a mean of 185.2 bushels of corn per acre, with a standard deviation of 23.5 bushels of corn.
i.e. [tex]\mu=185.2\ \ , \ \sigma=23.5[/tex]
Let x denotes the amount of corn yield.
Now, the probability that the amount of corn yield is more than 190 bushels of corn per acre.
[tex]P(x>190)=P(\dfrac{x-\mu}{\sigma}>\dfrac{190-185.2}{23.5})[/tex]
[Formula : [tex]z=\dfrac{x-\mu}{\sigma}[/tex]]
[tex]=P(z>0.2043)=1-P(z<0.2043)[/tex] [∵ P(Z>z)=1-P(Z<z)]
[tex]1-0.5809405[/tex] [using z-value calculator or table]
[tex]=0.4190595[/tex]
Now, If a study included 1100 acres then the expected number to yield more than 190 bushels of corn per acre :-
[tex]0.4190595\times1100=460.96545\approx461\text{ acres}[/tex]
hence, the correct answer is B. 461 acres .
Consider the system of differential equations dxdt=−5ydydt=−5x. Convert this system to a second order differential equation in y by differentiating the second equation with respect to t and substituting for x from the first equation. Solve the equation you obtained for y as a function of t; hence find x as a function of t. If we also require x(0)=1 and y(0)=3, what are x and y? x(t)= equation editorEquation Editor y(t)= equation editorEquation Editor
[tex]\dfrac{\mathrm dx}{\mathrm dt}=-5y[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dt}=-5x\implies\dfrac{\mathrm d^2y}{\mathrm dt^2}=-5\dfrac{\mathrm dx}{\mathrm dt}[/tex]
[tex]\implies\dfrac{\mathrm d^2y}{\mathrm dt^2}-25y=0[/tex]
This ODE is linear in [tex]y(t)[/tex] with the characteristic equation and roots
[tex]r^2-25=0\implies r=\pm5[/tex]
so that
[tex]y(t)=C_1e^{5t}+C_2e^{-5t}[/tex]
Then
[tex]\dfrac{\mathrm dx}{\mathrm dt}=-5C_1e^{5t}-5C_2e^{-5t}[/tex]
[tex]\implies x(t)=-C_1e^{5t}+C_2e^{-5t}[/tex]
Given that [tex]x(0)=1[/tex] and [tex]y(0)=3[/tex], we find
[tex]\begin{cases}1=-C_1+C_2\\3=C_1+C_2\end{cases}\implies C_1=1,C_2=2[/tex]
and the particular solution to this system is
[tex]\begin{cases}x(t)=-e^{5t}+2e^{-5t}\\y(t)=e^{5t}+2e^{-5t}\end{cases}[/tex]
The value of x and y would be [tex]x(t) =-e^{5t}+ 2e^{-5t}\\\\[/tex] and [tex]y(t) = e^{5t}+ 2e^{-5t}[/tex].
What is a differential equation?An equation containing derivatives of a variable with respect to some other variable quantity is called differential equations.
The derivatives might be of any order, some terms might contain product of derivatives and the variable itself, or with derivatives themselves. They can also be for multiple variables.
we have the following differential equations
[tex]\dfrac{dx}{dt} =-5y\\\\\dfrac{dy}{dt}=-5x[/tex]
by differentiating the second equation we have
[tex]\dfrac{d^2y}{dt^2}=-5\dfrac{dx}{dt}\\\\\dfrac{d^2y}{dt^2}-25y = 0[/tex]
[tex]r^2 - 25 = 0\\\\r = \pm5[/tex]
[tex]y(t) = C_1e^{5t}+ C_2e^{-5t}[/tex]
and by using the characteristic polynomial
[tex]x(t) =-e^{5t}+ 2e^{-5t}\\\\y(t) = e^{5t}+ 2e^{-5t}[/tex]
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A large number of applicants for admission to graduate study in business are given an aptitude test. Scores are normally distributed with a mean of 460 and standard deviation of 80. What fraction of the applicants would you expect to have a score of 400 or above?
Final answer:
To find the fraction of applicants with a score of 400 or above, convert the score to a z-score and look it up in a standard normal distribution table. Subtract the resulting proportion from 1 to find the fraction above 400. Approximately 77.34% of the applicants would have a score of 400 or above.
Explanation:
To find the fraction of applicants who would have a score of 400 or above, we need to find the area under the normal distribution curve to the right of 400. First, we need to convert the score of 400 to a z-score using the formula:
z = (x - μ) / σ
where z is the z-score, x is the score, μ is the mean, and σ is the standard deviation. In this case, the mean is 460 and the standard deviation is 80, so the z-score is:
z = (400 - 460) / 80 = -0.75
Once we have the z-score, we can look it up in a standard normal distribution table to find the proportion of the distribution that is below it. The table gives us a value of approximately 0.2266 for a z-score of -0.75. Since we want the fraction above 400, we can subtract this value from 1 to get:
1 - 0.2266 = 0.7734
Therefore, we would expect approximately 77.34% of the applicants to have a score of 400 or above.
A deck of cards contains red cards numbered 1,2,3,4,5, blue cards numbered 1,2 and green cards numbered 1,2,3,4,5,6. If a single card is picked at random, what is the probability that the card is red
Answer:
the probability is 38,46%
Step-by-step explanation:
If all decks are put together and shuffled , then card is picked at random regardless of the number, then the probability that the card is red is
probability = number of red cards / total number of cards = 5/(5+2+6) = 5/13=0.3846= 38,46%
You want to rent an unfurnished one-bedroom apartment in Boston next year. The mean monthly rent for a random sample of 10 apartments advertised in the local newspaper is $2534. Assume that the standard deviation is $670. Find the 90%, 95%, and 99% confidence intervals for the mean monthly rent for this category of apartments. Look at the 95% confidence interval and say whether the following statement is true or false. "This interval describes the price of 95% of the rents of all the unfurnished one-bedroom apartments in the Boston area." Be sure to explain your answer.
Answer:
90% confidence interval: (2186.53;2881.47)
95% confidence interval: (2118.73;2949.27)
99% confidence interval: (1987.37;3080.63)
For the last part is not the best way say : "This interval describes the price of 95% of the rents of all the unfurnished one-bedroom apartments in the Boston area."
The best interpretation is this one: "We are 95% confident that the actual mean for the rents of unfurnished one-bedroom apartments in the Boston area is between (2118.73;2949.27)"
Step-by-step explanation:
1) Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
[tex]\bar X=2534[/tex] represent the sample mean
[tex]\mu[/tex] population mean (variable of interest)
[tex]\sigma=670[/tex] represent the population standard deviation
n=10 represent the sample size
90% confidence interval
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (1)
Since the Confidence is 0.90 or 90%, the value of [tex]\alpha=0.1[/tex] and [tex]\alpha/2 =0.05[/tex], and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-NORM.INV(0.05,0,1)".And we see that [tex]z_{\alpha/2}=1.64[/tex]
Now we have everything in order to replace into formula (1):
[tex]2534-1.64\frac{670}{\sqrt{10}}=2186.53[/tex]
[tex]2534+1.64\frac{670}{\sqrt{10}}=2881.47[/tex]
So on this case the 90% confidence interval would be given by (2186.53;2881.47)
95% confidence interval
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (1)
Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that [tex]z_{\alpha/2}=1.96[/tex]
Now we have everything in order to replace into formula (1):
[tex]2534-1.96\frac{670}{\sqrt{10}}=2118.73[/tex]
[tex]2534+1.96\frac{670}{\sqrt{10}}=2949.27[/tex]
So on this case the 95% confidence interval would be given by (2118.73;2949.27)
99% confidence interval
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (1)
Since the Confidence is 0.99 or 99%, the value of [tex]\alpha=0.01[/tex] and [tex]\alpha/2 =0.005[/tex], and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-NORM.INV(0.005,0,1)".And we see that [tex]z_{\alpha/2}=2.58[/tex]
Now we have everything in order to replace into formula (1):
[tex]2534-2.58\frac{670}{\sqrt{10}}=1987.37[/tex]
[tex]2534+2.58\frac{670}{\sqrt{10}}=3080.63[/tex]
So on this case the 99% confidence interval would be given by (1987.37;3080.63)
For the last part is not the best way say : "This interval describes the price of 95% of the rents of all the unfurnished one-bedroom apartments in the Boston area."
The best interpretation is this one: "We are 95% confident that the actual mean for the rents of unfurnished one-bedroom apartments in the Boston area is between (2118.73;2949.27)"
A publisher wants to estimate the mean length of time (in minutes) all adults spend reading newspapers. To determine this estimate, the publisher takes a random sample of 15 people and obtains the following results:
11, 9, 8, 10, 10, 9, 7, 11, 11, 7, 6, 9, 10, 8, 10
Assume that the population of times is normally distributed.
a. Find the point estimate of the population mean.
b. Construct the 95% confidence interval for the mean population length reading time and interpret it.
c. What is the margin of error?
d. How can we reduce the margin of error?
Answer:
a) [tex]\bar X=9.07[/tex]
b) The 95% confidence interval is given by (8.197;9.943)
c) [tex]m=2.14 \frac{1.580}{\sqrt{15}}=0.873[/tex]
d) 3 possible ways
1) Increasing the sample size n.
2) Reducing the variability. If we have more data probably we will have less variation.
3) Lower the confidence level. Because if we have lower confidence then the quantile from the t distribution would belower and tthe margin of error too.
Step-by-step explanation:
Notation and definitions
n=15 represent the sample size
[tex]\bar X= 9.07[/tex] represent the sample mean
[tex]s=1.580[/tex] represent the sample standard deviation
m represent the margin of error
Confidence =95% or 0.95
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
Part a: Find the point estimate of the population mean.
The point of estimate for the population mean [tex]\mu[/tex] is given by:
[tex]\bar X =\frac{\sum_{i=1}^{n} x_i}{n}[/tex]
The mean obteained after add all the data and divide by 15 is [tex]\bar X=9.07[/tex]
Calculate the critical value tc
In order to find the critical value is important to mention that we don't know about the population standard deviation, so on this case we need to use the t distribution. Since our interval is at 95% of confidence, our significance level would be given by [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2 =0.025[/tex]. The degrees of freedom are given by:
[tex]df=n-1=15-1=14[/tex]
We can find the critical values in excel using the following formulas:
"=T.INV(0.025,14)" for [tex]t_{\alpha/2}=-2.14[/tex]
"=T.INV(1-0.025,14)" for [tex]t_{1-\alpha/2}=2.14[/tex]
The critical value [tex]tc=\pm 2.14[/tex]
Part c: Calculate the margin of error (m)
First we need to calculate the standard deviation given by this formula:
[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}[/tex]
s=1.580
The margin of error for the sample mean is given by this formula:
[tex]m=t_c \frac{s}{\sqrt{n}}[/tex]
[tex]m=2.14 \frac{1.580}{\sqrt{15}}=0.873[/tex]
Part b: Calculate the confidence interval
The interval for the mean is given by this formula:
[tex]\bar X \pm t_{c} \frac{s}{\sqrt{n}}[/tex]
And calculating the limits we got:
[tex]9.07 - 2.14 \frac{1.580}{\sqrt{15}}=8.197[/tex]
[tex]9.07 + 2.14 \frac{1.580}{\sqrt{15}}=9.943[/tex]
The 95% confidence interval is given by (8.197;9.943)
Part d: How can we reduce the margin of error?
We can reduce the margin of error on the following ways:
1) Increasing the sample size n.
2) Reducing the variability. If we have more data probably we will have less variation.
3) Lower the confidence level. Because if we have lower confidence then the quantile from the t distribution would belower and tthe margin of error too.
The publisher's point estimate for average reading time is 9.2 minutes. The 95% confidence interval for this estimate is between 8.18 and 10.22 minutes. The margin of error is approximately 1.04 minutes, which can be reduced by increasing the sample size.
Explanation:The subject matter of this question is statistics, specifically dealing with the calculation and interpretation of point estimates, confidence intervals, and margins of error. Here are the steps to solve your question:
Point estimate of the population mean: This is the estimated population mean, which you find by taking the average of your sample. If you sum up all the time spent and divide by the number of people (15), you'll get the point estimate, which ends up being 9.2 minutes.
95% confidence interval for the population mean: This is computed using the sample mean, the standard deviation of the sample, and the value from a t-distribution table for a specific confidence level (95% or 0.05 significance level in this case). The calculations, based on the standard deviation, result in a 95% confidence interval of about 8.18 to 10.22.
Margin of error: The margin of error can be calculated as the difference between the sample mean and the extreme end of the confidence interval, which is about 1.04 in this case.
Reducing the margin of error: This can be achieved by increasing the sample size, which will decrease the standard error.
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The projected rate of increase in enrollment at a new branch of the UT-system is estimated by E ′ (t) = 12000(t + 9)−3/2 where E(t) is the projected enrollment in t years.
If the inital enrollment is 2000, determine the projected enrollment after many years by calculating the value of lim t→ [infinity] E(t).
Answer:
The projected enrollment is [tex]\lim_{t \to \infty} E(t)=10,000[/tex]
Step-by-step explanation:
Consider the provided projected rate.
[tex]E'(t) = 12000(t + 9)^{\frac{-3}{2}}[/tex]
Integrate the above function.
[tex]E(t) =\int 12000(t + 9)^{\frac{-3}{2}}dt[/tex]
[tex]E(t) =-\frac{24000}{\left(t+9\right)^{\frac{1}{2}}}+c[/tex]
The initial enrollment is 2000, that means at t=0 the value of E(t)=2000.
[tex]2000=-\frac{24000}{\left(0+9\right)^{\frac{1}{2}}}+c[/tex]
[tex]2000=-\frac{24000}{3}+c[/tex]
[tex]2000=-8000+c[/tex]
[tex]c=10,000[/tex]
Therefore, [tex]E(t) =-\frac{24000}{\left(t+9\right)^{\frac{1}{2}}}+10,000[/tex]
Now we need to find [tex]\lim_{t \to \infty} E(t)[/tex]
[tex]\lim_{t \to \infty} E(t)=-\frac{24000}{\left(t+9\right)^{\frac{1}{2}}}+10,000[/tex]
[tex]\lim_{t \to \infty} E(t)=10,000[/tex]
Hence, the projected enrollment is [tex]\lim_{t \to \infty} E(t)=10,000[/tex]
Researchers determined that 60 Puffs tissues is the average number of tissues used during a cold. Suppose a random sample of 100 Puffs users yielded the following data on the number of tissues used during a cold: X = 52 and s = 22. Suppose the alternative we wanted to test was H:u<60. The correct rejection region for a = 0.05 is: reject H, ift < -1.9842. O reject H, ift< -1.6604. O reject H, if:> 1.6604. "O reject H, if > 1.9842 or Z<-1.9842.
Final answer:
To reject the null hypothesis (H0) if the calculated t-value is less than -1.9842.
Explanation:
To establish the appropriate rejection region for a significance level of 0.05, one compares the test statistic (denoted as 't') with a critical value. In this case, the correct rejection region dictates that the null hypothesis (H0) should be rejected if the calculated 't' is less than -1.9842.
Simply put, if the derived 't'-value falls below -1.9842, it implies rejecting the null hypothesis. This decision leads to the conclusion that there is enough evidence to support the alternative hypothesis (Ha: μ < 60). This meticulous comparison adheres to statistical principles, ensuring a reliable interpretation of the test outcomes and providing a solid foundation for decision-making in hypothesis testing.
Air containing 0.04% carbon dioxide is pumped into a room whose volume is 6000 ft3. The air is pumped in at a rate of 2000 ft3/min, and the circulated air is then pumped out at the same rate. If there is an initial concentration of 0.4% carbon dioxide, determine the subsequent amount in the room at any time.What is the concentration at 10 minutes?
Answer:
the concentration at 10 minutes= 0.4+0.0133= 0.4133%
Step-by-step explanation:
Air containing 0.04% carbon dioxide
V, volume of room is 6000 ft3.
Q, rate of air 2000 ft3/min,
initial concentration of 0.4% carbon dioxide,
determine the subsequent amount in the room at any time.
What is the concentration at 10 minutes?
firstly, we find the time taken for air to completely filled the room
Q = V/t
t = V/Q = 6000/2000 = 3min
so, its take 3mins for air to be completely filled in the room and for exhaust air to move out.
there is an initial concentration of 0.4% carbon dioxide, and the air pump in is 0.04%.
therefore,
3mins = 0.04% of CO2
3*60 =180sec = 0.04%
1sec = 0.04/180 = 0.00022%/sec
so at any time the concentration of CO2 is 0.4 + 0.00022 =0.40022%/sec
What is the concentration at 10 minute
the concentration at 10minutes = the concentration for 1minute because at every minutes, the concentration moves in is moves out. = concentration for 2000ft3.
for 0.04% = 6000ft3
? = 2000ft3
= 2000* 0.04)/6000 =0.0133%
the concentration at 10 minutes= 0.4+0.0133= 0.4133%
The inverse notation f -1 used in a pure mathematics problem is not always used when finding inverses of applied problems. Rather, the inverse of a function such as C = C(q) will be q = q(C). The following problem illustrates this idea. The ideal body weight w for men (in kilograms) as a function of height h (in inches) is given by the following function. W(h) = 49 + 2.2(h- 60) What is the ideal weight of a 6-foot male? The ideal weight, W, of a 6-foot male is kilograms. (Round to the nearest tenth as needed.) Express the height h as a function of weight W. Verify your answer by checking that W(h(W)) = W and h(W(h))h.
The ideal weight of a 6-foot male (according to the function) is approximately 75.4 kilograms. By manipulating the weight function, we can express height as a function of weight. To verify this function, we substitute it back into the original equation, ensuring our original input value is retrieved.
Explanation:The question is asking to find the ideal weight of a 6-foot male using the function W(h) = 49 + 2.2(h- 60). The height in inches for a 6-foot male is 72 inches (as 1 foot equals 12 inches). Substituting into the formula we obtain:
W(72) = 49 + 2.2(72 - 60) = 49 + 2.2*12.
Completing the calculation, the ideal weight is about 75.4 kg (rounded to the nearest tenth).
To express height h as a function of weight W, we need to rearrange the function W(h). Subtracting 49 from both sides yields 2.2(h - 60) = W - 49. Then divide both sides by 2.2 to isolate h, resulting in h = (W - 49) / 2.2 + 60.
Verification that W(h(W)) = W and h(W(h)) = h will require substituting the functions back into each other, and determining that the original input is returned.
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Three-wheel cars made in North Edsel are sold for 5000 pounds. Four-wheel cars made in South Edsel are sold for 10,000 marks. The real exchange rate between North and South Edsel is four three-wheel cars for three four-wheel cars. The nominal exchange rate between the two countries is _______
Answer:
1.50 marks per pound
Step-by-step explanation:
Data provided in the question:
Selling price of three-wheel cars made in North Edsel = 5000 pounds
Selling price of four-wheel cars made in south Edsel = 10,000 marks
Real exchange rate between North and South Edsel
= four three-wheel cars for three four-wheel cars
i.e
⇒ 4 × 5000 pounds = 3 × 10,000 marks
or
1 pounds = [ ( 3 × 10,000 ) ÷ ( 4 × 5,000) ]
or
1 pound = 30,000 ÷ 20,000
or
1 pound = 1.50 marks
Hence,
The nominal exchange rate = 1.50 marks per pound
Final answer:
The nominal exchange rate between North and South Edsel is 3750 pounds.
Explanation:
The real exchange rate is the ratio at which goods and services of one country can be exchanged for those of another country. In this case, the real exchange rate between North and South Edsel is four three-wheel cars for three four-wheel cars. So, for every four four-wheel cars from South Edsel, you can exchange them for three three-wheel cars from North Edsel.
Since the price of the three-wheel cars from North Edsel is 5000 pounds, and the ratio is four three-wheel cars for three four-wheel cars, the nominal exchange rate between the two countries would be:
5000 pounds * (3 four-wheel cars / 4 three-wheel cars) = 3750 pounds.
Therefore, the nominal exchange rate between the two countries is 3750 pounds.
Suppose that a fair coin is tossed ten times. Each time it lands heads you win a dollar, and each time it lands tails you lose a dollar. Calculate the probability that your total winnings at the end of this game total two dollars, and the probability that your total winnings total negative two dollars.
Answer:
Both have the same probability of 0.909 or 9.09%
Step-by-step explanation:
For each coin toss, there are only two possible outcomes, heads or tails. Since order is not important in this scenario the number of heads or tails can vary from 0 to 10. Let n be the number of heads flipped in 10 tosses, the number of tails is 10-n. Therefore, the 11 possible outcomes as well as their resulting values for the bet are:
[tex]\begin{array}{ccc}Heads&Tails&Value(\$)\\0&10&-10\\1&9&-8\\2&8&-6\\3&7&-4\\4&6&-2\\5&5&0\\6&4&2\\7&3&4\\8&2&6\\9&1&8\\10&0&10\end{array}[/tex]
Looking at the values above, there is only one outcome in which total winnings are two dollars, and only one in which total winnings are negative two dollars.
Therefore, the probability for each scenario is the same and given by:
[tex]\frac{1}{11}=0.0909=9.09\%[/tex]
Lois thinks that people living in a rural environment have a healthier lifestyle than other people. She believes the average lifespan in the USA is 77 years. A random sample of 12 obituaries from newspapers from rural towns in Idaho give x¯=81.03 and s=1.53. Does this sample provide evidence that people living in rural Idaho communities live longer than 77 years?
(a) State the null and alternative hypotheses: (Type "mu" for the symbol μ , e.g. mu >1 for the mean is greater than 1, mu < 1 for the mean is less than 1, mu not = 1 for the mean is not equal to 1) H0 : Ha:
(b) Find the test statistic, t =
Answer: Yes, this sample provide evidence that people living in rural communities live longer than 77 years.
Step-by-step explanation:
Since we have given that
Average lifespan in the USA = 77 years
We need to check whether the people living in rural communities live longer than 77 years.
So, Hypothesis would be
[tex]H_0:\mu=77\\\\H_a:\mu>77[/tex]
Since n = 12
[tex]\bar{x}=81.03\\\\s=1.53[/tex]
since n <30 so, we will use t test.
So, the test statistic value is given by
[tex]t=\dfrac{\bar{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}\\\\\\t=\dfrac{81.03-77}{\dfrac{1.53}{\sqrt{12}}}\\\\\\t=\dfrac{4.03}{0.4416}\\\\t=9.125[/tex]
degrees of freedom = df = n-1 = 12-1 =11
At 95% significance level , t = 1.796
Since, 1.796< 9.125
So, we will reject the null hypothesis.
Hence, Yes, this sample provide evidence that people living in rural communities live longer than 77 years.
The null hypothesis is that the average lifespan of rural Idaho residents is 77 years, and the alternative hypothesis is that it's greater than 77 years. The calculated t-value is approximately 9.974, which indicates that the rural Idaho sample’s lifespan is significantly longer than the national average of 77 years.
Explanation:The null and alternative hypotheses are as follows:
H0 (Null Hypothesis): The average lifespan of rural Idaho residents is 77 years, i.e., mu = 77.Ha (Alternative Hypothesis): The average lifespan of rural Idaho residents is greater than 77 years, i.e., mu > 77.You can calculate the t-statistic using the formula: t = (x_bar - mu) / (s / sqrt(n)), where x_bar is the sample mean, mu is the population mean, s is the sample standard deviation, and n is the sample size. Plugging in the given values, we get:
t = (81.03 - 77) / (1.53 / sqrt(12)) = 9.974, approx.
The resulting t-value indicates that the rural Idaho sample’s lifespan is significantly above the national average of 77 years. Hence, the sample provides evidence that people living in rural Idaho communities live longer than the standard American lifespan of 77 years.
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A single slit diffraction experiment performed with an argon laser of wavelength 454.6 nm produces a pattern on a screen with dark fringes (minima) separated by 10 mm. If we replace the slit with another slit of double the width of the original one, without changing anything else in the setup, what will be the new separation between dark fringes?
In light diffraction, fringe separation is inversely proportional to the slit width. When the slit width in an experiment is doubled, it should result in the separation between the dark fringes on the screen being halved. Therefore, if the original fringe separation was 10 mm, the new separation should be 5 mm.
Explanation:The subject question pertains to the topic of light diffraction, particularly through varying slit widths. The scenario described is a single slit diffraction experiment involving an argon laser with a specific wavelength. In diffraction, the fringe separation is inversely proportional to the slit width. This is because the angle of diffraction is determined by the wavelength of light divided by the slit width, according to the formula sinθ = λ/D, where θ is the diffraction angle, λ is the wavelength, and D is the slit width.
Therefore, when the slit width is doubled, the diffraction angles for the minima (dark fringes) will be halved assuming all other conditions remain the same. Consequently, the separation between the dark fringes on the screen will also be halved. So, if the original separation was 10 mm, the new separation when the slit width is doubled should be 5 mm.
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Let X1 be a normal random variable with mean µ1 and variance σ 2 1 , and let X2 be a normal random variable with mean µ2 and variance σ 2 2 . Assuming that X1 and X2 are independent, what is the distribution of X1 + X2? g
Answer:
[tex]X_1 + X_2 \sim (\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2 )[/tex]
Step-by-step explanation:
We are given the following in the question:
[tex]X_1[/tex] is a random normal variable with mean and variance
[tex]\mu_1\\\sigma_1^2[/tex]
[tex]X_1 \sim N(\mu_1,\sigma_1^2)[/tex]
[tex]X_2[/tex] is a random normal variable with mean and variance
[tex]\mu_2\\\sigma_2^2[/tex]
[tex]X_2 \sim N(\mu_2,\sigma_2^2)[/tex]
[tex]X_1, X_2[/tex] are independent events.
Let
[tex]Z =X_1 + X_2[/tex]
Then, Z will have a normal distribution with mean equal to the sum of the two means and its variance equal the sum of the two variances.
Thus, we can write:
[tex]\mu = \mu_1 + \mu_2\\\sigma^2 = \sigma_1^2 + \sigma_2^2\\Z \sim (\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2 )[/tex]
Final answer:
The sum of two independent normal random variables X1 and X2 is also a normal random variable with a mean of μ1 + μ2 and variance of σ²₁ + σ²₂, denoted by N(μ1 + μ2, σ²₁ + σ²₂).
Explanation:
The distribution of the sum of two independent normal random variables, X1 + X2, is itself a normal random variable. Given that X1 is a normal random variable with mean μ1 and variance σ²₁, and X2 is a normal random variable with mean μ2 and variance σ²₂, the sum X1 + X2 will have a mean of μ1 + μ2 and variance of σ²₁ + σ²₂ because of the properties of the normal distribution and the independence of X1 and X2.
The resulting distribution can be denoted as N(μ1 + μ2, σ²₁ + σ²₂). This conclusion comes from the central limit theorem, which states that the sum of independent random variables tends towards a normal distribution as the number of variables increases, and the means and variances add up.
The lengths of brook trout caught in a certain Colorado stream are normally distributed with a mean of 14 inches and a standard deviation of 3 inches. What proportion of brook trout caught will be between 12 and 18 inches in length?
Answer:
0.6563 or 65.63% of brook trout caught will be between 12 and 18 inches
Step-by-step explanation:
Mean trout length (μ) = 14 inches
Standard deviation (σ) = 3 inches
The z-score for any given trout length 'X' is defined as:
[tex]z=\frac{X-\mu}{\sigma}[/tex] e interval
For a length of X =12 inches:
[tex]z=\frac{12-14}{3}\\z=-0.6667[/tex]
According to a z-score table, a score of -0.6667 is equivalent to the 25.25th percentile of the distribution.
For a length of X =18 inches:
[tex]z=\frac{18-14}{3}\\z=1.333[/tex]
According to a z-score table, a score of 1.333 is equivalent to the 90.88th percentile of the distribution.
The proportion of trout caught between 12 and 18 inches, assuming a normal distribution, is the interval between the equivalent percentile of each length:
[tex]P(12\leq X\leq 18) = 90.88\% - 25.25\%\\P(12\leq X\leq 18) = 65.63\%[/tex]
To find the proportion of brook trout caught between 12 and 18 inches, calculate the z-scores for these values and find the area between them on a standard normal distribution curve.
Explanation:To find the proportion of brook trout caught between 12 and 18 inches in length, we need to calculate the z-scores for these values and then find the area between the z-scores on a standard normal distribution curve.
First, we calculate the z-score for 12 inches: z = (12 - 14) / 3 = -2/3.
Second, we calculate the z-score for 18 inches: z = (18 - 14) / 3 = 4/3.
Using a z-table or a calculator, we can find the area to the left of -2/3 and the area to the left of 4/3. Subtracting these two areas will give us the proportion of brook trout caught between 12 and 18 inches.
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Answer:
B. 0.10 - 0.20 = 0.10.
Step-by-step explanation:
100/1000 = 0.10 of the population were born , and
200/1000 = 0.20 of the population died so it is:
0.10 - 0.20 = 0.10.
Answer: B
Step-by-step explanation:
The total population was initially 1000 individuals. In this population, a total of of 100 new individuals were born over the course of one year. The proportion of new individuals that make up the population is 100/1000 = 0.1
Since they were added, then it is positive and it is a gain for the population.
A total of 200 individuals also died over the course of one year. The proportion that died = 200/1000 = 0.2. This would be negative because it is a loss for the population.
The population growth rate will be gain - loss. This becomes
0.1 - 0.2 = -0.1
A powder diet is tested on 49 people and a liquid diet is tested on 36 different people. Of interest is whether the liquid diet yields a higher average weight loss than the powder diet. The powder diet group had an average weight loss of 42 pounds with a standard deviation of 12 pounds. The liquid diet group had an average weight loss of 44 pounds with a standard deviation of 14 pounds. Conduct a hypothesis test at the 5% level. State the distribution to use for the test.
Answer:
add all of it
Step-by-step explanation:
A father and his two sons wanted to measure the distance between the two tallest trees in the City Garden by their footsteps. It was winter, and there was fresh snow in the garden, so they decided to start measuring from the same tree by walking one after another straight to the other tree. The father's footstep is 32 inches long, while the same for his sons is 28 inches and 24 inches. In what distance, in feet, the three steps would overlap the first time?
PLEASE SOLVE ASAP, YOU GET BRAINLIEST IF CORRECT
The distance (in feet) in which the three steps would overlap for the first time is 56 feet.
Calculations and ParametersUsing LCM method
We would find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 24:
24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 264, 288, 312, 336, 360, 384, 408, 432, 456, 480, 504, 528, 552, 576, 600, 624, 648, 672, 696, 720
Multiples of 28:
28, 56, 84, 112, 140, 168, 196, 224, 252, 280, 308, 336, 364, 392, 420, 448, 476, 504, 532, 560, 588, 616, 644, 672, 700, 728
Multiples of 32:
32, 64, 96, 128, 160, 192, 224, 256, 288, 320, 352, 384, 416, 448, 480, 512, 544, 576, 608, 640, 672, 704, 736
Therefore,
LCM(24, 28, 32) = 672
Then we convert to feet:
672/12= 56 feet.
Read more about LCM here:
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I need help on 4,6,7 please!
Answer:
Step-by-step explanation:
4) The shape in the figure is a parallelogram. In the parallelogram, the diagonals are equal and bisect each other at the midpoint, E. This means that AC is divided equally into AE and EC. Therefore
AC = 2EC
AC = 8x - 14 and EC = 2x + 11. So,
8x - 14 = 2(2x + 11)
8x - 14 = 4x + 22
8x - 4x = 22 + 14
4x = 36
x = 36/4
x = 9
6)The shape in the figure is a parallelogram. The opposite angles are equal. This means that
Angle BCD = angle BED
So angle BED = 51 degrees.
Since the sum of angles in a triangle is 180 degrees, then, angle BED + angle BDE + angle DBE = 180 degrees. It means
51 + 55 + 14x + 4 = 180
14x + 110 = 180
14x = 180 - 110 = 70
x = 70/14
x = 5
7) The shape in the figure is a parallelogram. The opposite angles are equal. Therefore,
angle VST = angle VUT
5x + 23 = 8x - 49
5x - 8x = -49 - 23
-3x = -72
x = -72 / -3
x = 24
angle VST = 5×24 + 23 = 143
angle VUT = 143
angle VST + angle VUT = 143 + 1143 = 286
Recall, the sum of angles in a parallelogram is 360 degrees. Therefore,
angle SVU + angle STU = 360 - 286 = 74 degrees
Angle SVU = 74/2 = 37 degrees.
Angle SVT + angle UVT = 37
angle SVT + 20 = 37
angle SVT = 37 - 20 = 17 degrees