The domain of a linear transformation is the set of all possible input vectors that can be operated on by the transformation, and in this case, T is defined for input vectors with three components (x1, x2, x3), indicating that the domain of T is R3, the three-dimensional real coordinate space.
The domain of T is R3.
Given that A is an m x n matrix, where m > n, and Ax = 0 does not have a non-trivial solution, we can determine the number of pivot columns in matrix A.
Understand the conditions
Since Ax = 0 does not have a non-trivial solution, it means that the only solution is the trivial solution (x = 0).
This implies that the columns of A are linearly independent.
Determine the number of pivot columns
Since the columns of A are linearly independent, the number of pivot columns in A will be equal to the number of columns in A.
Since A has more rows than columns, it cannot have full column rank (i.e., n pivot columns), and therefore it has at most n-1 pivot columns.
If Ax=0 does not have a non-trivial solution, then the null space of A is trivial, meaning that the only solution to Ax=0 is the zero vector.
By the rank-nullity theorem, we know that the dimension of the null space of A is n minus the rank of A.
Since the null space is trivial, its dimension is zero, which means that the rank of A is n.
Therefore, A has exactly n pivot columns, which is the maximum possible given that m>n.
These n pivot columns correspond to linearly independent columns of A, and their existence ensures that Ax=0 has only the trivial solution.
Matrix A has n pivot columns.
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What do you call a combination of numbers, symbols, or other values that produce a result when evaluated?
A combination of numbers, symbols, or other values that produce a result when evaluated is called an expression.
Expressions can be simple or complex, and they can include operations such as addition, subtraction, multiplication, and division, as well as functions, variables, and constants.
Expressions are an essential part of mathematics and are used in a wide range of applications, including algebra, calculus, and statistics.
They are often used to represent real-world situations and to solve problems in various fields such as engineering, physics, economics, and finance.
Evaluating expressions involves performing the indicated operations in the correct order, according to the rules of arithmetic, algebra, or calculus, and simplifying the result as much as possible.
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Prove the following identity:
(cos 2x + cos 4x)/(cos 2x - cos 4x) = (cot 3x/tan x)
To prove the identity (cos 2x + cos 4x)/(cos 2x - cos 4x) = (cot 3x/tan x), we can use the following trigonometric identities:
1. cos 2x = 2cos^2 x - 1
2. cos 4x = 8cos^4 x - 8cos^2 x + 1
3. cot 3x = (3cos^2 x - 1)/(3sin x cos x)
4. tan x = sin x/cos x
Starting with the left-hand side of the identity, we can substitute the expressions for cos 2x and cos 4x from identities 1 and 2:
(cos 2x + cos 4x)/(cos 2x - cos 4x) = ((2cos^2 x - 1) + (8cos^4 x - 8cos^2 x + 1))/((2cos^2 x - 1) - (8cos^4 x - 8cos^2 x + 1))
Simplifying this expression gives:
(cos 2x + cos 4x)/(cos 2x - cos 4x) = (6cos^2 x)/(2cos^2 x) = 3
Next, we can substitute the expressions for cot 3x and tan x from identities 3 and 4:
(cot 3x/tan x) = ((3cos^2 x - 1)/(3sin x cos x))/(sin x/cos x)
Simplifying this expression gives:
(cot 3x/tan x) = (3cos^2 x - 1)/sin x = 3cos^2 x/cos x = 3cos x
Therefore, (cos 2x + cos 4x)/(cos 2x - cos 4x) = (cot 3x/tan x) is equivalent to 3 = 3cos x, which is true for all values of x.
So, the identity is proven.
c. what part of the federal reserve’s congressional mandate does this scenario trigger (price stability and maximum sustainable employment
This scenario triggers the part of the Federal Reserve's congressional mandate related to price stability.
The Federal Reserve has a dual mandate set by Congress, which includes price stability and maximum sustainable employment. In this scenario, the increase in oil prices is likely to lead to an increase in inflation, which threatens price stability. The Federal Reserve may respond to this by implementing monetary policy tools to control inflation, such as raising interest rates or reducing the money supply. Therefore, this scenario triggers the part of the Federal Reserve's congressional mandate related to price stability. The other part of the mandate, maximum sustainable employment, refers to the Fed's goal of promoting maximum employment consistent with price stability. While this scenario may indirectly impact employment, it is not the primary concern in this case.
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3. Jyllina created this box plot representing the number of inches of snow that fell this winter in different nearby
cities.
(a) What was the greatest amount of snowfall in any of the cities?
(b) In which quarter is the data most concentrated? Explain how you know.
(c) In which quarter is the data most spread out? Explain how you know.
Answer:
a) The greatest amount of snowfall in any of the cities is given as follows: 48 inches.
b) The data is most concentrated on the third quarter, as Q3 - Median < Median - Q1.
c) The data is most spread out on the first quarter, as Q3 - Median < Median - Q1.
What does a box-and-whisker plot shows?A box and whisker plots shows these five features from a data-set, listed as follows:
The minimum non-outlier value.The 25th percentile, which is the median of the bottom 50%.The median, which splits the entire data-set into two halfs, the bottom 50% and the upper 50%.The 75th percentile, which is the median of the upper 50%.The maximum non-outlier value.The measures from the plot in this problem are given as follows:
Minimum of 5 inches.Q1 = 13 inches.Median of 35 inches.Q3 = 40 inches.Maximum of 48 inches.More can be learned about box plots at https://brainly.com/question/12343132
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What is the probability that at least 2 of a group of 4 persons were born on the same day of the week?
How long to double 325 at 4. 7% interest compounded monthly
The time it takes to double 325 at 4.7% interest compounded monthly is approximately 15 years.
Given that,
Principal amount = 325
Rate of interest = 4.7%
It is compounded monthly.
The formula to calculate the final amount in a compound interest is,
A = P(1 + r/n)^ (nt)
Here P is the principal amount, r is the rate of interest, n is the number of times interest is compounded per year and t is the number of years.
Here A = 2 × 325 = 650
r = 4.7% = 0.047
P = 325
n = 12
Substituting,
650 = 325 (1 + 0.047/12)^(12t)
2 = (1.048)^t
Taking logarithms,
log 2 = t log(1.048)
t = 14.777 ≈ 15 years
Hence the time is 15 years.
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Answering any questions will help
Answer:
SOH CAH TOA
Step-by-step explanation:
Okay use the formula above where by for example SOH means sin°=opposite/hypotenuse
in number 18 opposite is =30
and hypotenuse=50
then your answer is=36.87
Use the given information to find n(A x B) and n(B x A)
n(A)=84 and n(B) = 5
n(A x B) = ___ n(B x A)=
n(B x A) = n(A x B) = 420
find n(A x B) and n(B x A) n(A)=84 and n(B) = 5n(A x B) = ___ n(B x A)= iss
Since we know that n(A x B) = n(A) x n(B), we can substitute the given values to find:
n(A x B) = n(A) x n(B) = 84 x 5 = 420
Therefore, n(A x B) = 420.
To find n(B x A), we can use the fact that n(A x B) = n(B x A), which follows from the commutative property of set intersection.
So, the final answers are:
n(A x B) = 420
n(B x A) = 420
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Dr. Zadok's Museum has a collection of cameras. If a camera is selected at random from the museum's collection, the probability that it is digital is 0.43 and the probability that it is a single lens reflex (SLR) camera is 0.51. The probability that the randomly selected camera is both digital and an SLR is 0.19. Let the event that a camera is digital be D and the event that a camera is an SLR be S. Suppose that a camera is selected at random from the museum's collection. Find the probability that digital but not an SLR.
The probability that it is digital but nor SLR is 0.24.
How to solve for the probabilityP(D and not S) = P(D) - P(D and S)
We are given the following probabilities:
P(D) = 0.43 (probability that the camera is digital)
P(S) = 0.51 (probability that the camera is an SLR)
P(D and S) = 0.19 (probability that the camera is both digital and an SLR)
Now, we can plug these values into the formula:
P(D and not S) = P(D) - P(D and S)
P(D and not S) = 0.43 - 0.19
P(D and not S) = 0.24
So, the probability that a randomly selected camera is digital but not an SLR is 0.24.
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Suppose the total cost in Naira of manufacturing q units of a certain commodity is given by the function C(q) = q3 - 30q2 + 500q + 200. The cost of manufacturing 10 units of the commodity is
Answer:
3200
Step-by-step explanation:
You want the value of C(10) when C(q) is defined as q³ -30q² +500q +200.
CostThe cost of manufacturing 10 units is ...
C(10) = 10³ -30·10² +500·10 +200
C(10) = 3200 . . . . . . currency units
The cost of manufacturing 10 units of the commodity is 3200.
The product of the sum of 3 and 6 and the sum of 2 and 7
you are told that the perimeter of a circle is 24. you may use 22/7 or 3.14. what is the diameter of the circle?
The diameter of the circle is approximately 7.714 units. To find the diameter of the circle, we need to first know the formula for the perimeter of a circle.
The perimeter of a circle is given by the formula P = 2πr, where P is the perimeter, π is the mathematical constant pi (approximately equal to 3.14 or 22/7), and r is the radius of the circle.
Since we are given that the perimeter of the circle is 24, we can set up the equation as follows:
24 = 2πr
Next, we can solve for the radius by dividing both sides by 2π:
24 / 2π = r
Now, we can plug in the value of π (either 3.14 or 22/7) and simplify:
24 / (2 x 3.14) = r
24 / (2 x 22/7) = r
Simplifying the second equation, we get:
24 / (44/7) = r
24 x 7/44 = r
3.857 = r
So the radius of the circle is approximately 3.857 units.
Finally, we can find the diameter of the circle by multiplying the radius by 2:
Diameter = 2 x radius
Diameter = 2 x 3.857
Diameter = 7.714
Therefore, the diameter of the circle is approximately 7.714 units.
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Becky's statistics teacher was teaching the class how to perform the z-test for a proportion. Becky was bored because she had already mastered the test, so she decided to see if the coin she had in her pocket would come up heads or tails in a truly random fashion when flipped. She discretely flipped the coin 30 times and got heads 18 times. Becky conducts a one-proportion hypothesis test at the 5% significance level, to test whether the true proportion of heads is different from 50%
Becky conducted a one-proportion hypothesis test at the 5% significance level, to test whether the true proportion of heads is different from 50%.
Her null hypothesis was that the proportion of heads is equal to 50%, and her alternative hypothesis was that the proportion of heads is different from 50%. Using the z-test for a proportion, with a sample size of 30, Becky calculated a test statistic of z=1.05 and a p-value of 0.295. Since the p-value is greater than 0.05, she fails to reject the null hypothesis and concludes that there is not enough evidence to suggest that the true proportion of heads is different from 50%. Therefore, she cannot claim that the coin is biased towards heads or tails.
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F
10
Type here to search
5
A. The surface area of the triangular prism is
square units.
B. If the longest dimension of the prism is decreased by a factor of the resulting surface area is
square units.
A.) The surface area of the triangular prism = 184 unit ²
B.) The resulting surface area when length dimension us decreased =104units².
How to calculate the surface area of a triangular prism?For A.) To calculate the surface area of the triangular prism the formula that should be used is given below;
SA = b×h + (S1+S2+S3) ×L
where;
b = base = 6
S = base edge
L = length = 10
h = height = 4
S.A. = 6×4+(5+5+6)×10
= 24+160
= 184 units²
For question B.)
When length is reduced by a scale factor of 1/2, the resulting surface area =
SA = 6×4+(5+5+6) × 10/2
= 24+80
= 104units²
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19. In a right triangle a = 12, c = 20 find b. Round to the nearest tenth.
Answer: 16
Step-by-step explanation: In the Pythagorean theorem a^2 + b^2 = c^2, so 20^2 - 12^2 = b^2. Furthermore square root of 256 is our answer 16.
A baby takes 30 min to drink 150 ml of milk from the bottle. Calculate
milk flow in the bottle in millilitres per minute and millilitres per minute
second
The milk flow in the bottle is 0.0833 milliliters per second.
To calculate the milk flow in milliliters per minute, we can use the formula:
milk flow = milk volume / time
In this case, the milk volume is 150 ml and the time is 30 minutes, so:
milk flow = 150 ml / 30 min
milk flow = 5 ml/min
Therefore, the milk flow in the bottle is 5 milliliters per minute.
To convert milliliters per minute to milliliters per second, we need to divide by 60 (since there are 60 seconds in a minute). So:
milk flow = 5 ml/min / 60
milk flow = 0.0833 ml/s
Therefore, the milk flow in the bottle is 0.0833 milliliters per second.
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Andre is solving an equation 4(x+3/2)=7 where did he make his mistake
There doesn't seem to be any mistake in the given equation. Andre correctly simplified and solved the equation, and the solution is x = 1.
Let's solve the equation step by step to identify the mistake:
4(x + 3/2) = 7
First, we need to distribute the 4 to the terms inside the parentheses:
4x + 4(3/2) = 7
Simplifying the expression inside the parentheses:
4x + 6/2 = 7
4x + 3 = 7
Now, the equation becomes:
4x + 3 = 7
To isolate the variable x, we need to subtract 3 from both sides:
4x + 3 - 3 = 7 - 3
4x = 4
Finally, to solve for x, we divide both sides by 4:
(4x)/4 = 4/4
x = 1
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Which if the equations below is parallel to y = 3x - 9, and passes through the point (4, 5)?
An equation of the line that is parallel to y = 3x - 9 and passes through the point (4, 5) is y = 3x - 7.
We have,
The given line has a slope of 3 because it is in the form y = mx + b, where m is the slope and b is the y-intercept.
So, any line parallel to it must also have a slope of 3.
Now,
Using the point-slope form of the equation of a line.
y - y1 = m(x - x1)
where m = 3, x1 = 4, and y1 = 5. Substituting these values, we get:
y - 5 = 3(x - 4)
Expanding and simplifying this equation.
y - 5 = 3x - 12
y = 3x - 7
Therefore,
An equation of the line that is parallel to y = 3x - 9 and passes through the point (4, 5) is y = 3x - 7.
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Evaluate the expression shown below and write your answer as a fraction in simplest form.
\frac{8}{3}+\frac{11}{12}
3
8
+
12
11
A common denominator for the two fractions. The final answer in simplest form is [tex]\frac{43}{12}[/tex]
To evaluate the expression and write the answer as a fraction in simplest form, we need to find a common denominator for the two fractions. The smallest common denominator for 3 and 12 is 12.
[tex]\frac{8}{3} + \frac{11}{12} = \frac{32}{12} + \frac{11}{12}[/tex]
Now that we have a common denominator, we can add the numerators:
[tex]\frac{32}{12} + \frac{11}{12} = \frac{32+11}{12} = \frac{43}{12}[/tex]
Finally, we can simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 1 in this case. Therefore, the final answer in simplest form is [tex]\frac{43}{12}[/tex]
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Question
Marcus plays a game in which he spins a spinner over and over again. The spinner has 4 equally sized sections labeled E, F, G, and H. Spinning an E six times means the game is over.
Marcus uses a uniform probability model to predict the number of times the spinner will be spun before the letter E appears 6 times.
What is Marcus's prediction for the number of total spins before the letter E appears 6 times?
10 spins
18 spins
24 spins
30 spins
Marcus's prediction for the number of total spins before the letter E appears 6 times is 24.
How to find Marcus's prediction for the number of total spins before the letter E appears 6 timesThe probability of getting an E in one spin is 1/4.
Let X be the number of spins required to get 6 E's.
E(X) = r/p is the mean of a negative binomial distribution with parameters r and p.
In this scenario, the average number of spins required to obtain 6 E's is: E(X) = 6 / (1/4) = 24.
Therefore, Marcus's prediction for the number of total spins before the letter E appears 6 times is 24.
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Answer: 24
Step-by-step explanation: I did the test on k12
25. Determine which set of three numbers could represent the sides of a right triangle.
a. 9, 41, 41 c. 8, 41, 40
b. 9, 40, 41 d. 10, 40, 42
26. Determine which set of three numbers could NOT represent the sides of a right triangle.
a. 3, 4, 5 c. 14, 19, 24
b. 21, 28, 35 d. 30, 40, 50
Answers:
25. b. 9, 40, 41
26. c. 14, 19, 24
========================================
Explanation for problem 25
We use the converse of the pythagorean theorem.
If a^2+b^2 = c^2 is a true equation, then we have a right triangle with sides a,b,c where c is the longest side.
For part (a) we have a = 9, b = 41, c = 41
Then,
a^2+b^2 = c^2
9^2+41^2 = 41^2
1762 = 1681
The two sides aren't the same value at the end, so we have a false equation when (a,b,c) = (9,41,41). These three sides do NOT form a right triangle.
Repeat these steps for parts (b) through (d)
You should find that part (b) does lead to a true equation
a^2+b^2 = c^2
9^2+40^2 = 41^2
1681 = 1681
Therefore, a right triangle occurs when (a,b,c) = (9,40,41).
---------------------------------
Explanation for problem 26
We'll use the same idea as the previous problem. This time we're looking for non-right triangles.
Choice (a) can be ruled out because 3^2+4^2 = 5^2 leads to 25 = 25. Same goes for choice (d). That has been scaled up by 10.
Choice (b) gives 21^2+28^2=35^2 which boils down to 1225 = 1225, so this is ruled out as well.
Choice (c) on the other hand gives:
a^2+b^2 = c^2
14^2+19^2 = 24^2
557 = 576
Showing that a triangle with sides 14,19,24 is not a right triangle.
Job interview: Eight people, named Anna, Bob, Chandra, Darlene, Ed, Frank, Gina, and Hank, will be interviewed for a job. The interviewer will choose two at random to interview on the first day. What is the probability that Frank is interviewed first and Ed is interviewed second
Answer:
1/64 or 1.5625%
Step-by-step explanation:
To get to the answer shown above, we need to have a basic understanding of how probability works. Let's focus on the first part: "What is the probability that Frank is interviewed first" Since there are 8 people in the interview, that means that 8 people has to be in the denominator. Frank is one person and he represents 1/8th of the entire interviewees. So, the probability that he is chosen first is 1/8. Then, for the second part, we are also asked the probability of both frank being chosen first and Ed being chosen second. Ed is again only one person and thus has a 1/8 probability of being chosen. But, we are not done yet. We need to multiple the probabilities together because they have to happen simultaneously and in order. 1/8*1/8=1/64 or 1.5625%
Factor the expression completely 40x^2 - 48
Answer:
8(√5x - √6)(√5x + √6).
Step-by-step explanation:
First, we can factor out the greatest common factor, which is 8:
8(5x^2 - 6)
Next, we can use the difference of squares formula to factor further:
8(√5x - √6)(√5x + √6)
Therefore, the expression 40x^2 - 48 can be factored completely as 8(√5x - √6)(√5x + √6).
High school sponsored a tennis tournament. After each round one half of the players were eliminated. If there were 32 players at the start of the tournament which equation models the number of players left after the rounds
The equation that should model the number of players left after the tournament would be = f(X) = 32(1-0.5)³. That is option A.
How to determine the equation that represents the statement given above?To determine the equation that would represent the statement given above the following needs to be done.
The quantity of players that were eliminated for each round = 1-0.5.
The total number of players = 32
Therefore the correct equation that represents the number of players left after three round would be =
f(X) = 32(1-0.5)³.
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If a woman has s/13 055, and needs to convert it to dollars, 1 dolar = s/3.73how many dollars will she recieve?
If a woman has s/13,055 and needs to convert it to dollars, and 1 dollar equals s/3.73, she will receive approximately $3,498.69.
To convert s/13,055 to dollars, we need to divide by the exchange rate of s/3.73 per dollar. This gives us (s/13,055) / (s/3.73) = (s/13,055) * (3.73/s) = 3.73/13,055 dollars per s. Simplifying this expression, we get approximately 0.00028549 dollars per s. To find out how many dollars the woman will receive, we need to multiply this exchange rate by the amount of s she has: 0.00028549 dollars/s * s/13,055 = (0.00028549/13,055) * s = 0.00000002189 * s. If we substitute s/13,055 into this expression, we get approximately 0.00000002189 * (s/13,055) = 0.00000002189 * 1 = 0.00000002189 dollars. This is a very small amount, so we can convert it to a more usable value by multiplying by 1,000,000. This gives us approximately 0.022 dollars, which is equivalent to $0.022. Therefore, the woman will receive approximately $3,498.69 (rounded to the nearest cent).
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How many rolls of wallpaper, each 10 yards long and 24 inches wide, will paper the sides of a room 12. 1 feet by 17. 1 feet and 8 feet high, deducting 129 ft2 for doors and windows?
There are 3 feet in a yard and 12 inches in a foot, so there are 36 square inches in a square yard. 779.2 square feet = (779.2/9) square yards = 86.58 square yards. 5 rolls of wallpaper to cover the walls of the room.
First, we need to find the area of the walls to be covered. The room has dimensions of 12.1 feet by 17.1 feet and a height of 8 feet. The area of the walls is given by:
2h(w1 + w2) where h is the height, w1 and w2 are the widths of the walls
The width of one wall is the height of the room, so w1 = 8 feet. The other wall is the sum of the lengths of the adjacent walls minus the width of the door and window, so w2 = (12.1 + 17.1 + 12.1 + 17.1 - 129/144) = 48.7 feet.
Therefore, the area of the walls to be covered is:
2(8)(48.7) = 779.2 square feet.
To find how many rolls of wallpaper we need, we need to convert square feet to square yards, and then divide by the area of one roll of wallpaper.
There are 3 feet in a yard and 12 inches in a foot, so there are 36 square inches in a square yard. Therefore:
779.2 square feet = (779.2/9) square yards = 86.58 square yards
Each roll of wallpaper is 10 yards long and 24 inches (2 feet) wide, so the area of one roll is:
10 yards x 2 feet = 20 square yards.
Therefore, we need:
86.58 square yards / 20 square yards per roll = 4.33 rolls
We cannot buy a fraction of a roll, so we need to round up to the nearest whole number. Therefore, we need 5 rolls of wallpaper to cover the walls of the room.
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The temperature at the north pole is 30f and is expected to drop 5f per hour for the next several hours. write an equation that represents the relationship between temperature and time.
The equation that represents the relationship between temperature and time at the North Pole is T = -5t + 30, where T is the temperature in degrees Fahrenheit and t is the time in hours.
Since the temperature is expected to drop 5 degrees Fahrenheit per hour, we can use a linear equation in slope-intercept form, y = mx + b, where m is the slope (the rate of change) and b is the y-intercept (the initial temperature).
In this case, the slope is -5 (since the temperature drops 5 degrees Fahrenheit per hour) and the initial temperature (at t=0) is 30 degrees Fahrenheit.
Therefore, the equation that represents the relationship between temperature and time at the North Pole is T = -5t + 30. As time increases, the temperature will decrease by 5 degrees Fahrenheit per hour.
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In 2016, what was the probability that a randomly selected UC undergrad had experienced both food insecurity and homelessness at some point in the past 12 months
In 2016, the probability that a randomly selected UC undergrad had experienced both food insecurity and homelessness at some point in the past 12 months is 0.0385
How did we arrive at that?Since form the graph, we know that 77% experienced homelessness and
9% Food Insecurity
If the total sample is 100%
Then the the probability that a randomly selected UC undergrad had experienced both food insecurity and homelessness at some point in the past 12 months is:
0.05 x 0.77 = 0.0385
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Full Question:
See attached image.
In 2016, what was the probability that a randomly selected UC undergrad had experienced both food insecurity and homelessness at some point in the past 12 months?
If the heights of 99.7% of American men are between 5'0"
and 7'0", what is your estimate of the standard deviation of
the height of American men? (Assume the heights of
American Men are Normally Distributed).
A.) 1"
B.) 2"
C.) 4"
D.) 6"
E.) 12"
1 standard deviation = 8 inches. the answer is OPTION E
We know that 99.7% of American men have heights between 5'0" and 7'0". Since height is normally distributed, we can use the empirical rule to estimate the standard deviation.
According to the empirical rule, 99.7% of the data falls within 3 standard deviations of the mean. Therefore, we can say that the range of heights from 5'0" to 7'0" is 3 standard deviations.
We can use this information to estimate the standard deviation as follows:
Range of heights = 7'0" - 5'0" = 2 feet
3 standard deviations = 2 feet
1 standard deviation = 2/3 feet
1 foot = 12 inches
Therefore, 1 standard deviation = 2/3 * 12 inches = 8 inches
So, our estimate of the standard deviation of the height of American men is 8 inches. Therefore, the answer is OPTION E in the given options.
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Complete question
If the heights of 99.7% of American men are between 5'0"
and 7'0", what is your estimate of the standard deviation of
the height of American men? (Assume the heights of
American Men are Normally Distributed).
A.) 1"
B.) 2"
C.) 4"
D.) 6"
E.) 8"
Suppose that nominal gdp was $10 trillion in 2040 in atlantis. in 2050, nominal gdp was $15 trillion in atlantis. the price level fell 6% between 2040 and 2050, and population growth was 3%.
The real GDP per capita in Atlantis increased between 2040 and 2050.
Nominal GDP measures the total value of goods and services produced in an economy, without adjusting for changes in the price level. To calculate real GDP, we adjust nominal GDP for changes in the price level, in order to measure the actual changes in output. In this case, we know that the price level fell 6% between 2040 and 2050.
To calculate the real GDP in 2040 and 2050, we need to use the GDP deflator, which is a measure of the price level. The formula for calculating real GDP using the GDP deflator is:
Real GDP = Nominal GDP / GDP Deflator
We can calculate the GDP deflator for 2040 and 2050 as follows:
GDP Deflator 2040 = (Nominal GDP 2040 / Real GDP 2040) x 100
GDP Deflator 2050 = (Nominal GDP 2050 / Real GDP 2050) x 100
Assuming that the population growth rate is applied equally to both years, we can calculate the Real GDP for both years as:
Real GDP 2040 = Nominal GDP 2040 / (1 + (6% - 3%))
Real GDP 2040 = $10,000 billion / (1.03) = $9,708.74 billion
Real GDP 2050 = Nominal GDP 2050 / (1 - 6%)
Real GDP 2050 = $15,000 billion / (0.94) = $15,957.45 billion
Using the above formula for real GDP, we can calculate the real GDP per capita for 2040 and 2050 by dividing the real GDP by the population in each year. Assuming that the population in 2040 and 2050 are the same, the real GDP per capital in 2040 and 2050 is:
Real GDP per capita 2040 = Real GDP 2040 / Population
Real GDP per capita 2040 = $9,708.74 billion / Population
Real GDP per capita 2050 = Real GDP 2050 / Population
Real GDP per capita 2050 = $15,957.45 billion / Population
Since the population growth rate was 3%, we can assume that the population in 2050 was 103% of the population in 2040:
Population 2050 = Population 2040 x 1.03
Therefore, the real GDP per capita in 2040 and 2050 is:
Real GDP per capita 2040 = $9,708.74 billion / Population 2040
Real GDP per capita 2050 = $15,957.45 billion / (Population 2040 x 1.03)
Since the real GDP per capita in 2050 is greater than in 2040, we can conclude that the real GDP per capita in Atlantis increased between 2040 and 2050.
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Suppose that nominal gdp was $10 trillion in 2040 in atlantis. in 2050, nominal gdp was $15 trillion in atlantis. the price level fell 6% between 2040 and 2050, and population growth was 3%. What was the real GDP per capita in Atlantis in 2040 and 2050, using 2040 as the base year?
