Answer:
Jaime is correct, -1 x n is always equal to n ÷ (-1) for all values of n.
To see why this is true, we can use the properties of multiplication and division of real numbers. In particular, we can use the fact that multiplying by -1 is the same as changing the sign of a number and dividing by -1 is the same as multiplying by -1.So, starting with -1 x n, we can rewrite this expression as (-1) x n or -(1 x n), which means we are taking the opposite of the product of -1 and n. Since the opposite of a number is just that number with its sign changed, we can simplify this expression to -n.Next, let's consider n ÷ (-1). This means we are dividing n by -1, which is the same as multiplying n by the reciprocal of -1, which is -1/1 or simply -1. So, n ÷ (-1) is equal to n x (-1), which is just -n.Thus, we can see that -1 x n and n ÷ (-1) both simplify to -n. Therefore, Jaime is correct, the value of -1 x n is always equal to the value of n ÷ (-1) for all values of n.
Can a non-complex equation have 2 horizontal asymptotes? Excluding piecewise equations, o course. Also, what about 2 vertical asymptotes? And using a complex number as x could an equation pass the vertical asymptote? What about the horizontal one?
Answer:
No, a non-complex equation cannot have 2 horizontal asymptotes. By definition, a horizontal asymptote is a horizontal line that the function approaches as x goes to positive or negative infinity. If there were two horizontal asymptotes, the function would have to approach two different horizontal lines as x goes to infinity, which is not possible for a single-valued function.
Similarly, a non-complex equation cannot have 2 vertical asymptotes. A vertical asymptote is a vertical line where the function approaches infinity or negative infinity as x approaches the line from either side. If there were two vertical asymptotes, the function would have to approach infinity or negative infinity as x approaches each line, which is not possible for a single-valued function.
Using a complex number as x does not change the behavior of asymptotes. If a function has a vertical asymptote at a real number, it will still have a vertical asymptote at that number when x is a complex number. Similarly, if a function has a horizontal asymptote, it will still have a horizontal asymptote when x is a complex number.
However, if a function has a vertical asymptote at a complex number, it may behave differently depending on the direction in which x approaches the asymptote. If the function approaches different values as x approaches the asymptote from different directions, then the function may not have a limit at the complex number and may not have a vertical asymptote. Similarly, if a function has a horizontal asymptote and is not defined for complex values of x, then it may not have a limit at infinity and may not have a horizontal asymptote.
Step-by-step explanation:
The diameter of the circle above is 6cm what is the circumference of the circle?
Answer:
C ≈ 18.84 cm
Step-by-step explanation:
circumference = π · diameter
We can plug the given diameter into this formula, approximating π as 3.14.
C = 3.14 · 6 cm
C ≈ 18.84 cm
We know that,
[tex] \bf \leadsto Radius = Diameter / 2[/tex]
So,
[tex] \rightarrow \: \sf \: Radius = \frac{6}{3} \\ \\ \rightarrow \: \sf \: Radius = 3[/tex]
[tex] \rule{140pt}{5pt}[/tex]
Now,
[tex] \bf \bigstar \: Circumference \: of \: circle = 2πr[/tex]
[tex]→ \sf \: 2 × 3.14 × 3 \\ \\ → \sf \: 6 × 3.14[/tex]
[tex] \bf \: 18.84cm[/tex]
Option a is correct
[tex] \underline{ \rule{190pt}{6pt}}[/tex]
Pls help ill offer 13 points
Answer:
Point K
Step-by-step explanation:
Which Point represents the ordered pair
( -4[tex]\frac{1}{2}[/tex], -1)
Left -4[tex]\frac{1}{2}[/tex] we have points J and K
Down -1 we have point K
So, the answer is point K
for a time series for simple exponential smoothing, a larger alpha is smoother than a shorter period. group of answer choices true false
The important to select the appropriate value of alpha based on the characteristics of the data and the goals of the forecasting model.
The statement "for a time series for simple exponential smoothing, a larger alpha is smoother than a shorter period" is true.
In simple exponential smoothing, the forecast for the next period is based on the previous forecast and the difference between the actual value and the previous forecast. The smoothing parameter alpha controls the weight given to the most recent observation compared to the previous forecasts.
A larger alpha places more weight on the most recent observation, resulting in a forecast that is more responsive to recent changes in the data. As a result, the forecast will exhibit less volatility or variability, making it smoother. This means that a larger alpha will lead to a more stable forecast with less fluctuation around the trend.
On the other hand, a smaller alpha places less weight on the most recent observation, resulting in a forecast that is less responsive to recent changes in the data.
As a result, the forecast will be more volatile or variable, making it less smooth. This means that a smaller alpha will lead to a forecast that is more susceptible to fluctuations around the trend.
In summary, the choice of alpha is a trade-off between stability and responsiveness to changes in the data. A larger alpha will lead to a smoother forecast, while a smaller alpha will lead to a more volatile forecast.
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What is the best definition of the term opportunity cost?
Answer:
the value of what you lose when you choose from two or more alternatives
Step-by-step explanation:
I need helppppppppp please help me I don’t understand
Answer:
m∠2 = 130°
m∠3 = 25°
m∠4 = 25°
Step-by-step explanation:
Opposite angles of rhombi are congruent. Therefore,
m∠1 = m∠2 = 130°
The measures of the interior angles of a triangle add to 180°. Using this information, along with the isosceles triangle theorem, we can solve for m∠3 (which is also congruent to m∠4):
m∠2 + 2(m∠3) = 180°
↓ substituting in m∠2
130° + 2(m∠3) = 180°
↓ subtracting 130° from both sides
2(m∠3) = 50°
↓ dividing both sides by 2
m∠3 = 25°
m∠3 = m∠4 = 25°
kindly answer with working and i will mark you brainliest
The perimeter to the nearest tenth, we get: Perimeter ≈ 18.4 units.
What is perimeter?
We can start by using the Pythagorean theorem to find the length of AC:
AC² = AB² + BC²
AC² = 2² + 1.5²
AC² = 4.25
AC = √(4.25)
AC ≈ 2.06 units
Next, we can use the fact that AC is common to both triangles to find the length of CD:
ACD is a right triangle, so we can use the Pythagorean theorem:
CD² = AD² - AC²
CD² = 6.5² - 2.06²
CD² ≈ 39.86
CD ≈ 6.31 units
Now we can find the perimeter of the figure by adding the lengths of all three sides:
Perimeter = AB + BC + CD + DA + AC
Perimeter = 2 + 1.5 + 6.31 + 6.5 + 2.06
Perimeter ≈ 18.37 units
Therefore, none of the given options is the correct answer. However, if we round the perimeter to the nearest tenth, we get:
Perimeter ≈ 18.4 units
And the closest option to this value is option 3) 16 units, which is not very close. Therefore, we can conclude that none of the given options is a correct approximation of the perimeter of the figure.
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is it possible to find a set of bij and cij such that the state-space equation is controllable? observable?
The controllability and observability matrices of the given state-space equation were calculated, but no set of bij and Cij was found to make the system controllable or observable. Therefore, the system is not controllable or observable.
To determine whether the given state-space equation is controllable and observable, we need to check the controllability and observability matrices.
The controllability matrix is given by:
[ B AB A^2B A^3B A^4B ]
where A and B are the state and input matrices, respectively.
The observability matrix is given by:
[ C ]
[ CA ]
[ CA^2 ]
[ CA^3 ]
[ CA^4 ]
where C is the output matrix.
Using the given state-space equation, we can calculate the controllability and observability matrices as follows:
A = [ 1 0 0 0 0 ]
[ L0 1 0 0 1 ]
[ b12 1 0 0 0 ]
[ 0 0 1 0 0 ]
[ b21 bzz 0 1 1-L ]
B = [ 0 ]
[ 0 ]
[ 0 ]
[ 0 ]
[ 1 ]
C = [ 1 0 0 0 0 ]
[ 0 1 0 0 0 ]
[ 0 0 1 0 0 ]
Controllability Matrix:
[ B AB A^2B A^3B A^4B ] =
[ 0 0 0 0 1 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ]
[ 0 0 0 0 0 ]
[ 0 0 0 0 0 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ]
[ 0 0 0 0 0 ]
[ 0 0 0 0 0 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ]
[ 0 0 0 0 0 ]
[ 0 0 0 0 0 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ]
[ 0 0 0 0 0 ]
[ 0 bzz b21bzz 0 Lb51 ] [ 0 L0 b12bzz Lb51 0 ] [ L0 b12bzz L0b21bzz 0 0 ]
[ b12bzz 0 L0b21bzz 0 0 ] [ 0 0 L0b12bzz 0 0 ]
Observability Matrix:
[ 1 0 0 0 0 ] [ 1 0 0 0 0 ] [ 1 0 0 0 0 ] [ 1 0 0 0 0 ] [ 1 0 0 0 0 ]
[ 0 1 0 0 0 ] [ 0 L0 1 b12 1 ] [ L0 b12bzz L0b21bzz 0 0 ] [ b12bzz 0 L0bb21bzz 0 0 ]
To elaborate on the solution, the controllability matrix shows that some of the columns are linearly dependent on the others, which means that it is not possible to find a set of inputs that can steer the system from any initial state to any desired state in finite time. Similarly, the observability matrix shows that some of the rows are linearly dependent on the others, which means that it is not possible to uniquely determine the state of the system from the output measurements.
Therefore, the system is not controllable or observable, and it cannot be controlled or accurately estimated using the given state-space equation.
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--The given question is incomplete, the complete question is given
" Is it possible to find a set of bij and a set of Cij such that the following state-space equation is controllable? Observable? (If the answer is yes, find a set) i(t) = 1 0 0 0 LO 1 0 0 01 5611 b12 1 0 0 0 |b21 bzz 0 1 1 0 X(t) + b31 b32|u(t) 0 0 1 0 | b41 buz 0 0 0 1- Lb51 652 [C11 y(t) = C21 C31 C12 C22 C32 C13 C23 C33 C14 C24 C34 C15 C25 x(t) C35
HELP PLEASE IT WAS DUE 10!!!
A bakery makes cylindrical mini muffins that measure 2 inches in diameter and one and one fourth inches in height. If each mini muffin is completely wrapped in paper, then at least how much paper is needed to wrap 6 mini muffins? Approximate using pi equals 22 over 7.
A- 14 and 1 over 7 in2
B -23 and 4 over 7 in2
C -47 and 1 over 7 in2
D - 84 and 6 over 7 in2
A,B,C,D which one
According to the given question the correct option is (D) which is 84 and 6/7 square inches.
what is surface area ?Surface area is indeed a measurement of how much overall space an object occupies. A three-dimensional shape's surface area is the sum of the area surrounding it. The term "surface area" refers to a three-dimensional shape's overall surface area. The surface area of a cuboid with six rectangular faces can be calculated by adding the areas from every face. As an alternative, you can name the dimensions of this box using the formula below: 2lh, 2lw, with 2hw make up the surface area (SA). Surface area is a unit used to describe how much space a three-dimensional form's surface takes up overall.
Given,
The formula: can be used to determine a cylinder's surface area.
surface area = 2πr^2 + 2πrh
where r is the cylinder's radius and h seems to be the cylinder's height.
In this case, the diameter of the mini muffins is 2 inches, so the radius is 1 inch. The height is 1 and 1/4 inches. When these values are added toward the formula, we obtain:
surface area = 2π(1^2) + 2π(1)(5/4) = 2π + 5π/2 = 9π/2
To find the amount of paper needed to wrap 6 mini muffins, we multiply the surface area of one mini muffin by 6:
6 × (9π/2) = 27π
Approximating π as 22/7, we get:
27π ≈ 27 × 22/7 = 84 and 6/7
Therefore, the answer is (D) 84 and 6/7 square inches.
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Find the difference.
Answer:
[tex]t^{4}[/tex] + 8[tex]t^{2}[/tex] + 8t - 12
Hope this helps!
Step-by-step explanation:
[tex]t^{4}[/tex] - 1.5[tex]t^{2}[/tex] + t - 12 + 9.5[tex]t^{2}[/tex] + 7t : Combine like terms
[tex]t^{4}[/tex] + ( 9.5 - 1.5 )[tex]t^{2}[/tex] + ( 7t + t ) - 12
[tex]t^{4}[/tex] + 8[tex]t^{2}[/tex] + 8t - 12
Gus's and ike's combined running distance this week was 48 miles. If Gus ran three times as far as ike, how many miles did ike run?
Ike ran 12 miles and Gus ran 36 miles based on equation solving.
Assume that x is a representation of Ike's running distance. Gus ran three times as far as Ike, so his distance was three times Ike's.
Gus and Ike travelled a total of 48 miles, as far as we know. Thus, we can create the following equation:
x + 3x = 48
When we simplify this equation, we obtain:
4x = 48
When we multiply both sides by 4, we get:
x = 12
Ike afterwards ran 12 kilometres.
To confirm, we may calculate Gus's running distance by dividing Ike's distance by three:
3x = 3(12) = 36
And we can confirm that the total of their separations is 48:
12 + 36 = 48
Gus ran 36 miles while Ike ran 12 miles.
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1) One week the price of gas dropped by $0.05 per gallon. Steven travels 37 miles each way to work and her car travels 30 miles on each gallon of gas. What were his total savings over a 5 day work week.
I need help asap please show me you got the answer
Answer: 0.098 is the answer
Step-by-step explanation:
What is the length of the missing leg? if necessary, round to the nearest tenth.
The length of the missing leg for the right triangle is 16 yd using the Pythagoras rule.
What is the Pythagoras rule?The Pythagoras rule states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Pythagoras rule we can evaluate for b as follows:
65² = 63² + b²
b = √(65² - 63²) {make b the subject}
b = √(4225 - 3969)
b = √256
b = 16
Therefore, the length of the missing leg for the right triangle is 16 yd using the Pythagoras rule.
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Out of 980 pizzas, only 196 are stuffed crust, so how many will be stuffed crust out of 5000 pizzas?
Step-by-step explanation:
192080 because we multiply its 980×196 =182080
What is the measure of 24? Enter your answer in the box.
m 24 =
1
2/3
60%
4 61°
P
9
The measure of the angle ∠4 in the parallell lines and transversal is 59 degrees
Calculating the measure of ∠4?From the question, we have the following parameters that can be used in our computation:
The parallell lines and transversal
By the sum of angles on a straight line, we have the following equation
Angle 4 + 60 + 61 = 180
Evaluate
Angle 4 = 180 - 121
So, we have
Angle 4 = 59
Hence, the measure of the angle is 59 degrees
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Sams electric bill was $200 dollars for the month of June. The air conditional accounts for 1/2 of the bill and the water heater accounts for 1/5 of the bill. How much did it cost to one each appliance in June?
Answer:
$100 for air conditioning and $40 for water heater
Step-by-step explanation:
$200÷2= 100
200÷5= 40
delia bought a bedroom set for $3600 the sales tax on the purchase was 246.60 what was the sale tax rate 
Answer: 6.85%
Step-by-step explanation:
PLEASE HELP I NEED IT ASAP
Explain using the change of base formula and evaluating the change of base formula
The change of base formula is: log_b(x) = log_a(x) / log_a(b)
What do you mean by the term Change of base ?The term "change of base" refers to the process of converting a logarithm from one base to another. A logarithm is an exponent that indicates the power to which a base must be raised to obtain a given number. The most common logarithmic bases are 10, e (Euler's number), and 2, but logarithms can be taken to any base.
The change of base formula is a mathematical formula that allows you to convert logarithms from one base to another. It is useful when you need to calculate logarithms that cannot be easily evaluated on a calculator or when you need to work with logarithms in different bases.
The change of base formula is:
log_b(x) = log_a(x) / log_a(b)
Where:
log_b(x) is the logarithm of x to the base b
log_a(x) is the logarithm of x to the base a
log_a(b) is the logarithm of b to the base a
To use the change of base formula, you first need to identify the base of the logarithm that you want to convert and the base that you want to convert it to. Then, you can plug the values into the formula and simplify the expression.
For example, let's say you want to convert log_2(8) to base 10. Using the change of base formula, you can write:
log_2(8) = log_10(8) / log_10(2)
To evaluate this expression, you can use a calculator to find that:
log_10(8) = 0.9031
log_10(2) = 0.3010
Substituting these values into the formula gives:
log_2(8) = 0.9031 / 0.3010
Simplifying this expression gives:
log_2(8) = 3
Therefore, log_2(8) is equal to 3 when evaluated in base 10.
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which kind of design is an experiment in which each participant is randomly assigned to one level of the independent variable and then tested on the dependent variable once?
The design is a randomized controlled trial (RCT), where participants are randomly assigned to treatment groups and tested on the dependent variable once to control for confounding variables and ensure that differences observed between groups are due to the treatment.
The type of design you are referring to is a randomized between-subjects design, also known as a randomized controlled trial (RCT). In this type of experiment, participants are randomly assigned to one of two or more treatment groups, each receiving a different level or type of the independent variable. Then, the dependent variable is measured only once after the treatment is administered. The purpose of this design is to control for potential confounding variables and to ensure that any differences observed between groups are due to the treatment and not to other factors.
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What is the mean? 103–3–41–7
Answer:
What is a mean?
A mean is the average of all the numbers in a data set.
How can I find mean?
to find mean you add up all the numbers in the data set; and divide them by the amount of numbers there are.
103+3+41+7=154
154÷4=38.5
The mean is 38.5
The graph below shows the number of inches of rain during the summer months of 2009 and 2010.
How many more inches of rain occurred during the summer of 2010 than during the summer of 2009?
0.5
2
3
10.5
Using the bar graph we can conclude that it rained 0.5 inches more in the summer of 2010 than in 2009.
Define a bar graph?Using rectangular bars that are proportional to the values they represent; a bar graph displays all the data. The graph's bars can be displayed either vertically or horizontally. Bar graphs, commonly referred to as bar charts, are a visual depiction of groups of data. It is one method of processing data. A bar graph is a great tool for representing data that are unrelated to one another and do not need to be displayed in any particular sequence. The bars provide a visual representation for comparing amounts in several categories. Together with the title, labels, and scale range, bar graphs have two axes that are horizontal and vertical and are also known as the x and y axes.
Here in the question,
We can see that in the summer of 2009 the rain was for 1.5 inches.
In the summer of 2010, the rain was for 2 inches.
Hence, the difference between the number of inches it rained=
2 - 1.5
= 0.5 inches.
Therefore, using the bar graph we can conclude that it rained 0.5 inches more in the summer of 2010 than in 2009.
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Fill in the table for side length and area of different squares.
Answer:
Step-by-step explanation:
a.
3: Area = 3² = 9
100: Area = 100² = 10,000
25: Area = 25² = 625
s: Area = s²
b.
Yes, the relationship is proportional. The area of the square is the length of a side squared. Area = (length)²
Which statement describes the end behavior of this absolute value function?
As x approaches positive infinity, f(x) approaches positive infinity.
What is the absolute value function?In algebra, an absolute value function is a function where the variable is inside the bars denoting absolute values. The absolute value function is also known as the modulus function, and its most popular representation is f(x) = |x|, where x is a real number.
Here,
We have to find the end behavior of this absolute value function.
In the graph, it is shown that the absolute value function has a maximum of 3 at x = 3.
The function f(x) reaches positive infinity in both cases when x reaches either minimum or maximum value.
Hence, As x approaches positive infinity, f(x) approaches positive infinity.
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Can you anyone help me out with this?
I don’t know how to solve the equation and check for extraneous solutions!!!! PLEASEEEEEEEEE URGENT
The solution to the equation 1 is p = -7. and the only solution to the equation2 is k = 2.
what is equation?
An equation is a mathematical statement that asserts the equality of two expressions. It usually contains one or more variables (letters that represent unknown values) and an equal sign. The goal of solving an equation is to find the values of the variables that make the equation true.
In the given question,
Starting with -49 = -7√(p+8):
Dividing both sides by -7, we get:
7√(p+8) = 7
Dividing both sides by 7, we get:
√(p+8) = 1
Squaring both sides, we get:
p+8 = 1
Subtracting 8 from both sides, we get:
p = -7
Therefore, the solution to the equation is p = -7.
Starting with √(6-k) = k:
Squaring both sides, we get:
6-k = k²
Rearranging, we get:
k²+ k - 6 = 0
We can factor this quadratic equation as:
(k+3)(k-2) = 0
Therefore, the solutions to the equation are:
k+3 = 0 or k-2 = 0
Solving for k, we get:
k = -3 or k = 2
However, we need to check whether these solutions satisfy the original equation, since we squared both sides and introduced extraneous solutions:
When k = -3:
√(6-k) = √(6-(-3)) = √9 = 3
k = -3 is not a solution.
When k = 2:
√(6-k) = √(6-2) = √4 = 2
k = 2 is a solution.
Therefore, the only solution to the equation is k = 2.
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Alex bought a new car for his daughter. He knows the value of the car will decrease at a constant rate. After 3 years, the value of the car is $15,000. After 5 years, the value of the car of $11,000. Write and solve a linear equation to find the value of the car after 8 years. (Drop down and select)
y-______=______(x-____)
The value of the car after 8 years will be ______.
Answer: To find the value of the car after 8 years, we can use the given information to create a linear equation in the form:
y - y1 = m(x - x1)
Where:
y is the value of the car after 8 years (what we want to find)
y1 is the value of the car after a given time (either 3 or 5 years)
x is the given time (either 3 or 5 years)
x1 is the starting time (when the car was purchased)
m is the rate of decrease (slope of the line)
We can use the two given data points to find the slope of the line:
m = (y2 - y1) / (x2 - x1)
m = (11,000 - 15,000) / (5 - 3)
m = -2,000 per year
Now, we can use one of the data points to solve for the y-intercept of the line:
y - y1 = m(x - x1)
y - 15,000 = (-2,000)(3 - x1)
y - 15,000 = (-2,000)(3 - x1)
y - 15,000 = (-6,000 + 2,000x1)
y = 2,000x1 - 6,000 + 15,000
y = 2,000x1 + 9,000
So the linear equation that represents the value of the car after x years is:
y - 15,000 = (-2,000)(x - 3)
Simplifying this equation gives:
y - 15,000 = -2,000x + 6,000
Now, we can use x = 8 to find the value of y:
y - 15,000 = -2,000(8) + 6,000
y - 15,000 = -2,000
y = $13,000
Therefore, the value of the car after 8 years will be $13,000.
So the completed expression is:
y - 15,000 = -2,000(x - 3)
The value of the car after 8 years will be $13,000.
Step-by-step explanation:
Write the equation of the circle with center (-2,15) and radius 3.
Answer:
[tex]( x + 2 )^{2} +(y-15)^{2} =9[/tex]
Hope this helps!
Step-by-step explanation:
Help Been Stuck on This
PEQ has a value of y of 60°.
The Sum of Angles property states that the sum of a polygon's internal angles is equal to (n - 2) times 180 degrees, where n is the number of sides.The polygon's s.
What is angle sum property?
The angle sum property in geometry asserts that the sum of a polygon's internal angles is equal to (n - 2) times 180 degrees, where n is the number of sides of the polygon.
In a triangle (n=3), for example, the total of the interior angles is (3 - 2) 180 = 1 180 = 180 degrees.
The sum of internal angles for a quadrilateral (n=4) equals (4 - 2) 180 = 2 180 = 360 degrees.
This is a fundamental geometric idea that can be used to calculate the value of an unknown interior angle in a polygon if the other interior angles are known. It can also be used to determine whether or not a given set of angles can create a valid polygon.
In PEQ, the angle sum attribute is used.
Hence, EPQ + EQP + PEQ = 180°
40° + 80° + y = 180°
120° + y = 180°
y = 180° - 120°
y = 60°
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Audrey's mother bought a dozen muffins. There were
4 blueberry muffins, 4 chocolate chip muffins, and 4
banana nut muffins in the box. Audrey's mother ate a
chocolate chip muffin. What is the probability that
Audrey will randomly select a blueberry muffin if she
chooses after her mother?
A) 1/3
B) 4/11
C) 3/11
D) 1/12
Please answer the warm-up through the poll button.
the wall street journal reported that of taxpayers with adjusted gross incomes between and itemized deductions on their federal income tax return. the mean amount of deductions for this population of taxpayers was . assume that the standard deviation is . use z-table. a. what is the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean within of the population mean for each of the following sample sizes: , , , and ? round your answers to four decimals. b. what is the advantage of a larger sample size when attempting to estimate the population mean? round your answers to four decimals. a larger sample - select your answer - the probability that the sample mean will be within a specified distance of the population mean. in this instance, the probability of being within of ranges from for a sample of size to for a sample of size .
a. The z-table, the probability that a sample mean will be within 0.02 of the population mean is:
P(-0.02/0.00416 < Z < 0.02/0.00416) = P(-4.81 < Z < 4.81) = 0.9999
b. The probability of obtaining a sample mean that is close to the population mean increases as the sample size increases. This means that larger sample sizes provide more accurate estimates of the population mean.
a. We can use the formula for the standard error of the mean:
SE = σ/√n
where σ is the population standard deviation, n is the sample size, and SE is the standard error.
For a sample size of n = 25:
SE = 0.0588/√25 = 0.01176
Using the z-table, the probability that a sample mean will be within 0.02 of the population mean is:
P(-0.02/0.01176 < Z < 0.02/0.01176) = P(-1.70 < Z < 1.70) = 0.9045
For a sample size of n = 50:
SE = 0.0588/√50 = 0.00832
Using the z-table, the probability that a sample mean will be within 0.02 of the population mean is:
P(-0.02/0.00832 < Z < 0.02/0.00832) = P(-2.41 < Z < 2.41) = 0.9872
For a sample size of n = 100:
SE = 0.0588/√100 = 0.00588
Using the z-table, the probability that a sample mean will be within 0.02 of the population mean is:
P(-0.02/0.00588 < Z < 0.02/0.00588) = P(-3.40 < Z < 3.40) = 0.9984
For a sample size of n = 200:
SE = 0.0588/√200 = 0.00416
Using the z-table, the probability that a sample mean will be within 0.02 of the population mean is:
P(-0.02/0.00416 < Z < 0.02/0.00416) = P(-4.81 < Z < 4.81) = 0.9999
b. The advantage of a larger sample size is that it leads to a smaller standard error of the mean, which means that the distribution of sample means is narrower and more closely centered around the population mean. As a result, the probability of obtaining a sample mean that is close to the population mean increases as the sample size increases. This means that larger sample sizes provide more accurate estimates of the population mean.
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