The radius of convergence is R = 7/6.
To find the radius of convergence of the function f(x) = 7x^3 - 5x^2 - 6x^5, we can use the ratio test.
The ratio test states that if the limit of |a_{n+1}/a_n| as n approaches infinity is L, then the series converges absolutely if L < 1, diverges if L > 1, and the test is inconclusive if L = 1.
We can apply this test to the power series representation of f(x) as follows:
f(x) = 7x^3 - 5x^2 - 6x^5
= 0 + 0x + 0x^2 + 7x^3 - 5x^4 + 0x^5 + 0x^6 + ...
The coefficients of x^n for n > 2 are all zero, so we can write the power series as:
f(x) = 7x^3 - 5x^2 - 6x^5 + 0x^6 + ...
Using the ratio test, we have:
|a_{n+1}/a_n| = |(-6(x+1)^5)/((n+1)(7/n)^3 - 5(n/n)^2 - 6n^5)|
= 6(n+1)^5/(n^5(7n^3 - 5n^2(n+1) - 6(n+1)^5))
Taking the limit as n approaches infinity, we get:
L = lim_{n->∞} |a_{n+1}/a_n|
= lim_{n->∞} 6(n+1)^5/(n^5(7n^3 - 5n^2(n+1) - 6(n+1)^5))
= 6/7
Since L < 1, the series converges absolutely for |x| < 7/6. Therefore, the radius of convergence is R = 7/6.
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A bag contains 8 red marbles, 9 yellow marbles, and
7 green marbles. How many additional red marbles
must be added to the 24 marbles already in the bag so
that the probability of randomly drawing a red marble
is 3/5?
A. 11
B. 16
C. 20
D. 24
E. 32
Answer:
x = 16
Step-by-step explanation:
[tex]\frac{8+x}{24+x}=\frac{3}{5}\\ \\ 5(8+x)=3(24+x)\\\\40+5x=72+3x\\\\5x=32+3x\\\\2x=32\\\\x=16[/tex]
Use a trigonometric ratio to solve for y. Round to two decimal places as necessary!!
HELP, and can someone explain how to get the answer?
On solving the provided question, we can say that by trigonometry so sin 90 = p/h => y = 19
what is trigonometry?The area of mathematics known as trigonometry examines the correlation between triangle side lengths and angles. The area first appeared in the Hellenistic era, around the third century BC. from the use of geometry in astronomical study. The area of mathematics known as exact methods deals with specific trigonometric functions and how they might be used in calculations. There are six popular trigonometric functions in trigonometry. Sine, cosine, tangent, cotangent, secant, and cosecant are their respective names and acronyms (csc). Studying the characteristics of triangles, particularly right triangles, is called trigonometry. The study of geometry, however, is the characteristics of all geometric figures
here,
it is a right angled triangle
by trigonometry
so sin 90 = p/h
1 = 19/y
y = 19
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find the y-intercept, the equation of the axisof symmetry, and the x-coordinate of vertex for the graph of f(x)= x^2-3x+5. use this information to graph the function.
The y-intercept of the graph of f(x) = x^2 - 3x + 5 is (0, 5).
The equation of the axis of symmetry is x = (3/2).
The x-coordinate of the vertex is (3/2), and the y-coordinate of the vertex is (5 - (9/4)) = (-1/4).
So the vertex of the graph of f(x) = x^2 - 3x + 5 is (-1/4, (3/2)).
To graph the function, you can start by plotting the vertex, then use the axis of symmetry to reflect the parabola across the line x = (3/2). The y-intercept is the highest point of the parabola if it opens upwards, and the lowest if opens downwards. This can be used to confirm the direction of the parabola.
A quadratic function's U-shaped curve is seen on a parabola graph. A right circular cone and a plane surface intersect to generate one of the conic sections known as a parabola in mathematics. It has a U-shaped symmetry in the plane. The conventional form of a parabola is a graph whose equation takes the form f(x) = ax2+bx+c. A parabola's vertex is its most extreme point, and the vertical line that runs through it forms its axis of symmetry.
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(5,3)(-1,3)(2,-2)(-4,-2) Find the perimeter
Answer：[tex]2\sqrt34+12[/tex]
Step-by-step explanation:Draw a planar Cartesian coordinate system and that's it
Why is geometry so hard?
Due to the lack of spatial thinking instruction in most math classes, geometry is hard. They should acquire geometrical ideas easily instead.
Who named geometry?The father of geometry, Euclid was a brilliant mathematician. The terms "ge" and "metria," which both imply "measuring," are the origins of geometry. For more than 2000 years, the Greeks' method of approaching geometry has served as the foundation for geometry.
The study of various mathematical or actual-world forms, figures, and sizes is known as Geometry. We learn about various angles, transformations, and similarities between the forms in geometry. The point, line, and plane are the main building blocks of geometry.
Everyone finds it difficult to dive into the issue of proofs. Children must realize that this issue affects everyone, even the mathematically skilled. Because they have trouble seeing how 3D, and 2D, forms, angles, and lines connect from various viewpoints, most youngsters struggle with geometry issues. Even kids who are really good at solving sum problems in math might be utterly stumped by this.
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Due to the lack of spatial thinking instruction in most math classes, geometry is hard. They should acquire geometrical ideas easily instead.
What are some reasons that children think geometry is hard?Visual Ability: While some kids think in terms of forms and proportions, others think visually. Children must comprehend geometry in order to visualize, or else they may not be able to see things in a larger context.
Numbers & Measurements: Children will learn about the link between shapes and sizes as numbers and measures come to life in shapes and sizes. Children will be able to comprehend the measurements and connections of lines, angles, surfaces, and solids using numbers and measures.
3D Thinking: Since these ideas originated in geometry, children cannot possibly comprehend 2 and 3 dimensions without having a working knowledge of geometry! The only way to visualize X, Y, and Z axis, triangles, and pyramids are with the aid of geometry.
Spatial Understanding: If it weren't for geometry, kids could have trouble comprehending the notion of spatial comprehension. Children will learn to appreciate the wonders of the world by studying concepts relating to space, location, size, and shape of objects.
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Which type of polynomial is 2 − x2 x3?
Answer:
cubic polynomial
What is the image of (12,-12) after a dilation by a scale factor of 1/3 centered at the origin?
The image of (12,-12) after a dilation by a scale factor of 1/3 centered at the origin is (4, -4).
What is dilation at origin?
On the coordinate plane, figures can be magnified. The centre of dilation is frequently the origin. The coordinates of the points in the preimage can be multiplied by the scale factor to get the points in the image if you are distorting a figure that is centered at the origin.
We have to find the image of (12,-12) after a dilation by a scale factor of 1/3 centered at the origin.
To find the image after dilation multiply both coordinates (12, -12) by 1/3 which is the same as dividing by 3.
⇒ (12, -12)*1/3 = (4, -4)
Hence, the image of (12,-12) after a dilation by a scale factor of 1/3 centered at the origin is (4, -4).
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Mei and her friends went to the theater to see the latest action movie. After Mei spent $12.00 on the ticket, she had $9.25 left to buy snacks. How much money did Mei start with?
Answer: Mei had $21.25
Step-by-step explanation: if you add $12 to $9.25 you get $21.25
Answer:
The money she had in the beginning was $21.25
Step-by-step explanation:
Given ,
The cost of the ticket Mei spent = $12.00
Money remaining = $9.25
To Find : The Money she had in the beginning
When we add the given values we get the money she had in the beginning
which is,
Amount = Amount Spent on a Ticket + Money Remaining
Amount = [tex]$12.00 + 9.25[/tex]
Amount = [tex]21.25[/tex]
Hence , the amount she had in the beginning was $21.25
Which system of equations can be graphed to find the solution(s) to x2 = 2x + 3?A.y=x2+2x+3/y=2x+3B.y=x2-3/y=2x+3C.y=x2-2x-3/y=2x+3D.y=x2/y=2x+3
The correct choice is:
y= x^2 and y= 2x+3
Given equation,
x^2=2x+3
this can be used for factorization by taking all the factors in one side of the equation
x^2-2x-3=0
or, x^2-3x+x-3=0
or, x(x-3)+x(x-1)=0
or, (x-3)(x-1)=0
Therefore,
x=-1 or x= 3
The result indicates,
when we plot the equation x^2=2x+3 in graph, as a function of y=x^2-3x+x-3, the x axis will be intersected on x=-1 and x=3
Now, if we consider
y=2x+3 and y=x^2
it gives x^2-2x-3=0 all over again, when combined.
The solution to,
x^2=2x+3 is the intersection point of the graph of straight line alongside the graph for the quadratic equation curve.
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The correct answer is y= [tex]x^{2}[/tex] and y= 2x+3 that can be graphed to find the solution(s) to [tex]x^{2}[/tex]= 2x + 3 .
The Given equation was ,
[tex]x^{2}[/tex]= 2x + 3 .
this can be used for factorization by taking all the factors in one side of the equation
[tex]x^{2}[/tex] - 2x - 3 =0
or, [tex]x^{2}[/tex]-3x+x-3=0
x(x-3)+x(x-1)=0
(x-3)(x-1)=0
∴x=-1 or x= 3
The result indicates,
when we plot the equation [tex]x^{2}[/tex]=2x+3 in graph, as a function of y=[tex]x^{2}[/tex]-3x+x-3, the x axis will be intersected on x=-1 and x=3
Now, if we consider
y=2x+3 and y=x^2
it gives [tex]x^{2}[/tex]-2x-3=0 all over again, when combined.
The solution to,
[tex]x^{2}[/tex]=2x+3 is the intersection point of the graph of straight line alongside the graph for the quadratic equation curve.
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What is the area of a triangle with the base of 7 cm and the height of 12 cm?
The area of the triangle is 42 cm².
To calculate the area of a triangle, we need to use the formula:
Area = (base x height) ÷ 2
Therefore, the area of a triangle with a base of 7 cm and a height of 12 cm is:
Area = (7 cm x 12 cm) ÷ 2
Area = 42 cm²
The area of a triangle with a base of 7 cm and a height of 12 cm is 42 cm². This can be calculated by using the formula: Area = (base x height) ÷ 2. The base and the height of the triangle are multiplied together and then divided by two to get the area. Therefore, the area of the triangle is 42 cm².
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for the function f(x)= 1-2x/5x-1, find f^-1(x)
'Therefore , the solution of the given problem of function comes out to be f⁻¹(x) = 1 + x / 5x+2 .
What exactly does the word function mean?Mathematics is the study of number, their variations, the geographical tissue, architecture, and both both actual and fictitious locations. A equation is a depiction of the relationship here between number of inputs and the corresponding outputs for each one. A function is just a collection of input that, when put together, provide one variable distinct output with each input. Every function is assigned a city, county, or scope, sometimes known as a realm.
Here,
Given:
=> f(x) = (1 -2x)/(5x-1)
To find f⁻¹(x)
So,
=> y = (1 -2x)/(5x-1)
=> x = (1 -2y)/(5y-1)
=> y = 1 + x / 5x+2
=> f⁻¹(x) = 1 + x / 5x+2
Therefore , the solution of the given problem of function comes out to be f⁻¹(x) = 1 + x / 5x+2 .
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Is it possible to draw a triangle with sides of lengths 2cm 3cm and 7 cm?
It is not possible to construct a triangle whose sides are 2 cm 3 cm and 7 cm because the sum of two sides(2 cm and 3 cm) are smaller than third sides(7 cm).
In the given question, we have to check that it is possible to construct a triangle whose sides are 2 cm 3 cm and 7 cm.
To check whether the given sides can make a triangle or not, we have to check that the sum of two sides always grater than the third side.
To check this we firstly add the 2 and 3
2 + 3 < 7
5 > 7
Now we add 2 and 7
2 + 7 > 3
9 > 3
Now we add 3 and 7
7 + 3 > 2
10 > 2
It is not possible to construct a triangle whose sides are 2 cm 3 cm and 7 cm because the sum of two sides(2 cm and 3 cm) are smaller than third sides(7 cm).
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What is the domain and range of modulus function Class 11?
The domain of the modulus function is the set of all real numbers. The range of the modulus function is the set of non-negative real numbers.
The modulus function, also known as the absolute value function, is a function that takes any real number as input and returns its distance from zero on the number line.
In mathematical notation, the modulus function can be represented as |x|, where x is the input.
For example, if x = -3, then |x| = 3, and if x = 5, then |x| = 5.
In class 11, the domain of the modulus function is all real numbers, which is represented by (-∞, +∞) and the range is non-negative real numbers represented by [0, +∞)
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Determine the partial fraction expansion for the rational function below: s/(s – 7)(s^2 - 49) S/(s - 7) (s2 - 49) Determine the partial fraction expansion for the rational function below. -25-56/ (5+3) (s^2+ 16) - 2s - 56 / (s +3)(s^2 + 16)
The partial fraction expansion for the rational function
S/ (s - 7) (s^2 - 49) = -25-56/ (5+3) (s^2+ 16) - 2s - 56 / (s +3) (s^2 + 16)
can be found by decomposing it into the sum of simpler fractions.
One way to do this is to use the method of long division and then match the coefficients of the resulting polynomials. The process is as follows:
Divide S by (s - 7) to get a quotient of S1 and a remainder of R1.
Divide R1 by (s^2 - 49) to get a quotient of S2 and a remainder of R2.
Write the expansion as S1 + S2 + R2/(s - 7)(s^2 - 49)
Alternatively, we can use the method of partial fraction decomposition by inspection.
To do this, you will write the rational function as the sum of simpler fractions with the same denominator.
S/ (s - 7) (s^2 - 49) = A/(s-7) + B/ (s^2 - 49) + (C*s + D)/(s^2 - 49)
where A, B, C, and D are unknowns that you will find by matching coefficients.
S = (A(s^2-49) + B(s-7) + C*s + D) / (s-7) (s^2-49)
You can use this equation to solve for A, B, C, and D by matching coefficients.
-25-56/ (5+3) (s^2+ 16) - 2s - 56 / (s +3) (s^2 + 16) = A/(s-7) + B/(s^2 - 49) + (C*s + D)/ (s^2 - 49)
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3X + 1 i need help with this asp
Answer:
In algebra, the 3x + 1 semigroup is a special subsemigroup of the multiplicative semigroup of all positive rational numbers. The elements of a generating set of this semigroup are related to the sequence of numbers involved in the still open Collatz conjecture or the "3x + 1 problem".
Step-by-step explanation:
True/False 1. The complement of a non-regular language can never be regular.
Answer: false
Step-by-step explanation:
False
70x-40x^4 factor the expression
The expression 70x-40x^4 if factor is:-10x (4x^3-7).
What is expression?Expressions can be defined as equation that has both number that are known as well as numbers that are not known .
Example of expression is the sum of 9 and 1 will gives us 10 which is calculated as (9+1 =10).
Given data:
Expression: 70x-40x^4
Now let factor the expression using the given expression
70x-40x^4
Rearrange the expression
-40x^4 +70x
Common factor
-10(4x^4 -7x)
So the expression is:
-10x (4x^3-7)
Therefore we can conclude that the expression is -10x (4x^3-7).
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What is the formula of substitution property?
The substitution property states that if two expressions are equal, then they remain equal when the same values are substituted into them.
The formula of substitution propertyThis is usually expressed as:
If a = b, then a[x/y] = b[x/y]The substitution property states that if two expressions are equal, then they remain equal when the same values are substituted into them. This means that if one expression is equal to another, then the same value can be swapped in for any of the variables within them and the equality will still hold.
For example, if we have the equation 4x + 7 = 11, then we can swap in any value for x and the equation will still be true. This is because the substitution property allows us to make a substitution for any of the variables and the equation will remain true.
This property is useful in solving equations and is a key part of algebra.
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I need help finding the answer for number 17!!
Answer:
39°
Step-by-step explanation:
[tex]angle \: 1 + 2 = 180(co \: interior \: angles \: \:in \: parallel \: lies \: are \: add \: up \: t \: equal \: 180) \\ [/tex]
13x + 24 + 5x -6 = 180
18x+ 18 =180
18x +18 -18 =180-18
18x = 162
18 18
x = 9
Therefore,
m<2 = 5x- 6
= 5(9) -6
= 45-6
= 39°
hELP QUICK PLEASEE!!! 30 points!
The solution to the following linear equation problem can be seen in the graph showing how the width and diameter of one face are linearly related.
What exactly is a linear equation?In a linear model, one level is the maximum amount of levels that may be used. Any equation that has a theoretical extension of at least two is said to be nonlinear. There is a straight line on a map with a simple equation. A nonlinear equation's curve is depicted on the graph.
Here,
one face's perimeter and side lengths are linearly related.
line-drawing formula.
The formula y = mx + b can be used to describe a regular model.
If m is the rising rate and b is in reality the y intercept, then x and y are variables.
The graph shows a linear relationship between the length and perimeter of one face.
Therefore, the graph showing how the perimeter or edge length of one face are linearly connected is the solution to the provided linear equation problem.
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At a local fitness center,members pay a $10 members pay $3 for each class nonmembers pay $5 for each aerobics class. For what number of aerobics classes will the cost of members and nonmembers be the same
The number of aerobics classes that will cost the same for members and non-members is 5 classes.
When would the cost be equal?The linear equation that represents the amount paid by members is:
membership fee + (amount per class x number of classes)
= $10 + ($3 x m)
= $10 + $3m
The linear equation that represents the amount paid by no-members is: (amount per class x number of classes)
$5 x m
= $5m
When the two costs are the same, the two equations would be equal:
$5m = $10 + $3m
$5 - $3m = $10
$2m = $10
m = $10 / $2
m = 5
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London went shopping for a new video game because of a sale. The store was offering a 20% discount. If the price on the tag was $50, what would be the price after the discount but before tax, to the nearest dollar and cent?
The price of a new video game, before tax in the shop that London visited is $10.00
How to determine the price of a new video gameThe question is a percentage problem, percent is a fraction of hundred and the calculations will be using 100
In the store that London visited, they were offering a 20% discount
= 20 percent
= 20 / 100
= 0.2
discount factor = 1 - 0.2 = 0.8
the discount factor will be used to multiply by the price of the video game to get a new price
selling price of the new video game before tax = $50 * 0.2 = $10
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Please solve the question and show work.
Answer:
all i could do was simplify it
Step-by-step explanation:
15.
Marco is making blueberry muffins. The ratio between flour and blueberries for the recipe being used is shown in the table below.
Cups of Cups of
Flour Blueberries
1-2
24
A.Ocup
2
8
How many cups of blueberries will Marco need if he has 2 cups of flour to use?
B.
cup
D
1
2
3
4
1
c. 11 cups
6 cups
The magic ratio for muffins in 2:2:1:1.
What is the appropriate liquid-to-flour ratio while baking muffins?The magic muffin ratio is 2:2:1:1. This equates to 2 parts flour, 2 parts liquid, 1 part fat, and 1 part eggs. The most crucial aspect is to remember to utilize the same unit of weight for each component!
Rinse frozen blueberries many times in cold water until the water turns lighter in color to prevent your batter and baked items from developing a purple-blue tint. Then, blot them dry with a paper towel and gently fold them into the batter. The goal of this procedure is to maintain fast breads light and delicate by limiting the quantity of gluten produced during the mixing process. To do this, the dry components are mixed in a one bowl, the wet ingredients are combined in a separate bowl, and then the two mixtures are combined together.
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Alicia went to the supermarket with $200. She bought 3 cans of peas for $9 each 5 bags of flour at $15 each and 7 packs of milk at $12.50 each. Calculate the change she received.
The 3 cans of peas cost 39 = $27
The 5 bags of flour cost 515 = $75
The 7 packs of milk cost 7*12.50 = $87.50
The total cost of all the items is 27+75+87.50 = $189.50
To calculate the change she received, you subtract the total cost from the amount she had:
change = 200 - 189.50 = $10.50
So Alicia received $10.50 change.
HELP THIS IS DUE TOMORROW Tamika throws a ball up in the air. The graph below shows the height of the ball hin
feet after t seconds. How long is the ball in the air?
Answer:
3 seconds?
Step-by-step explanation:
I mean he throws it up at x= 0 and it drops at x=3 So technically it must be 3?
This seems too simple, but see if it is correct.
if JKLM is a parallelogram what is measure angle K
For the shape to be a quadrilateral, then the measure of angle k must 128 degrees
What is JKLM parallelogram measure angle K?For quadrilateral PQRS to be an actual quadrilateral, then:
K=M opposite angles of quadrilateralJ+M=180 adjacent angles of quadrilateralJ+M=180 implies that:
2x+9+6x-1=180
Collect like terms:
2x+6x=180-9+1
8x=172
Divide both sides by 8
x=21.5
We have:
K=M
This becomes
K=6x-1
Substitute 21.5 for x
K=6*21.5-1
K=129-1
Subtract 1 from 129
K=128
Hence, the measure of angle K must be 128 degrees
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Ray AT bisects MAR. If m MAT = (6x - 4) and m RAT = (2x + 8), what is is the m MAR
Answer:
∠MAR = 28
Step-by-step explanation:
Given angle MAR bisected by ray AT to form angle MAT = (6x-4) and RAT = (2x+8), you want to know the measure of MAR.
Angle bisectorAn angle bisector divides an angle into two congruent parts:
∠MAT = ∠RAT
6x -4 = 2x +8
4x = 12 . . . . . . . . add 4-2x
x = 3 . . . . . . . divide by 4
Then the measure of one of the halves is ...
∠RAT = 2x +8 = 2·3 +8 = 14
So, the measure of the whole is ...
∠MAR = 2·14
∠MAR = 28
x^4+4x^3+7x^2+16x+12
The factoring of the polynomial x^4+4x^3+7x^2+16x+12 = 0, considering it's linear factors, is given as follows:
(x² + 5.4x + 11.29)(x + 1)(x + 3).
How to factor the polynomial?The polynomial for this problem is defined as follows:
x^4+4x^3+7x^2+16x+12 = 0.
We want to factor the polynomial according to it's linear factors, hence the Factor Theorem is used.
The Factor Theorem states that a polynomial function with roots(also called zeros) [tex]x_1, x_2, \codts, x_n[/tex], is represented by the product of it's linear factors [tex]x - x_1, x - x_2, \cdots x - x_n[/tex], according to the rule the rule presented as follows:
[tex]f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)[/tex]
The coefficients of the polynomial in this problem are given as follows:
a = 1, b = 4, c = 7, d = 16, e = 12.
Inserting these coefficients into a 4th degree equation solver, the roots are given as follows:
x = -2.7 + 2i.x = -2.7 - 2i.x = -1.x = -3.Hence the polynomial is factored as follows:
(x + 2.7 - 2i)(x + 2.7 + 2i)(x + 1)(x + 3).
Applying the subtraction of perfect squares for the first two linear factors, considering that i² = -1, we have that:
(x + 2.7 - 2i)(x + 2.7 + 2i) = (x + 2.7)² - (2i)²
(x + 2.7 - 2i)(x + 2.7 + 2i) = (x² + 5.4x + 11.29).
Hence the factored polynomial is of:
(x² + 5.4x + 11.29)(x + 1)(x + 3).
Missing informationThe problems asks to factor the polynomial according to it's linear factors.
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Estimate the level of DVD sales (measured in million dollars) for movies, by using the amount of annual revenue (measured in million dollars) generated by the box office gross. We randomly selected a sample of 43 movies for the same year.
Excel (for α = 0.05) printed out: r^2 = 0.5123, SY.X = 12.2279, tcritical = 2.0195, hin = 0.026249, b0 = 4.8445, b1 = 0.1631. Assume normality and homoscedasticity are satisfied.
Source df Sum of Squares Mean Squares F Stat P-value F Crit
Regression 1 6439.6133 ? ? N/A 4.0785
Error ? ? ?
Total 42 12570.025
a) Interpret the meaning for the coefficient of determination and also the standard error of prediction, for this problem.
b) At the 0.05 level of significance, is there evidence of a linear relationship between the revenues of DVD sales and the box-office gross?
c) Construct the 95% confidence interval of the actual mean revenue of DVD sales for all the movies that generated 100 million dollars box-office gross.
d) Construct the 95% prediction interval for the actual revenue of DVD sales for a movie that generated 100 million dollars box-office gross.
please show all work and calculations
The average deviation of predicted DVD sales from the actual is 12.2279 million dollars. F-critical value is 4.0785>4.0785 we can reject the null hypothesis. The 95% confidence interval is (17.1536, 25.1554). The 95% prediction interval for the given value of x is (-3.8623, 46.1713).
Estimate the level of DVD sales for movies, by using the amount of annual revenue generated by the box office gross
We randomly selected a sample of 43 movies for the same year.
Excel α = 0.05, r^2 = 0.5123
=51.23%
51.23% of the DVD sales variation is explained by annual revenue.
SY.X = 12.2279.
Therefore, the average deviation of predicted DVD sales from the actual is 12.2279 million dollars.
b. The hypothesis being tested is:
Null Hypothesis, H0: β1 = 0
Alternative Hypothesis, H1: β1 ≠ 0
The test statistic, F = (6439.6133/1)/((12570.025 - 6439.6133)/(42 - 1)) = 43.07
The F-critical value is 4.0785.
Since 43.07 > 4.0785, we can reject the null hypothesis.
Therefore, we can conclude that there is a linear relationship between the revenues of DVD sales and the box-office gross.
c. y = bo + b1*x
y = 4.8445 + 0.1631*x
Put x = 100
y = 4.8445 + 0.1631*100 = 21.1545
The 95% confidence interval for the given value of x is:
= y ± t*s*[tex]\sqrt{0.026249}[/tex]
= 21.1545 ± 2.020*12.2279*[tex]\sqrt{0.026249}[/tex]
= (17.1536, 25.1554)
d. The 95% prediction interval for the given value of x is:
= y ± t×s×[tex]\sqrt{}[/tex]1 + hin [The t-value is calculated using =TINV(0.05,42-1) in Excel]
= 21.1545 ± 2.020*12.2279*[tex]\sqrt{1 + 0.026249}[/tex]
= (-3.8623, 46.1713)
Therefore, the value is (-3.8623, 46.1713).
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