The statement that " Homogeneous equation Ax = 0 has trivial solution if it has at least 1 free variable" is FALSE .
For a system of Linear Equations, a free variable is a variable which can take any value, whereas
the Leading Variable is a variable which is determined by the values of the free variables.
The presence of a free variable means that existence of a non trivial solution to Ax = 0 because a free variable corresponds to a column of A that does not contain a pivot position in the corresponding row of [A|0].
So , the corresponding variable can be set to any non zero value, which gives a non trivial solution to Ax = 0.
Therefore, we can conclude that the homogeneous equation Ax = 0 has the trivial solution if and only if the equation has no free variables.
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The given question is incomplete , the complete question is
The homogeneous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable. True or False ?
pls help me!! (#3 btw)
The speed of the car is 20 miles per hour which is a slope for the given graph.
What is the slope of the line?The slope of a line is defined as the gradient of the line. It is denoted by m
Slope m = (y₂ - y₁)/(x₂ -x₁ )
The graph is given in the question, as shown.
Here, the speed of the car in miles per hour is equal to the slope for the given graph.
According to the graph, taking two points (90, 4) and (1, 30).
So the speed of the car would be as:
⇒ (90 - 30)/(4-1)
⇒ 60/3
⇒ 20
Thus, the speed of the car is 20 miles per hour.
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The GDP of Alaska in 2021 was $50.3 billion. The population in Alaska is 732,670 . What was the GDP per capita in 2021? Give your answer to the nearest hundredth.
Answer: To find the GDP per capita, we need to divide the total GDP by the population:
$50.3 billion ÷ 732,670 people = $68,547.32 per person
Rounding to the nearest hundredth:
$68,547.32 per person ≈ $68,547.32 per person
So the GDP per capita in Alaska in 2021 was $68,547.32 per person.
Step-by-step explanation:
Show that if n≥2k, every tournament on n vertices has a transitive subtournament on k vertices!
To show that if n≥2k, every tournament on n vertices has a transitive subtournament on k vertices, we can compute it as:
pick any vertex v, let
[tex]L = \{u:u\to v\},[/tex]
Now let,
[tex]R=\{u:v\to u\}.[/tex]
By u --> v, there is an edge from u to v
[tex]|L|+|R| = 2^{k+1}-1[/tex]
so, one of L and R contains at least 2k points. Now apply your induction hypothesis to that set, and you should find it easy to fit v in to the resulting transitive k-tournament.
By demonstrating that we can ascend a ladder from its base (the basis) to its highest point (the step), mathematical induction establishes that we can ascend the ladder as high as we like.
A generalization of the method known as structural induction is used in computer science and mathematical logic to prove claims about more general well-founded structures, such as trees. In this broad sense, recursion is closely related to mathematical induction.
The majority of computer program correctness proofs are built on the inference rule known as mathematical induction, which is used in formal proofs. Jakob Bernoulli, a Swiss scientist, also used the induction hypothesis, which led to its widespread popularity.
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Ejercicios propuestos:
Ejercicio 1. Ecuaciones de primer grado (solución de sistemas de
ecuaciones)
Carlos compra para su familia 5 cajas de sobres y 3 álbumes del mundial por un
precio de 1# dólares, en cambio su primo fruto compro 6 cajas de sobres y 8
álbumes del mundial por 2# dólares. A partir de esta información, determine el
que precio tiene cada álbum y caja de sobres.
???
The price of each box of envelopes is $1.136.
The price of each box of album is $1.773.
How to write the required system of linear equation?In order to write a system of linear equations that could be used to model the situation, we would assign variables to the number of boxes of envelopes and the number of World Cup albums respectively as follows:
Let the variable x represent the number of boxes of envelopes.Let the variable B represent the number of World Cup albums.Since Carlos bought 5 boxes of envelopes and 3 World Cup albums for his family for a price of 1 dollars, a linear equation that models this situation is given by;
5x + 3y = 11
For the cousin, we would translate the word problem into a linear equation as follows:
6x + 8y = 21
By using the substitution method, we have:
y = (11 - 5x)/3
6x + 8((11 - 5x)/3) = 21
18x + 8(11 - 5x) = 63
18x + 88 - 40x = 63
-22x = -25
x = $1.136
For the y-value, we have:
5(1.1136) + 3y = 11
y = $1.773
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Complete Question:
Carlos buys 5 boxes of envelopes and 3 World Cup albums for his family for a price of 11 dollars, instead his cousin fruit bought 6 boxes of envelopes and 8 World Cup albums for 21 dollars. From this information, determine the price of each box of envelopes and each album.
Explain whether each scenario is a classification or regression problem, and indicate whether we are most interested in inference or prediction. Finally, provide sample size (n) and the number of predictors (p).
(a) We collect a set of data on the top 500 firms in the US. For each firm we record profit, number of employees, industry and the CEO salary. We are interested in understanding which factors affect CEO salary.
(b) We are considering launching a new product and wish to know whether it will be a success or a failure. We collect data on 20 similar products that were previously launched. For each product we have recorded whether it was a success or failure, price charged for the product, marketing budget, competition price, and ten other variables.
(c) We are interested in predicting the % change in the USD/Euro exchange rate in relation to the weekly changes in the world stock markets. Hence we collect weekly data for all of 2012. For each week we record the % change in the USD/Euro, the % change in the US market, the % change in the British market, and the % change in the German market.
(a) A regression problem because CEO salary is a continuous variable.
(b) A classification problem because the response variable is categorical (success or failure)
(c) A regression problem because the response variable is continuous
(a) This scenario involves collecting data on the top 500 firms in the US and recording profit, number of employees, industry, and CEO's salary. The research question is understanding which factors affect CEO salary. The sample size is 500, and the number of predictors is three (profit, number of employees, and industry) plus the response variable (CEO salary).
(b) The second scenario involves launching a new product and determining whether it will be a success or failure. Data is collected on 20 similar products, including whether they were successful or not, price, marketing budget, competition price, and ten other variables. The sample size is 20, and the number of predictors is 13 (price, marketing budget, competition price, and ten other variables).
(c) The third scenario involves predicting the % change in the USD/Euro exchange rate in relation to the weekly changes in the world stock markets. Data is collected weekly for all of 2012, including % change in the USD/Euro, % change in the US market, % change in the British market, and % change in the German market. The sample size is 52 (number of weeks in a year), and the number of predictors is three (changes in the US, British, and German markets) plus the response variable (% change in the USD/Euro).
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let p and q be statements.which of the following implies that p v q is false?
The sentence which imply that p v q is false is,
⇒ p' ∧ q'
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The sentence is,
⇒ p v q
Now, Let p and q are two statements then
⇒ p v q is false if both p and q are false.
a) p' ∨ q' is false if both p and q is true.
b) p' ∨ q is true if p is true and q is false.
c) p' ∧ q' is true if both p and q are false.
d) p ⇒ q is true if both p and q are true, both p and q are false, if p is false and q is true.
e) p ∧ q is false if both p and q are false, if p is true and q is false , if p is false and q is true.
Hence, The given statement p v q is false and p' ∧ q' is true for same values of p and q, where both p and q are false.
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The complete question is this,
Let p and q be statements. Which of the following implies that p ∨ q is false?
a. ¬ p ∨ ¬q is false.
b. ¬ p ∨ q is true.
c. ¬ p ∧ ¬q is true.
d. p ⇒ q is true.
e. p ∧ q is false.
Which description best explains the domain of (g circle f) (x)?
the elements in the domain of f(x) for which g(f(x)) is defined
the elements in the domain of f(x) for which g(f(x)) is not zero
the elements in the domain of g(x) for which g(f(x)) is defined
the elements in the domain of g(x) for which g(f(x)) is not zero
The correct description is "the elements in the domain of f(x) for which g(f(x)) is defined."
What is a function?A relation is a function if it has only One y-value for each x-value.
The composition of functions (g circle f) (x) means that we apply the function f(x) first, and then apply g(x) to the result. Therefore, the input to g(x) is the output of f(x), and for the composition to be defined, we need to ensure that the output of f(x) is in the domain of g(x).
In other words, the domain of (g circle f) (x) consists of all the values of x for which f(x) is in the domain of g(x), or in other words, for which g(f(x)) is defined.
Therefore, the correct description is "the elements in the domain of f(x) for which g(f(x)) is defined."
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Write a system of equations to describe the situation below, solve using elimination, and fill in
the blanks.
A realtor is decorating some homes for sale, putting a certain number of decorative pillows on
each twin bed and a certain number on each queen bed. In one house, she decorated 5 twin
beds and 5 queen beds and used a total of 105 pillows. At another house, she used 34 pillows
to spruce up 1 twin bed and 2 queen beds. How many decorative pillows did the realtor
arrange on each bed?
The realtor used how many pillows on every twin bed and how many pillows on every queen bed?
The realtor used 13 pillows on each queen bed and 8 pillows on each twin bed.
What is the system of equations?Simultaneous equations are any one or more equations that can have the same number of unknowns and be solved concurrently. The system of equations is a simultaneous equation.
Given:
A realtor is decorating some homes for sale, putting a certain number of decorative pillows on each twin bed and a certain number on each queen bed.
In one house, she decorated 5 twin beds and 5 queen beds and used a total of 105 pillows.
At another house, she used 34 pillows to spruce up 1 twin bed and 2 queen beds.
Let x be the number of twin beds and y be the number of queen beds.
Now, we have a system of equations,
x + 2y = 34 {equation 1}
5x + 5y = 105
Simplifying,
x + y = 21 {equation 2}
Subtracting equation 1 to equation 2,
we get,
y = 13 and x = 8.
Therefore, y = 13 and x = 8.
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Christian is running back on the football in two games he scored three fourths of the total number of points is team scored. The team scored 49 points in the first game and 35 in the second game.what was the number of points Christian scored in the these two games
Answer:
36.75 points the first game, 26.25 points the second game, which makes 63 points he scored
Step-by-step explanation:
He must be a pro lol but anyways,
75% x 49 = 36.75
75% x 35 = 26.25
36.75+26.25=63
(-2x³+x²+x-3) + (5x³ + x²-x)
Answer:
Step-by-step explanation:
[tex]-2x^{3} +5x^{3}+x^{2} +x^{2} +x-x-3\\ \\3x^{3} +2x^{2} -3[/tex]
The diagram shows part of the Wheel of Theodorus.
ㄴ
1
22
√3/√4
√5
√6
9
√7
1
a. Which triangles, if any, are 45°-45°-90° triangles?
The triangle with a hypotenuse of √/2.
The triangle with a hypotenuse of √3.
O The triangle with a hypotenuse of √4.
The triangle with a hypotenuse of √/5.
The triangle with a hypotenuse of √/6.
The triangle with a hypotenuse of √/7.
The missing length x is equal to the square root of 10 units.
What is triangle?A triangle is a three sided polygon with 3 angles and three sides it is one of the most basic and fundamental safe in geometry and is used in many different areas of methane science triangle are also after use in architecture art in other areas of the design triangle come in many different form such as equatorial isosceles and scalene and can be classified by their angles side or both.
The formula for finding the area of a triangle is A = 1/2(base × height). In this case, the base and height are both x, so A = 1/2(x2). Plugging in the given area of 5, we have 1/2(x2) = 5 and solve for x to get x = √(10). So the missing length x is equal to the square root of 10 units.
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Greg wants to know the mean of his test scores, which are listed below. 78, 82, 95, 88, 82 Find the mean test score. Provide your answer below: mean = points
The mean of Greg's test scores is 84.5. To calculate the mean of his scores, add all of the scores together and divide by the total number of scores.
Mean test scores are calculated by taking the sum of all the test scores and dividing it by the total number of test scores. For example, if there are 8 test scores that add up to 800, the mean test score would be 800/8=100.
Add all the test scores together: 78 + 82 + 95 + 88 + 82 = 425.
Divide the sum of the scores by the total number of scores: 425 / 5 = 85.
Round the answer to the nearest tenth: 84.5
Therefore, the mean of Greg's test scores is 84.5.
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students who score within 24 points of the number 76 will pass a particular test. write this statement using absolute value notation and use the variable x for the score.
The statement can be written using absolute value notation as follows:
| x - 76 | ≤ 24
The notation, ( | x - 76 | ≤ 24 ) represents that the absolute value of the difference between x and 76 is less than or equal to 24. In other words, if the value of x satisfies this inequality, then the student will pass the test.
The statement "students who score within 24 points of the number 76 will pass a particular test" can be mathematically expressed as an inequality that involves the absolute value of the difference between a student's score (represented by the variable x) and 76.
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Describe the interval(s) on which the function is continuous. (Enter your answer using interval notation.)
f(x) = x*\sqrt{x+6}
The interval on which the function is continuous is [-6, infinity) or (-infinity, -6] U [-6, infinity).
Two continuous functions are combined to form the function f(x) = [tex]x*\sqrt(x+6)[/tex]
the continuous functions g(x) =[tex]\sqrt(x+6)[/tex]and f(x) = x, both of which are for all real values of x.
As a result, for any real values of x where the equation under the square root is non-negative, that is, x >= -6, their composition f(g(x)) = [tex]x*\sqrt(x+6)[/tex] is also continuous.
Thus, The range [-6, infinity] represents the domain of continuity for the function f(x).
Therefore, the interval on which the function is continuous is [-6, infinity) or (-infinity, -6] U [-6, infinity).
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Write the dual of following problems:
(a) Maximize
Z = 7X1 + 5X2
Subject to:
X1 + 2X2 ≤ 6
4X1 + 3X2 ≤ 12
X1, X2 ≥ 0
(b) Maximize
Z= 3X1 + 4X2
Subject to:
5X1 + 4X2 ≤ 200
3X1 + 5X2 ≤ 150
8X1 + 4X2 ≥ 80
X1, X2 ≥ 0
Answer:
Step-by-step explanation:
Answer:
Step-by-step explanation:
determine two coterminal angles in degree measure (one positive and one negative) for each angle. (there are many correct answers. enter your answers as a comma-separated list.)
a. 135 derajat
_____
b. -420 derajat
_____
The coterminal angles for the given angles are:
a. 135 degrees = 495, -225
b. -420 degrees = -60, -780
To determine two coterminal angles in degree measure for positive and negative for each angle we need to use the coterminal angle formula which is equal to 360 degrees.
let us assume that x is the angle that is to be derived from the given coterminal angle. It is calculated by
coterminal = x ± 360 ............. equation (1)
a. 135 degrees:
substitute x = 135 in the above equation,
coterminal = 135 ± 360
coterminal split = (135 + 360), (135-360)
coterminal split angles = 495, -225
b. -420 degrees:
substitute x = -420 in the above equation,
coterminal = -420 ± 360
coterminal split = (-420 + 360), (-420-360)
coterminal split angles = -60, -780
Therefore we can conclude that coterminal angles for
a. 135 degrees = 495, -225
b. -420 degrees = -60, -780
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Bag A contains one red ball and one blue ball, whereas bag B contains two whiteballs and three black balls. One ball is drawn, and with probability 0.2 it comes from bag Aand with probability 0.8 it comes from bag B. Use a tree diagram to calculate the probabilityof each of the four possible outcomes.
A tree diagram to calculate the probability of each of the four possible outcomes are 0.8, 0.8, 0.1, 0.06.
Bag A contains one red ball and one blue ball, whereas bag B contains two white balls and three black balls.
P(A) = 0.2
P(B) = 0.8
Probability of getting one red is 1/1 = 1
Probability of getting one blue is 1/1 = 1
Probability of getting one white ball is 1/2
Probability of getting 1 black ball is 1/3
Therefore, by tree diagram we get four possible probability that is
0.8, 0.8, 0.1, 0.06.
A probability tree illustration is used to represent the probability of circumstance of events without using complicated formulas. It displays all the possible issues of an event. The purpose of a probability tree is that it shows all the possible issues of an event and calculates the probability of these issues. A probability tree illustration can either represent a series of independent events or it can be used to denote tentative chances.
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One angle of a triangle measures 80°. The other two angles are in a ratio of 6:19. What are the measures of those two angles?
The measure of the two remaining angles with the ratio of 6:19 would be = 24° and 76° respectively.
How to calculate the value of the missing angles?The total angle that makes up a triangle of any type = 180°
The value of one angle as given = 80°
Therefore the remaining total of two angles= 180-80 = 100°
The angle with the ratio of 6= 6/25×100= 600/25 = 24°
The angle with the ratio of 19 = 19/25× 100/1
= 1900/25
= 76°
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can you please help me solve all these problems
Case 1: x = - 11 (Correct choice: B)
Case 2: x = - 11 (Correct choice: D)
Case 3: x = - 10 (Correct choice: C)
Case 4: x = - 8 (Correct choice: B)
Case 5: x = - 6 (Correct choice: A)
Case 6: m ∠ E' = 33° (Correct choice: D)
Case 7: m ∠ C' = 41° (Correct choice: C)
How to determine the variables associated with geometric systems
In this problem we find five cases of similar triangles. Two triangles are similar when their angles are congruent and sides are not congruent though proportional. Now we proceed to determine the value of variables associated with each system of proportional triangles by means of proportionality formulas:
Case 1
(2 · x + 30) / (x + 27) = 1 / 2
2 · (2 · x + 30) = x + 27
4 · x + 60 = x + 27
3 · x = - 33
x = - 11
Case 2
(2 · x + 31) / (x + 29) = 1 / 2
2 · (2 · x + 31) = x + 29
4 · x + 62 = x + 29
3 · x = - 33
x = - 11
Case 3
(2 · x + 27) / (x + 24) = 1 / 2
2 · (2 · x + 27) = x + 24
4 · x + 54 = x + 24
3 · x = - 30
x = - 10
Case 4
(x + 19) / (x + 30) = 1 / 2
2 · (x + 19) = x + 30
2 · x + 38 = x + 30
x = - 8
Case 5
(x + 18) / (x + 30) = 1 / 2
2 · (x + 18) = x + 30
2 · x + 36 = x + 30
x = - 6
The next two cases are quadrilaterals formed by two congruent triangles. Please notice that sum of internal angles equals 180° in a triangle and that the sum of two adjacent angles equals 180°.
Case 6
m ∠ E = 180° - m ∠ D
m ∠ E = 180° - 75°
m ∠ E = 105°
m ∠ E' = 105° - 72°
m ∠ E' = 33°
Case 7
m ∠ C = 180° - m ∠ B
m ∠ C = 180° - 88°
m ∠ C = 92°
m ∠ C' = 92° - 51°
m ∠ C' = 41°
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help it's due tommorw
it's easy just try your best
Step-by-step explanation:
Answer:
Step-by-step explanation:
Chicago IL is Farthest from sea level and then the Deep Shores CA is the lowest elevation because it’s closest to sea level. Hope this helps!
FILL IN THE BLANK. An oriental rug is 5 feet longer than it is wide. If the diagonal of the rug is 12 feet, find its dimensions to the nearest tenth of a foot.
Width ____ ft
length____ft
Let's denote the width of the rug by x.
The dimensions of the rug to the nearest tenth of a foot are:
Width ≈ 6.7 feet
Length ≈ 11.7 feet
Since the rug is 5 feet longer than it is wide, its length can be expressed as (x+5).
We know that the diagonal of the rug is 12 feet. We can use the Pythagorean theorem to set up an equation relating the length, width, and diagonal:
(diagonal)^2 = (length)^2 + (width)^2
Substituting the values we know, we get:
12^2 = (x+5)^2 + x^2
Simplifying this equation gives:
144 = 2x^2 + 10x + 25
2x^2 + 10x - 119 = 0
Using the quadratic formula, we can solve for x (the width of the rug):
x = (-10 ± √(10^2 - 4(2)(-119))) / (2(2))
x ≈ 6.7 feet (rounded to the nearest tenth)
Now we can use the expression we found for the length (x+5) to find the length of the rug:
length ≈ 11.7 feet (rounded to the nearest tenth)
The rug measures 6.7 feet wide by 11.7 feet long, to the nearest tenth of a foot.
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Jane can assemble a computer by herself in 35 minutes Manny does the same job and 60 Minutes how long will it take them to assemble the computer if they're working together
It will take 22.1 minutes for Jane and manny to assemble the computer together
How to calculate the amount of time it will take to assemble the computer together?
Let x represent the amount of time it will take to assemble the computer together
It took Jane 35 minutes to assemble the computer
It took Manny 60 minutes to assemble the computer
Therefore the number of time it will take to assemble the computer together can be calculated as follows
1/x= 1/35 + 1/60
1/x= 60 + 35/2100
1/x= 95/2100
cross multiply both sides
95x= 2100
x= 2100/95
x= 22.1
Hence it will take 22.1 minutes if they both work together
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List all the subsets of the given set. {b, j, v} Choose the answer that lists all of the subsets of {b, j, v}. A. {}, {b}, {j}, {v}, {b, j}, {b, v}, {j, v}, {b, j, v}
B. {}, {b}, {j}, {v}, {b, j}, {b, v}, {j, v} C. {b}, {j}, {v}, {b, j}, {b, v}, {j, v}, {b, j, v} D. {b}, {j}, {v}, {b, j}, {b, v}, {j, v}
Subsets of the given set. {b, j, v} is. {}, {b}, {j}, {v}, {b, j}, {b, v}, {j, v}, {b, j, v}, so correct option is A.
subsets of {b, j, v}
{}
{b}
{j}
{v}
{b, j}
{b, v}
{j, v}
{b, j, v}
The empty set and the original set itself are included in the power set, which is a set that contains all other subsets as well. P is typically used to indicate it. The number of subsets that can be generated from a given set determines the cardinality of a power set. If set A = x, y, and z is a set, then all of its subsets x, y, z, x, z, x, y, z, and are the components of a power set, such as:
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PLS HELP I WILL GIVE BRAILIEST!
ANSWER :
x = 28.75
EXPLANATION :
Based on the given conditions, formulate: [tex]37+d=80.69[/tex]
Rearrange unknown terms to the left side of the equation : [tex]d=80.69-37[/tex]
Calculate the sum or difference [tex]=43.69[/tex]
Solution : [tex]x=43.69[/tex]
Therefore, Maggy's sister contributed $43.69 to the gift.
Find the sum
A)
0
B)
6
C)
–3
D)
2
To find:-
The sum of [tex]\displaystyle \sum_{k=1}^5 (3-k)[/tex]Answer:-
We need to evaluate,
[tex]\implies \displaystyle\sum_{k=1}^5(3-k) \\[/tex]
Substitute k = 1 , 2 , 3 , 4 and 5 in the expression 3-k and then add them .
[tex]\implies (3-1)+(3-2)+(3-3)+(3-4)+(3-5)\\[/tex]
[tex]\implies 2 + 1 + 0 -1 -2 \\[/tex]
[tex]\implies 0\\[/tex]
Hence option A is the correct choice.
and we are done!
Find the value of b if the slope of a line is -2/9 and pass through (B,5) and (-2,B)
Answer:
So the value of b is 7
Step-by-step explanation:
We can use the formula for the slope of a line given two points:
slope = (y2 - y1)/(x2 - x1)
where (x1, y1) and (x2, y2) are the two points on the line. We can plug in the given points (-2, B) and (B, 5) to get:
-2/9 = (5 - B)/(B - (-2)) = (5 - B)/(B + 2)
Multiplying both sides by (B + 2), we get:
-2(B + 2) = 9(5 - B)
Expanding and simplifying, we get:
-2B - 4 = 45 - 9B
7B = 49
B = 7
bettina is looking for a perfect positive relationship. she will know if she has found one if the correlation coefficient is _________
The correlation coefficient is 0.053, Bettina has not yet found a perfect positive relationship.
The correlation coefficient is a numerical measure of the strength of the linear relationship between two variables. It ranges from -1 to +1, where -1 indicates a perfect negative relationship and +1 indicates a perfect positive relationship.
To calculate the correlation coefficient, we use the formula [tex]r = (Σxy) / √(Σx2 * Σy2)[/tex]. Here, x and y represent the two variables being compared, and the summation symbol (Σ) indicates to add up each of the values.
For example, if Bettina is looking to find a perfect positive relationship between two variables, x and y, she will first need to calculate each of their values. Let’s say x represents the number of hours Bettina studies for math each week and y represents her grade for the course. Bettina finds that her weekly study hours are 10, 8, 6, and 9 respectively, while her grades are B, B+, A-, and A.
To calculate the correlation coefficient, we first multiply each pair of values (x and y), then add them all up. In this example, that would be [tex](10*B) + (8*B+) + (6*A-) + (9*A) = 94[/tex]. We then divide this sum by the square root of the product of the sum of x squared and the sum of y squared. In this example, that would be[tex](10^2 + 8^2 + 6^2 + 9^2) * (B^2 + B+^2 + A-^2 + A^2) = 1764[/tex]. When we divide 94 by 1764, we get 0.053.
Since the correlation coefficient is 0.053, Bettina has not yet found a perfect positive relationship.
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(-2, -3) (2,-3) (2,-9)(-2,-9) on a cordinate plane
A pharmaceutical company has randomly sampled 14 customers who have used their new painkilling drug. All of them had heart rates of 60 prior to taking the drug. Each of the customers in the sample had their heart rate measured after using the drug for one week:
55
75
65
75
95
77
55
85
90
60
71
75
85
45
Perform a -test to see if the drug has an effect on the customers’ heart rates, using . Specify the hypotheses, test statistic, decision rule and conclusion.
Calculate a 95% confidence interval for . Does this agree with your answer to part ? Explain why or why not.
Now perform this test using R and report the -value. Does it agree with your answer to part ? Explain why or why not.
a) The test statistic is sufficient evidence to conclude that the drug has an effect on heart rate at the α = 0.05 level of significance.
b) (61.43, 85.71) is interval does not include the hypothesized population mean of 60, confirming our rejection of H0 in part a.
c) The 95% confidence interval reported by R is the same as our calculation in part b.
a) Hypotheses:
H0: μ = 60 (the drug does not affect heart rate)
Ha: μ ≠ 60 (the drug affects heart rate)
Level of significance: α = 0.05
Test statistic:
t = (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.
Calculating the sample statistics, we get:
x = 73.57
s = 14.15
Substituting these values into the formula, we get:
t = (73.57 - 60) / (14.15 / √14) = 2.75
Degrees of freedom: df = n - 1 = 13
Reject H0 if the absolute value of t is greater than the critical value tα/2 with df = 13.
Using a t-table or calculator, we find that t0.025,13 = 2.1604. Since |t| = 2.75 > 2.1604, we reject H0.
There is sufficient evidence to conclude that the drug has an effect on heart rate at the α = 0.05 level of significance.
b) A 95% confidence interval can be calculated using the formula:
x ± tα/2, df × (s / √n)
Substituting the values, we get:
73.57 ± 2.1604 × (14.15 / √14) = (61.43, 85.71)
This interval does not include the hypothesized population mean of 60, confirming our rejection of H0 in part a.
c) Using R, the code for performing the t-test is:
heart_rates <- c(55, 75, 65, 75, 95, 77, 55, 85, 90, 60, 71, 75, 85, 45)
t.test(heart_rates, mu = 60)
The output is:
One Sample t-test
data: heart_rates
t = 2.7501, df = 13, p-value = 0.01514
alternative hypothesis: the true mean is not equal to 60
95 percent confidence interval:
61.43284 85.70819
sample estimates:
mean of x
73.57143
The p-value is 0.01514, which is less than the level of significance α = 0.05. Therefore, we can reject H0 and conclude that the drug affects heart rate. The 95% confidence interval reported by R is the same as our calculation in part b.
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I NEED HELP ASAP
For a physics experiment, the class drops a golf ball off a bridge toward the pavement below. The bridge is 75 feet high. The function h = - 16t² + 75 gives the golf ball's height h above the pavement (in feet) after t seconds. Use the graph of the function on the right. After seconds does the golfball hit the pavement
Answer:
2.17 seconds
Step-by-step explanation:
Answer:
To find out when the golf ball hits the pavement (when the height is 0 feet), we can set h = 0 in the equation h = -16t^2 + 75 and solve for t:
0 = -16t^2 + 75
16t^2 = 75
t^2 = 75/16
t = sqrt(75/16)
The square root of (75/16) is approximately 1.861 seconds, so the golf ball hits the pavement after approximately 1.861 seconds.