A college finds that 10% of students have taken a distance learning class and that 40% of students are part time students. Of the part time students, 20% have taken a distance learning class. Let D = event that a student takes a distance learning class and E = event that a student is a part time student
a. Find P(D AND E).
b. Find P(E|D).
c. Find P(D OR E).
d. Using an appropriate test, show whether D and E are independent.
e. Using an appropriate test, show whether D and E are mutually exclusive.

The value of the test statistic z is, 2.31.

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

It is claimed that less than 25% of adults prefer Italian food as their favorite ethnic food.

And,  A survey of 1122 adults resulted in 314 who say that Italian food is their favorite ethnic food.

Now,

⇒ P = 25%

⇒ q = 100% - 25% = 75%

⇒ n = 1122

Hence, We get;

⇒ z = (314/1122 - 25%) / √(25% 75% / 1122)

⇒ z = 2.31

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The way to transform the function such that the bond goes from quarterly compounding interest to annually compounding interest is = 1 + ( 1 + rate / 4 )ⁿ ˣ ⁴.

The bond is said to be compounded quarterly. There would therefore be four compounding periods in a single year because there are four quarters in a year.

The formula would have been:

= Initial investment + ( 1 + rate ) ^ number of years

The adjusted formula to account for the quarters becomes :

= 1  + ( 1 + rate / 4 )ⁿ ˣ ⁴

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As per the information provided, option B) The errors e1,e2, e; have the same variance is the assumptions for linear regression.

Regression analysis is a linear technique used to model the relationship between a scalar response and/or one or more explanatory variables, also known as the dependent and independent variables here.

Simple linear regression and multiple linear regression are two terms used to describe the process when there is only one explanatory variable present. This statement is different from multivariate linear regression, which thus predicts multiple correlated dependent variables as opposed to a single scalar variable present.

The first regression analysis technique that attracted a lot of research attention and was frequently applied in practical situations was linear regression. This is because models with linear rather than non-linear dependence on their unknown parameters are easier to fit, and it is also easier to determine the statistical properties of the resulting estimators.

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120 is the number divisible by 6, 8, and 10.

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

The common multiples to 6, 8, and 10.

6 x 20 =  120

8 x 15 = 120

10 x 12 = 120

So,

120 is divisible by 6, 8, and 10.

Thus,

120 is the number.

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Using unitary method, we can calculate the number of 0.04s boxes between R-R intervals on an ECG trace for our chosen event.

The unitary method is a mathematical technique that involves finding the value of one unit of a given quantity and then using it to find the value of a different quantity. In this case, we want to find the change in heart rate per second, so we need to determine how many beats per second our chosen event increases our heart rate by.

To calculate the duration of each R-R interval, we can use the following formula:

R-R interval duration = 1 / heart rate (in seconds)

Assuming our starting heart rate is 90 bpm and our event increases our heart rate by y beats per second, we can calculate the new heart rate as follows:

New heart rate = 90 bpm + (y * 60) bpm

We can then calculate the duration of each R-R interval using the formula above.

Finally, to find the number of 0.04s boxes between R-R intervals on an ECG trace, we can convert the R-R interval duration to a number of 0.04s boxes. Each 0.04s box on an ECG trace represents 0.04 seconds, so we can use the following unitary method:

1 second = 25 0.04s boxes

R-R interval duration (in seconds) = x seconds

Number of 0.04s boxes = x * 25

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The height of flagpole c is given as follows:

66 ft.

Similar triangles are triangles that share these two features presented as follows:

For each flagpole, the parameters are given as follow:

The triangles are similar, hence the proportional relationship to obtain the height of flagpole's c is given as follows:

2/10 = 13.2/h

Applying cross multiplication, the height is obtained as follows:

2h = 10 x 13.2.

h = 132/2

h = 66 ft.

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The hypothesis tested are given as follows:

[tex]H_0: \mu = 3.2, H_a: \mu \neq 3.2[/tex]

The claim for this problem is given as follows:

"The mean salary for the baseball players on the East Coast is different from $3.2 million".

At the null hypothesis, we test if the claim is false, that is, if the mean salary is equals to 3.2 million, hence:

[tex]H_0: \mu = 3.2[/tex]

At the alternative hypothesis, we test if the claim is true, that is, if the mean salary is different of 3.2 million, hence:

[tex]H_a: \mu \neq 3.2[/tex]

The [tex]\mu[/tex] symbol, without the bar, is used, as we are testing for the population mean, not sample mean, hence the first option is the correct option.

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The proof method to show that the statement "The average of any three odd integers is an odd integer" is true is "Prove true by direct proof."

To prove that the statement is true by direct proof, we can assume that the three odd integers are a, b, and c, where a = 2k + 1, b = 2m + 1, and c = 2n + 1 for some integers k, m, and n. Then, the average of these three odd integers is:

(a + b + c)/3 = (2k + 1 + 2m + 1 + 2n + 1)/3

= (2k + 2m + 2n + 3)/3

= 2(k + m + n) + 1

Since k, m, and n are integers, k + m + n is also an integer. Therefore, the average of any three odd integers is of the form 2x + 1, where x is an integer, which is an odd integer.

The reason the attempted proof of the statement "The difference between any odd integer and any even integer is odd" is not correct is "The proof violates the parity property." The student incorrectly subtracted an even integer from an odd integer and concluded that the result is odd. However, the difference between an odd integer and an even integer is always odd, which can be proven directly as follows:

Let n be any odd integer and m be any even integer. Then, by definition, n = 2k + 1 and m = 2j for some integers k and j. The difference between n and m is:

n - m = (2k + 1) - 2j

= 2k - 2j + 1

= 2(k - j) + 1

Since k and j are integers, k - j is also an integer. Therefore, the difference between any odd integer and any even integer is of the form 2x + 1, where x is an integer, which is an odd integer.

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The total area of both triangles is the product of the height and base of the flag.

The connecting of three noncolinear points results in a triangle, which is a three-sided closed-plane object. Triangles can be classified into three categories based on their side properties: equilateral, scalene, and isosceles.

Given, Anthony painted 2triangles of the same size on the top corners of his house flag for sports day.

We are aware of A parallelogram can be divided into two comparable triangles by dividing it along its diagonal.

A planar figure with opposite sides that are parallel and of equal length is called a parallelogram.

Since the area of a parallelogram is equal to its base times its height, the combined area of the two triangles will be the same.

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Rounded to the nearest tenth, the area of the region that receives the radio signal is approximately 6,366.2 square miles.

Consider a plane region described by a xy-plane graph. The signed area of the region is defined as the area of the graph on or above the x-axis. The negative area of the graph is defined as the area below the x-axis.

The formula A = πr² gives the area of a circle with radius r. In this case, the radio station broadcasts over a 45-mile radius, so the region that receives the radio signal is:

A = π(45)²

A ≈ 6,366.2 square miles

The area of the region that receives the radio signal is approximately 6,366.2 square miles when rounded to the nearest tenth.

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Answer:

--------------------------------

Given line:

We know that perpendicular lines have opposite-reciprocal slopes.

Perpendicular line to the given has a slope of m = - 1 / (1/4) = - 4.

Let the y-intercept be b, then the line is:

Given line:

We know parallel lines have equal slopes.

The parallel line to the given has a slope of m = - 3 and passes through point (3, 8).

Set the point-slope equation:

Convert this into slope-intercept form:

If W is a subspace of V, then ax + by is an element of W since it satisfies the closure axioms of a vector space. Specifically, scalar multiplication closure confirms that ax and by are in W, and additive closure confirms that their sum, ax + by, is also in W.

Conversely, if ax + by is an element of W for all scalars a and b and all vectors x and y in W, then W must be closed under both addition and scalar multiplication.

To demonstrate this, we can assign a particular value to a or b, set the other to 1, and substitute the corresponding vectors x and y from W to get the equation ax + by = a(x) + b(y). This equation confirms that W is closed under addition and scalar multiplication, making it a subspace of V.

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The expression "D(50) - (D(15) ∪ D(20))" refers to the set difference between the set D(50) and the union of sets D(15) and D(20).

The set difference is the operation that gives you the elements that are in the first set (D(50)) but not in the second set (D(15) ∪ D(20)). Note that this prompt draws on Set Theory.

Set Theory is a branch of mathematics concerned with the study of sets, their properties, and their relationships.

In set theory, a set is a collection of distinct objects, and the set difference operation, represented by the minus sign (-), returns the elements that are in the first set but not in the second set.

The union of sets, represented by the symbol ∪, is the operation that gives you the set of all elements that are in at least one of the sets being considered.

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The value of the probabilities are

a. P(x ≥ 12.5) is 6.68%

b. P(x ≤ 9) is 2.28%

c. P(10.88 ≤ x ≤ 15.3) is 61.38%

d. P(4.72 ≤ x ≤ 12.46) is 99.45%

Probability is the measure of the likelihood of an event occurring. It is a way of quantifying uncertainty and predicting the likelihood of an outcome. In this question, we will use the normal distribution to calculate probabilities.

The mean is the center of the distribution, and the standard deviation measures the spread of the distribution.

a. P(x ≥ 12.5):

Using a standard normal distribution table, we find that the area to the right of 12.5 is 0.0668. This means that the probability of x being greater than or equal to 12.5 is 0.0668 or 6.68%.

b. P(x ≤ 9):

Using a standard normal distribution table, we find that the area to the left of 9 is 0.0228. This means that the probability of x being less than or equal to 9 is 0.0228 or 2.28%.

c. P(10.88 ≤ x ≤ 15.3):

Using a standard normal distribution table, we find that the area to the left of 15.3 is 0.9987 and the area to the left of 10.88 is 0.3849. Therefore, the area between 10.88 and 15.3 is

=> 0.9987 - 0.3849 = 0.6138.

This means that the probability of x being between 10.88 and 15.3 is 0.6138 or 61.38%.

d. P(4.72 ≤ x ≤ 12.46):

Using a standard normal distribution table, we find that the area to the left of 12.46 is 0.9946 and the area to the left of 4.72 is 0.0001. Therefore, the area between 4.72 and 12.46 is

=> 0.9946 - 0.0001 = 0.9945.

This the probability of x being between 4.72 and 12.46 is 0.9945 or 99.45%.

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The height of the triangle is 20 and base of the triangle is 12.

"Information available from the question"

The base of a triangle is 8 ft. less than its height.

If the area is 120 [tex]ft^2.[/tex]

We have to find the base and the height of this triangle.

Now, According to the question:

Let the height be x then if the base is 8 ft. less, then the base is x - 8.

We know that:

Area of the triangle is :  (b × h)/2

Area of the triangle is : (x (x - 8)) /2

120 [tex]ft^2[/tex] = ([tex]x^{2} -8x[/tex])/2

240 = [tex]x^{2} -8x[/tex]

[tex]= > x^{2} -8x-240=0[/tex]

For solving equation, Use the quadratic formula :

x = (-b ± [tex]\sqrt{b^2-4ac}[/tex]) / 2a

a = 1

b = -8

c = -240

x = (-(-8) ± [tex]\sqrt{(-8)^2-4.1(-240)})/2.1[/tex]

By solving, we get

x = (8 ± 32)/2

We get the values :

x = 20 and x = -12

We know x = −12 can't be a solution as length can't be negative.

Base = x - 8

Base = 20 - 8

Base = 12

Hence, The height of the triangle is 20 and base of the triangle is 12.

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The slope of line q is equal to 3.

The slope of line q is equal to -1/3.

An equation for line p in point-slope form is y + 5 = -1/3(x - 6).

An equation for line p in slope-intercept form is y = -1/3x - 3.

Mathematically, the slope-intercept form of the equation of a straight line is given by this mathematical expression;

y = mx + c   ....equation 1.

Where:

From the information provided, we have the following linear equation in slope-intercept form for line q:

y = 3x + 5  ....equation 2.

By comparing equation 1 and equation 2, we can logically deduce the following:

Slope (m) = 3

Since line p is perpendicular to line q, the slope of line p is the negative reciprocal of the slope of line q;

m₁ × m₂ = -1

3 × m₂ = -1

Slope, m₂ of line p = -1/3.

At data point (6, -5), a linear equation of this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - (-5) = -1/3(x - 6)

y + 5 = -1/3(x - 6)

By simplifying further, the slope-intercept form is given by;

y + 5 = -1/3(x - 6)

3y + 15 = -x + 6

3y = -x - 9

y = -1/3x - 3

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Complete Question:

Line p contains point (6, -5) and is perpendicular to line q. The equation for line q is y = 3x + 5. Write an equation for line p.

Find the slope of line q.

Find the slope of line p. (Write the negative reciprocal of the slope you found in Part I.)

Use the point given for line p and the slope you found in Part II to write an equation for line p in point-slope form: y - y1 = m (x - x1)

Use your equation from Part III to write an equation for line p in slope-intercept form: y = mx + b.

Answer:

x = 12

Step-by-step explanation:

The exterior angle is equal to the sum of the opposite interior angles.

5x+20 = 20 + 6x-12

Combine like terms.

5x+20 = 6x +8

Subtract 5x from each side.

5x+20-5x = 6x+8-5x

20 = x+8

Subtract 8 from each side.

20-8 = x+8-8

12 =x

In accordance with Side-Angle-Side theorem of congruence, the triangles seen in choices A and C are congruent.

In this problem we need to determine what triangles are congruent, that is, two triangles whose sides and angles are the same in measure and distribution. We can prove congruence by means of Side-Angle-Side theorem, where given angle is between two given sides.

Then, by direct inspection, the triangles from choices A and C are congruent, because sides and in-between angle are the same in measure and distribution.

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The triangles that are certainly congruent in the diagram are triangle A and triangle C.

Congruent triangles are triangles that have the same size and shape, such that the lengths of the three sides in one triangle are equivalent or have same length as the lengths of the three sides of the other triangle.

The congruent triangles can be obtained by using the congruence rules, or conditions for congruence, which includes; ASA, SAA, SSS, SAS

Where;

ASA is an acronym for Angle-Side-Angle

SAA is an acronym for Side-Angle-Angle

SSS is an acronym for Side-Side-Side

SAS is an acronym for Side-Angle-Side

The lengths of two sides and the measure of the included angle in triangle A are congruent to the length of two sides and the measure of the included side between the two sides in triangle C.

Therefore, the triangle in option A is congruent to the triangle in option C by the Side-Angle-Side, SAS, congruency rule.

The angle 62° is the non included angle to the 14 m and 10 m long side in option B and D, while the 14 m long side is the adjacent side to the 62° angle in option B. In option D the 10 m long side is adjacent to the 62° angle, which the orientation of triangles B and D to be different

The only two congruent triangles are triangles in option A and option C

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The system of equations has infinitely many solutions.

What is a system of equations?

A finite set of equations for which common solutions are sought is referred to in mathematics as a set of simultaneous equations, often known as a system of equations or an equation system.

Here, we have

Given:

-2x - 3y + z = 7

-x - y + 3z = -10

-2x - y - z = 13

We have to how many solutions the system of equations has.

Arranging the above equations in form of a matrix and finding the determinant, we get,

Δ = [tex]\left|\begin{array}{ccc}-2&-3&1\\-1&-1&3\\-2&-1&-1\end{array}\right|[/tex]

= -2(1+3) +3(1+6) +1(1-2)

= - 4 + 21 -1 = 16

Δ₁ = [tex]\left|\begin{array}{ccc}7&-3&1\\-10&-1&3\\13&-1&-1\end{array}\right|[/tex]

= 7(1+4) +3(10-39) +1(10+13)

= 35 - 87 + 23

= -29

Hence, the system of equations has infinitely many solutions.

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The value of the evaluated function f(x) = -3x² + 2x  at  

(i) f(-2) is -16   ;

(ii) f(a) is -3a² + 2a .

The Function is represented as : f(x) = -3x² + 2x  ;

(i) We have to evaluate the above function at f(-2),

So, we substitute x = -2 into the function:

we get ;

⇒ f(-2) = -3(-2)² + 2(-2)

⇒ f(-2) = -12 - 4 = -16  .

So ,  f(-2) = -16  .

(ii) Next we have to evaluate f(a),

So , we substitute x = a into the function:

⇒ f(a) = -3a² + 2a

So , f(a) = -3a² + 2a   .

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The given question is incomplete , the complete question is

For the following exercises, evaluate the function f(x) = -3x² + 2x ; at the given input: (i) f(-2) (ii) f(a)  .

The area of the figure is 596.625 in².

A rectangle is a two-dimensional shape where the length and width are different.

The area of a rectangle is given as:

Area = Length x width

We have,

We will consider the figure to have a rectangle and a circle.

Rectangle:

Length = 28 in

Breadth = 15 in

Area = 28 x 15 = 420 in²

Semicircle:

Diameter = 15 in

Radius = 7.5 in

Area = 1/2 πr²

= 1/2 x 3.14 x 7.5²

Area = 176.625/2

Area = 88.3125

There are two semicircles so,

= 2 x 88.3125

= 176.625 in²

Now,

Area of the figure.

= 420 + 176.625

= 596.625 in²

Thus,

596.625 in² is the area of the figure.

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Point of  y-intercept is (0,1)

In a graph, the point where the line or curve crosses the axis is called the intercept. If a point crosses the x-axis it is x-intercepts. If a point crosses y-axis it is y-intercepts.

Given the function is [tex]y=2.5^{x}[/tex]

y-intercept is the point where the graph of the function crosses the y-axis.

To find the point of y-intercept put x=0 in the given function, we get

if x=0  then [tex]y=2.5^{0}[/tex] ⇒ y=1.

The function crosses at the point 1 in the  y-axis.

Hence, point of  y-intercept is (0,1)

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Answer:

[tex]l = 30[/tex]     OR     [tex]l=-60[/tex]

Step-by-step explanation:

We can solve this problem with a system of equations by creating 2 equations from the information given.

1. "a length of 30 feet less than 5 times its width"

[tex]l = 5w - 30[/tex]

2. "the area of the rectangle is 360 ft²"

[tex]l\cdot w = 360[/tex]

To solve for the rectangle's length, we can create a substitution for w by dividing both sides of the second equation by length ([tex]l[/tex]).

[tex]l\cdot w = 360\\\overline{\ \ l\ \ } \ \ \ \: \ \overline{\ l \ \ }[/tex]

[tex]w = \dfrac{360}{l}[/tex]

Then, we can substitute this value back into the first equation and solve for [tex]l[/tex].

[tex]l = 5(\dfrac{360}{l}) - 30[/tex]

↓ distribute the 5

[tex]l(l) = \left(\dfrac{1800}{l} - 30\right)l[/tex]

↓ multiply both sides by [tex]l[/tex]

[tex]l^2 = 1800 - 30l[/tex]

↓ move all [tex]l[/tex]'s to one side

[tex]l^2 + 30l = 1800[/tex]

To solve this quadratic, we can complete the square.

    ↓ add (30/2)² to both sides

    [tex]l^2 + 30l + \left(\dfrac{30}{2}\right)^2 = 1800 + 15^2[/tex]

    ↓ simplify 15²

    [tex]l^2 + 30l + 225 = 1800 + 225[/tex]

    ↓ factor the left side as a perfect square

    [tex](l + 15)^2 = 2025[/tex]

    ↓ take the square root of both sides

    [tex]l + 15 = \pm \sqrt{2025}[/tex]

    [tex]l + 15 = \pm 45[/tex]

↓ split into 2 equations

[tex]l + 15 = 45[/tex]     OR     [tex]l + 15 = -45[/tex]

[tex]\boxed{l = 30}[/tex]          OR     [tex]\boxed{l=-60}[/tex]

The total area of the canvas needed would be 21.4 square yards.

The formula to calculate the total surface area of a triangular pyramid is -

T.S.A. = {1/2}(a × b) + {1/2}(b × s).

Given is a tent is in the shape of a triangular prism.

The total area of the canvas would be -

A{C} = 2{1/2 x 2 x 1.7} + 2{3 x 2} + {3 x 2}

A{C} = (2 x 1.7) + 12 + 6

A{C} = 3.4 + 18

A{C} = 21.4 square yards

Therefore, the total area of the canvas needed would be 21.4 square yards.

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Answer:6

Step-by-step explanation:beacuse it it

The probability she pulls out a purple piece of candy would be 0.22.

In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.

Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.

Given is Sam's fathers collection.

We can write the equations for upstream and downstream as -

x - y = 7/2

x + y = 21/10

Solving the equations graphically -

{x} = 2.8

{y} = 0.7
In still water, the speed would be -

S = 3 - 0.7

S = 2.3 Km/h

Distance peddled upstream -

D = 2.8 x 3.5 = 9.8 Km

Therefore, the speed in still water would be 2.3 Km/h and the distance peddled upstream would be 9.8 Km.

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Answer:

7/3>x

Step-by-step explanation:

-5<9-6x

subtract 9 from both sides

-14<-6x

divide by 6

14/6<x

simplified its 7/3<x

but since we divided by a negative we switch the sign

final answer is 7/3>x

Answer:

60 dollars

Step-by-step explanation:

I say this because it's simple math 199.90 - 139.99 its obviously 60

The ratio of the volume of a paraboloid (a parabola rotated about the y axis) to the volume of a cylinder with the same height and base radius is 1/2

We know that the formula for the volume of a paraboloid is:

V = 1/2 × π × r²  × h

where r is the Radius of Paraboloid

and h is Height of Paraboloid

Let us assume that V₁ represents the volume of a paraboloid (a parabola rotated about the y axis)

So, V₁ = 1/2 × π × r²  × h

Also, we know that the formula for the volume of a cylinder is:

V = π × r²  × h

Let us assume that V₂ be the volume of a cylinder with the same height h and base radius r.

So, V₂ =  π × r²  × h

consider the ratio of the volume of a paraboloid to the volume of a cylinder with the same height and base radius,

V₁ / V₂

= (1/2 × π × r²  × h) / (π × r²  × h)

= 1/2

Therefore, the required ratio is 1/2

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