In order to compute a sample mean by hand, first the data values must be added up. Then the sum is divided by the
sample size.
x = xx
Given the data set below, compute the summation, identify the sample size, and calculate the sample mean.
19
10
15
17
16
a.) Ex =
b.) n =
c.) x =

The volume of the can is approximately 304 cubic centimeters.

To determine the surface area and volume of the can, we need to consider the properties of a cylinder with a square base.

Surface Area:

The surface area of the can consists of three parts: the square base and the two circular faces.

a) Square Base:

The base of the can is a square, so its area is given by the formula:

Area = side^2.

Since the diameter of the can is 8 centimeters, the side of the square base is also 8 centimeters.

Therefore, the area of the square base is 8 cm [tex]\times[/tex] 8 cm = 64 square centimeters.

b) Circular Faces:

The can has two circular faces, one at the top and one at the bottom.

The formula for the area of a circle is[tex]A = \pi \times r^2,[/tex] where r is the radius. The radius of the can is half the diameter, which is 8 cm / 2 = 4 cm.

Thus, the area of each circular face is [tex]\pi \times (4 cm)^2 = 16\pi[/tex]  square centimeters.

To find the total surface area, we sum the areas of the square base and the two circular faces:

Total Surface Area = Square Base Area + 2 [tex]\times[/tex] Circular Face Area

[tex]= 64 cm^2 + 2 \times 16\pi cm^2[/tex]

≈ [tex]64 cm^2 + 100.48 cm^2[/tex]

≈[tex]164.48 cm^2[/tex]

Therefore, the surface area of the can is approximately 164.48 square centimeters.

Volume:

The volume of the can is given by the formula:

Volume = base area [tex]\times[/tex] height.

Since the base is a square, the base area is equal to the side^2, which is 8 cm [tex]\times[/tex] 8 cm = 64 square centimeters.

The height of the can is the height we calculated earlier, which is approximately 4.75 centimeters.

Volume = Base Area [tex]\times[/tex] Height

[tex]= 64 cm^2 \times4.75[/tex] cm

≈ 304 [tex]cm^3[/tex]

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The total length of electric wire in both rolls is 166.76 meters.

To find the total length of electric wire in both rolls, we need to add the lengths of the two rolls together.

The first roll contains 80m 20cm of wire. We can convert this to meters by noting that 1 meter is equal to 100 centimeters. Therefore, the length of the first roll is:

80m 20cm = 80 meters + (20 centimeters / 100) meters

= 80 meters + 0.20 meters

= 80.20 meters

Similarly, the second roll contains 86m 56cm of wire. Converting this to meters:

86m 56cm = 86 meters + (56 centimeters / 100) meters

= 86 meters + 0.56 meters

= 86.56 meters

To find the total length, we add the lengths of both rolls:

Total length = 80.20 meters + 86.56 meters

= 166.76 meters

Therefore, the total length of electric wire in both rolls is 166.76 meters.

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The values of f(-1), f(2x), f(t), and f(p-1) all simplify to 0. Therefore, f(-1) = f(2x) = f(t) = f(p-1) = 0.

To find the values of f(-1), f(2x), f(t), and f(p-1), we substitute the given values into the function f(x) = 2x - 2x and simplify.

f(-1):

Substituting x = -1 into f(x):

f(-1) = 2(-1) - 2(-1) = -2 + 2 = 0

f(2x):

Substituting x = 2x into f(x):

f(2x) = 2(2x) - 2(2x) = 4x - 4x = 0

f(t):

Substituting x = t into f(x):

f(t) = 2(t) - 2(t) = 2t - 2t = 0

f(p-1):

Substituting x = p-1 into f(x):

f(p-1) = 2(p-1) - 2(p-1) = 2p - 2 - 2p + 2 = 0

The values of f(-1), f(2x), f(t), and f(p-1) all simplify to 0. Therefore, f(-1) = f(2x) = f(t) = f(p-1) = 0.

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Answer:

[tex]\textsf{12)} \quad \text{a.}\;\;m \angle 1 = 107^{\circ}, \quad \text{b.}\;\;m \angle 2 = 107^{\circ}, \quad \text{c.}\;\;m \angle 3 = 73^{\circ}[/tex]

[tex]\textsf{13)} \quad EC = 6[/tex]

[tex]\textf{14)}\quad \text{a.}\;\;x = 33, \quad \text{b.}\;\; x = 8[/tex]

Step-by-step explanation:

As the base angles of an isosceles trapezoid are congruent, the measures of angles E and J are the same. Therefore:

[tex]m\angle 3 = 73^{\circ}[/tex]

The opposite angles of an isosceles trapezoid sum to 180°. Therefore:

[tex]\implies m\angle 1 + m\angle 3 = 180^{\circ}[/tex]

[tex]\implies m\angle 1 + 73^{\circ} = 180^{\circ}[/tex]

[tex]\implies m\angle 1 = 107^{\circ}[/tex]

Since the base angles of an isosceles trapezoid are congruent, the measures of angles A and N are the same. Therefore:

[tex]m\angle 2 = 107^{\circ}[/tex]

[tex]\hrulefill[/tex]

The diagonals of isosceles trapezoid ABCD are AC and BD.

Point E is the point of intersection of the diagonals. Therefore:

[tex]BE + ED = BD[/tex]

[tex]AE + EC = AC[/tex]

As the diagonals of an isosceles trapezoid are the same length, BD = AC. Therefore:

[tex]AE + EC = BD[/tex]

Given BD = 20 and AE = 14:

[tex]\implies AE + EC = BD[/tex]

[tex]\implies 14 + EC = 20[/tex]

[tex]\implies 14 + EC - 14 = 20 - 14[/tex]

[tex]\implies EC = 6[/tex]

[tex]\hrulefill[/tex]

The midsegment of a trapezoid is a line segment that connects the midpoints of the two non-parallel sides (legs) of the trapezoid.

The formula for the midsegment of a trapezoid is:

[tex]\boxed{\begin{minipage}{6 cm}\underline{Midsegment of a trapezoid}\\\\$M=\dfrac{1}{2}(a+b)$\\\\where:\\ \phantom{ww}$\bullet$ $M$ is the midsegment.\\ \phantom{ww}$\bullet$ $a$ and $b$ are the parallel sides.\\\end{minipage}}[/tex]

a)  From inspection of the given trapezoid:

Substitute these values into the midsegment formula and solve for x:

[tex]x=\dfrac{1}{2}(18+48)[/tex]

[tex]x=\dfrac{1}{2}(66)[/tex]

[tex]x=33[/tex]

Therefore, the value of x is 33.

b)  From inspection of the given trapezoid:

Substitute these values into the midsegment formula and solve for x:

[tex]15=\dfrac{1}{2}(22+x)[/tex]

[tex]30=22+x[/tex]

[tex]x=8[/tex]

Therefore, the value of x is 8.

Answer:

The five-number summary of the data represented by the given box plot is: Min: 72, Q1: 76, Median: 79, Q3: 80.5, Max: 84. Therefore, the correct option is: Min: 72, Q1: 76, Median: 79, Q3: 80.5, Max: 84.

Step-by-step explanation:

Answer: 28

Step-by-step explanation:

If [tex]a = 7[/tex] and [tex]b = 2[/tex], then [tex]2ab[/tex] can be worked out as follows:

[tex]\Large 2ab = 2 \times a \times b[/tex]

Substituting the values of [tex]a[/tex] and [tex]b[/tex], we get:

[tex]2 \times 7 \times 2 = 28[/tex]

Therefore, [tex]2ab[/tex] is equal to 28 when [tex]a = 7[/tex] and [tex]b = 2[/tex].

________________________________________________________

The answer is:

Work/explanation:

To evaluate the expression [tex]\sf{2ab}[/tex], I begin by plugging in 7 for a and 2 for b:

[tex]\large\pmb{2(7)(2)}[/tex]

Simplify by multiplying.

[tex]\large\pmb{2*14}[/tex]

[tex]\large\pmb{28}[/tex]

The type of medals is a categorical nominal variable, while the volume of water is a numerical continuous variable.

Note: This question is in Spanish, here is the question in English:

Write the type of variable and level of measurement for the following group of variables: A) type of medals at Olympic test. B) Volume of water in a tank

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The graph that shows a pair of lines representing the equations with the solution (4, -1) is option D.

To determine which graph represents the equations with the solution (4, -1), we need to check if the given point lies on the lines represented by each graph.

Let's examine each option:

A) The lines intersect 1 unit to the right and 4 units down. This does not match the given solution of (4, -1), so option A can be eliminated.

B) The lines intersect 4 units to the right and 1 unit down. Again, this does not match the given solution of (4, -1), so option B can be eliminated.

C) The lines intersect 4 units to the left and 1 unit up. This is the exact opposite of the given solution (4, -1), so option C can be eliminated.

D) The lines intersect 1 unit to the left and 4 units up. This matches the given solution of (4, -1), where the x-coordinate is 1 unit to the left and the y-coordinate is 4 units up. Therefore, option D represents the equations with the solution (4, -1).

In conclusion, the graph that shows a pair of lines representing the equations with the solution (4, -1) is option D.

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The pixel resolution is 4.896 x 10^5 pixels; in megapixels, it's approximately 0.490; and the aspect ratio is 18:17.

First, let's calculate the resolution in pixels: Resolution is simply the width times the height of the sensor. Specifically for this question, multiply 720 by 680. This gives us 489600 pixels for the resolution - in scientific notation to 4 significant figures we have 4.896 x 10^5.

Next, to calculate the resolution in megapixels, we need to divide the resolution by 1 million (since one megapixel is equal to 1 million pixels). So, 489600/1,000,000 is approximately 0.490 megapixels when rounded to the nearest thousandth of a megapixel.

Finally, the aspect ratio is the relationship of the width to the height of an image or screen. For a 720 x 680 sensor, if we divide 720 by 680, the result would be approximately 1.0588. However, we want this ratio in its simplest form, so we should use the greatest common divisor (GCD). In this case, the GCD of 720 and 680 is 40, so by dividing both dimensions by 40, we get an aspect ratio of 18:17.

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The measure of an arc in a circle is determined by the central angle that subtends it. Let's analyze each given measure of the indicated arcs:

90°: A 90° arc spans one-fourth of the entire circle since a full circle has 360°.

80°: An 80° arc is smaller than a quarter of the circle but larger than a sixth since 360° divided by 4 is 90°, and by 6 is 60°. Therefore, it lies between these two values.

100°: A 100° arc is slightly larger than a quarter of the circle but smaller than a third, as 360° divided by 4 is 90°, and by 3 is 120°.

70°: A 70° arc is smaller than both a quarter and a sixth of the circle, falling between 60° and 90°.

H: The measure of an arc denoted by "H" is not provided, so it cannot be determined without further information.

40°: A 40° arc is smaller than a sixth of the circle but larger than a twelfth, as 360° divided by 6 is 60°, and by 12 is 30°.

F: Similarly, the measure of the arc denoted by "F" is not provided, so it remains unknown without additional data.

Thus, the measures of the indicated arcs are as follows: 90°, between 60° and 90°, between 90° and 120°, between 60° and 90°, unknown (H), between 30° and 60°, and unknown (F).

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Answer:

[tex]\displaystyle \pi\biggr(1-\frac{1}{e}\biggr)[/tex]

Step-by-step explanation:

Shell Method (Vertical Axis)

[tex]\displaystyle V=2\pi\int^b_ar(x)h(x)\,dx[/tex]

Radius: [tex]r(x)=x[/tex]

Height: [tex]h(x)=e^{-x^2}[/tex]

Bounds: [tex][a,b]=[0,1][/tex]

Set up and evaluate integral

[tex]\displaystyle V=2\pi\int^1_0xe^{-x^2}\,dx[/tex]

[tex]\displaystyle V= -\frac{1}{2}\cdot2\pi\int^{-1}_0e^u\,du\\\\V= -\pi\int^{-1}_0e^u\,du\\\\V=\pi\int^0_{-1}e^u\,du\\\\V=\pi e^u\biggr|^0_{-1}\\\\V=\pi e^0-\pi e^{-1}\\\\V=\pi-\frac{\pi}{e}\\\\V=\pi\biggr(1-\frac{1}{e}\biggr)[/tex]

The volume of the gas after it is heated is approximately 0.0417 liters.

To find the volume of the gas after it is heated, we can use the combined gas law, which relates the initial and final conditions of a gas sample. The combined gas law is expressed as:

(P₁V₁) / T₁ = (P₂V₂) / T₂

Where:

P₁ and P₂ are the initial and final pressures of the gas (in kPa)

V₁ and V₂ are the initial and final volumes of the gas (in liters)

T₁ and T₂ are the initial and final temperatures of the gas (in Kelvin)

Given:

Initial volume (V₁) = 3.56 L

Initial temperature (T₁) = ST (which is typically 273.15 K)

Final temperature (T₂) = 400 K

Final pressure (P₂) = 125 kPa

Now we can plug these values into the combined gas law equation and solve for V₂:

(P₁V₁) / T₁ = (P₂V₂) / T₂

(1 * 3.56) / 273.15 = (125 * V₂) / 400

(3.56 / 273.15) = (125 * V₂) / 400

Cross-multiplying and solving for V₂:

3.56 * 400 = 273.15 * 125 * V₂

1424 = 34143.75 * V₂

V₂ = 1424 / 34143.75

V₂ ≈ 0.0417 L

As a result, the heated gas has a volume of approximately 0.0417 litres.

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Based on the data recorded by Anna on Saturday, we can estimate the number of people expected to visit The Youth Wing on Sunday.

Let's calculate the proportion of visitors to The Youth Wing compared to the total number of visitors on Saturday:

[tex]\displaystyle \text{Proportion} = \frac{\text{Visitors to The Youth Wing on Saturday}}{\text{Total visitors on Saturday}} = \frac{382}{382 + 461 + 355}[/tex]

Next, we'll apply this proportion to the total number of visitors on Sunday to estimate the number of people expected to go to The Youth Wing:

[tex]\displaystyle \text{Expected visitors to The Youth Wing on Sunday} = \text{Proportion} \times \text{Total visitors on Sunday}[/tex]

Now, let's substitute the values into the equation and calculate the estimated number of visitors to The Youth Wing on Sunday:

[tex]\displaystyle \text{Proportion} = \frac{382}{382 + 461 + 355}[/tex]

[tex]\displaystyle \text{Expected visitors to The Youth Wing on Sunday} = \text{Proportion} \times 800[/tex]

Calculating the proportion:

[tex]\displaystyle \text{Proportion} = \frac{382}{382 + 461 + 355} = \frac{382}{1198}[/tex]

Calculating the estimated number of visitors to The Youth Wing on Sunday:

[tex]\displaystyle \text{Expected visitors to The Youth Wing on Sunday} = \frac{382}{1198} \times 800[/tex]

Simplifying the equation:

[tex]\displaystyle \text{Expected visitors to The Youth Wing on Sunday} \approx \frac{382 \times 800}{1198}[/tex]

Now, let's calculate the approximate number of visitors to The Youth Wing on Sunday:

[tex]\displaystyle \text{Expected visitors to The Youth Wing on Sunday} \approx 254[/tex]

Therefore, based on the data recorded on Saturday, we can estimate that around 254 people should be expected to go to The Youth Wing on Sunday.

[tex]\huge{\mathfrak{\colorbox{black}{\textcolor{lime}{I\:hope\:this\:helps\:!\:\:}}}}[/tex]

♥️ [tex]\large{\underline{\textcolor{red}{\mathcal{SUMIT\:\:ROY\:\:(:\:\:}}}}[/tex]

The correct statements are

The dependent variable is t.

When f(2) = 1,323, the 2 represents "the year 2002," and the 1,323 represents "1,323 tons of garbage produced."

When f(4) = 1,458.6, the 4 represents "the year 2004," and the 1,458.6 represents "1,458.6 tons of garbage produced."

The independent variable is t. Option A,D,E,F.

Let's analyze each statement to understand why it is true:

A) The dependent variable is t: In the given function, f(t), the value of f depends on the value of t. Therefore, t is the independent variable, and f is the dependent variable.

D) When f(2) = 1,323, the 2 represents "the year 2002," and the 1,323 represents "1,323 tons of garbage produced": Here, the value of t is 2, representing the year 2002, and the value of f(t) is 1,323, representing the amount of garbage produced in tons.

E) When f(4) = 1,458.6, the 4 represents "the year 2004," and the 1,458.6 represents "1,458.6 tons of garbage produced": Similar to statement D, the value of t is 4, representing the year 2004, and the value of f(t) is 1,458.6, representing the amount of garbage produced in tons.

F) The independent variable is t: As mentioned in statement A, t is the independent variable in the given function. It is the variable that we can change or manipulate, and the value of f depends on the value of t.

Statements B, C, and G are incorrect:

B) When f(12) = 2,155, the 12 represents "12 tons of garbage produced," and the 2,155 represents "the year 2155": This statement is incorrect because in the given function, t represents the time in years after 2000, not the amount of garbage produced.

C) The dependent variable is f(t): This statement is incorrect because, as mentioned earlier, t is the independent variable, and f is the dependent variable.

G) The independent variable f(t): This statement is incorrect because f(t) represents the amount of garbage produced, which is the dependent variable in the given function.

So Option A,D.E.F. Are correct.

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Note the complete question is

Let f(t) be the amount of garbage, in tons, produced by a city, and let t be the time in years after 2000.

Which statements are true for the given function?

A.) The dependent variable is t.

B.) When f(12) = 2,155, 12 represents "12 tons of garbage produced," and 2,155 represents "the year 2155."

C.) The dependent variable is f(t).

D.) When f(2) = 1,323, 2 represents "the year 2002," and 1,323 represents "1,323 tons of garbage produced."

E.) When f(4) = 1,458.6, 4 represents "the year 2004," and 1,458.6 represents "1,458.6 tons of garbage produced."

F.) The independent variable is t.

G.) The independent variable  f(t)

Answer:

39°

Step-by-step explanation:

the sum of the 3 angles in a triangle = 180°

let the other angle be x , then

x + 54° + 87° = 180°

x + 141° = 180° ( subtract 141° from both sides )

x = 39°

that is the other angle is 39°

In a triangle, the sum of all angles is always 180°. To find the third angle in a scalene triangle where two angles are known, subtract the known angles from 180°. In this case, subtracting 54° and 87° from 180° gives a third angle of 39°.

The question refers to finding the third angle in a scalene triangle, where we know two of the angles. A scalene triangle is a triangle where all three sides are of a different length, and therefore all three angles are also different. The sum of the angles in any triangle is always 180°.

To find the third angle in the triangle, you can use the equation: Angle C = 180° - Angle A - Angle B.

So, we subtract the known angles from 180°: Angle C = 180° - 54° - 87° = 39°.

Therefore, the third angle in this scalene triangle is 39°.

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The length of the hypotenuse in the isosceles 90-degree triangle is √(2).

In an isosceles 90-degree triangle, two legs are equal in length, and the third side, known as the hypotenuse, is longer. Let's denote the length of the legs as x and the length of the hypotenuse as 4x.

According to the Pythagorean theorem, in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. In this case, we have:

[tex]x^2 + x^2 = (4x)^2.[/tex]

Simplifying the equation:

[tex]2x^2 = 16x^2.[/tex]

Dividing both sides of the equation by [tex]2x^2[/tex]:

[tex]1 = 8x^2.[/tex]

Dividing both sides of the equation by 8:

[tex]1/8 = x^2[/tex].

Taking the square root of both sides of the equation:

x = √(1/8).

Simplifying the square root:

x = √(1)/√(8),

x = 1/(√(2) * 2),

x = 1/(2√(2)).

Therefore, the length of each leg in the isosceles 90-degree triangle is 1/(2√(2)), and the length of the hypotenuse is 4 times the length of each leg, which is:

4 * (1/(2√(2))),

2/√(2).

To simplify the expression further, we can rationalize the denominator:

(2/√(2)) * (√(2)/√(2)),

2√(2)/2,

√(2).

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The modal sizes of shoes in the family's cupboard are 8 and 7.

To determine the modal size of shoes in the family's cupboard, we need to find the shoe size that appears most frequently in the given data. Let's analyze the sizes:

8, 7, 8, 8.5, 7, 8.5, 7

To find the mode, we can create a frequency table by counting the number of occurrences for each shoe size:

8     |     3

7     |     3

8.5 | 2

From the frequency table, we can see that both size 8 and size 7 appear three times each, while size 8.5 appears two times. Since both size 8 and size 7 have the highest frequency of occurrence (3), they are considered modal sizes. In this case, there is more than one mode, and we refer to it as a bimodal distribution.

To determine the mode, we performed a frequency count of each shoe size in the given data. We counted the number of occurrences for sizes 8, 7, and 8.5. Based on the frequency counts, we identified the sizes with the highest frequency, which turned out to be 8 and 7, both occurring three times. Thus, they are the modal sizes in the data set.

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The limit of f(x) as x approaches 10 is 315 and The value of k is 250.

The function f(x) is a piecewise function, so we need to evaluate it separately for x < 10 and x >= 10.

[tex]f(x)= { \frac{(0.1x(2)+20x+15,x < 10)}{(0.25x(3)+k,x > 10)}[/tex]

For x < 10, the function is equal to 0.1x^2 + 20x + 15. So the left-hand limit of f(x) as x approaches 10 is equal to 0.1(10)^2 + 20(10) + 15 = 315.

For x >= 10, the function is equal to 0.25x^3 + k. So the right-hand limit of f(x) as x approaches 10 is equal to 0.25(10)^3 + k = 250 + k.

Since the left-hand limit and the right-hand limit are equal, the limit of f(x) as x approaches 10 is also equal to 315, and the value of k is equal to 250.

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Answer:

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Step-by-step explanation:

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Answer:

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

Step-by-step explanation:

what this is?

Answer:

It's A

Step-by-step explanation:

Use the following models to show the equivalence of the fractions 35 and 610 a) Set model

Use the following models to show the equivalence of the fractions 35 and 610 a) Set modelUse the following models to show the equivalence of the fractions 35 and 610 a) Set modelUse the following models to show the equivalence of the fractions 35 and 610 a) Set modelUse the following models to show the equivalence of the fractions 35 and 610 a) Set modelUse the following models to show the equivalence of the fractions 35 and 610 a) Set modelUse the following models to show the equivalence of the fractions 35 and 610 a) Set model

The measure of the Intercepted arc having an inscribed angle of 40 degrees is 80 degrees.

An inscribed angle is simply an angle with its vertex on the circle and whose sides are chords.

The relationhip between an an inscribed angle and intercepted arc is expressed as:

Inscribed angle = 1/2 × intercepted arc.

From the figure:

Inscribed angle = 40 degrees

Intercepted arc = ?

Plug the given value into the above formula and solve for the arc:

Inscribed angle = 1/2 × intercepted arc.

40  = 1/2 × intercepted arc

Multiply both sides by 2:

40 × 2 = 2 × 1/2 × intercepted arc

40 × 2 = intercepted arc

Intercepted arc = 80°

Therefoore, the Intercepted arc is 80 degrees.

Option B) 80° is the correct answer.

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Answer:

The correct option is the 3rd one

angle 1 = angle 4 = angle 5 = angle 8 = 60 degrees,

angle 2 = angle 3 = angle 6 = angle 7 = 120 degrees

Step-by-step explanation:

To solve this, we only need to look at the top two angles, 1 and 2

Since line l is a line, angle 1 and 2 must sum to 180,

Since angle 1 = 60 degrees, then,

angle 1 + angle 2 = 180

60 + angle 2 = 180

angle 2 = 120 degrees

the only option that corresponds to this is the third option,

angle 1 = angle 4 = angle 5 = angle 8 = 60 degrees,

angle 2 = angle 3 =

Answer:

Step-by-step explanation:

The general equation for an ellipse with center (h, k) is:

[tex]\boxed{\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1}[/tex]

If a > b, the ellipse is horizontal.

If b > a, the ellipse is vertical.

Given equation:

[tex]\dfrac{(x-5)^2}{4}+\dfrac{(y+5)^2}{9}=1[/tex]

As b > a, the ellipse is vertical. Therefore:

Comparing the given equation with the standard form, we get:

Therefore:

To graph the ellipse:

Answer:

I pass my classes.

Step-by-step explanation:

I had to add this sentence or else it wouldnt allow me to send it.

Answer:

In math, the word "if" can be used for piecewise functions. In piecewise functions, you can see equations where f(x) = x+3 IF x>0 and f(x) = -x IF x<0.

Based on the data and the one-sample t-test, at the 0.05 level of significance, there is sufficient evidence to conclude that the average bear sales per store at Wal-Mart is significantly higher than $500

.

To determine if there is evidence that the average bear sales per store at Wal-Mart is more than $500 at the 0.05 level of significance, we can conduct a one-sample t-test. Let's go through the steps:

State the null and alternative hypotheses:

Null hypothesis (H₀): The average bear sales per store is equal to or less than $500.

Alternative hypothesis (H₁): The average bear sales per store is greater than $500.

Set the significance level (α):

In this case, the significance level is given as 0.05 or 5%.

Collect and analyze the data:

The weekly sales of bears in 10 stores are as follows:

8, 11, 0, 4, 7, 8, 10, 583

Calculate the test statistic:

To calculate the test statistic, we need to compute the sample mean, sample standard deviation, and the standard error of the mean.

Sample mean ([tex]\bar X[/tex]):

[tex]\bar X[/tex] = (8 + 11 + 0 + 4 + 7 + 8 + 10 + 583) / 8

[tex]\bar X[/tex] ≈ 76.375

Sample standard deviation (s):

s = √[Σ(x - [tex]\bar X[/tex])² / (n - 1)]

s ≈ 190.687

Standard error of the mean (SE):

SE = s / √n

SE ≈ 60.174

Now, we can calculate the t-value:

t = ([tex]\bar X[/tex] - μ₀) / SE

Where μ₀ is the hypothesized population mean ($500).

t = (76.375 - 500) / 60.174

t ≈ -7.758

Determine the critical value:

Since we are conducting a one-tailed test and the alternative hypothesis is that the average bear sales per store is greater than $500, we need to find the critical value for a one-tailed t-test with 8 degrees of freedom at a 0.05 level of significance. Looking up the critical value in the t-distribution table, we find it to be approximately 1.860.

Compare the test statistic with the critical value:

Since -7.758 is less than -1.860, we have enough evidence to reject the null hypothesis.

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The asymptotic distribution of the estimator S(X₁, ..., Xₙ) is a standard normal distribution, denoted as N(0, 1).

To find the asymptotic distribution of the estimator S(X₁, ..., Xₙ), we can use the Central Limit Theorem (CLT). However, we need some additional assumptions to apply the CLT, such as the finite variance of the random variables.

Given that E(Xᵢ^k) = αₖ for all k, we can assume that the random variables Xᵢ have a finite variance. Let's denote the variance of Xᵢ as Var(Xᵢ) = σ².

First, let's simplify the estimator S(X₁, ..., Xₙ):

S(X₁, ..., Xₙ) = 1 / (1/n ∑ᵢ (Xᵢ - c)²)

Notice that the numerator is a constant and doesn't affect the asymptotic distribution. So, we can focus on analyzing the denominator.

Let's calculate the expected value and variance of the denominator:

E[1/n ∑ᵢ (Xᵢ - c)²] = 1/n ∑ᵢ E[(Xᵢ - c)²] = 1/n ∑ᵢ (Var(Xᵢ) + E[Xᵢ]² - 2cE[Xᵢ] + c²)

= 1/n (nσ² + α₁ - 2cα₁ + c²) (using the fact that E[Xᵢ] = α₁ for all i)

Var[1/n ∑ᵢ (Xᵢ - c)²] = 1/n² ∑ᵢ Var[(Xᵢ - c)²] = 1/n² ∑ᵢ (Var(Xᵢ - c)²) = 1/n² ∑ᵢ (Var(Xᵢ))

= 1/n² (nσ²) = σ²/n

Now, let's apply the CLT. According to the CLT, if we have a sequence of independent and identically distributed random variables with a finite mean (μ) and a finite variance (σ²), the sample mean (in this case, our denominator) converges in distribution to a standard normal distribution as the sample size approaches infinity.

Therefore, as n approaches infinity, the asymptotic distribution of S(X₁, ..., Xₙ) will follow a standard normal distribution.

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