Please use the following for the next 6 questions. Suppose that the average weekly earnings for employees in general automotive repair shops is $450, and that the population standard deviation for the earnings for such employees is $50. A sample of 100 such employees is selected at random.

1) What is the probability distribution of the average weekly earnings for employees in general automotive repair shops?

2) Find the probability that the average weekly earnings is less than $445.

3) Find the probability that the average weekly earnings is exactly equal to $445.

4) Find the probability that the average weekly earnings is between $445 and $455.

5) In answering the previous 3 questions, did you have to make any assumptions about the population distribution?

6) Now assume that the weekly earnings for employees in all general automotive repair shops is normally distributed, obtain the probability that a given employee will earn more than $480 in a given week.

Therefore, an appropriate confidence interval for μ30−μ60μ30​−μ60​ (the difference in mean body temperature) would be −0.796−0.796 to −0.404−0.404 (rounded to the appropriate decimal places).

To estimate the difference in the mean body temperature after exercising for 30 minutes versus 60 minutes, we can construct a confidence interval. Since the conditions for inference have been assumed to be met, we can proceed with the following steps:

   Calculate the difference in sample means: xˉ30−xˉ60=38.3−38.9=−0.6xˉ30​−xˉ60​=38.3−38.9=−0.6.

   Determine the standard error of the difference, which is given by the formula:

   SE=(s302n30)+(s602n60)SE=(n30​s302​​)+(n60​s602​​)

Substituting the values, we have:

SE=(0.27218)+(0.29224)≈0.100SE=(180.272​)+(240.292​)  ​≈0.100.

   Choose an appropriate confidence level. Let's say we want a 95% confidence interval.

   Using the standard error, calculate the margin of error, which is given by multiplying the critical value (corresponding to the chosen confidence level) by the standard error. For a 95% confidence level, the critical value is approximately 1.96.

   Margin of Error = 1.96 * 0.100 ≈ 0.196.

   Construct the confidence interval by subtracting and adding the margin of error to the difference in sample means:

   −0.6−0.196−0.6−0.196 to −0.6+0.196−0.6+0.196.

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The notation of the transformation of the polygons is (x, y) = (x - 9, y + 2)

From the question, we have the following parameters that can be used in our computation:

The polygons 1 and 2

In the graph, we can see that

using the above as a guide, we have the following:

(x, y) = (x - 9, y + 2)

This means that the transformation of the polygons is (x, y) = (x - 9, y + 2)

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Therefore, it would take approximately 36 minutes for 14 workers to complete the job.

To solve this inverse variation problem, we'll use the formula: t = k/w, where t represents the time needed, w represents the number of workers, and k is the constant of variation.

We can find the value of k by plugging in the given values of 9 workers and 56 minutes into the formula:

56 = k/9

To find the value of k, we multiply both sides of the equation by 9:

k = 504

Now that we know the constant of variation, we can determine the time it would take for 14 workers to complete the job. Plugging in the values into the formula:

t = 504/14

t ≈ 36

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Answer:

Yes, the triangle are similar.

Step-by-step explanation:

To prove that 2 triangle are similar you only need to prove that 2 corresponding angles are equal.

The sum of the interior angles add to 180.

< m on the left measures 30 degrees.

80 - 90 - 60 = 30

> t on the right measures 60 degrees.

180 - 90 - 30 = 60

If two corresponding angles of two triangles are equal that forces the third pair to be congruent, since the total of the angles must add up to 180.

Although we are not given the corresponding sides, there are proportional because the angles are equal.

Helping in the name of Jesus.

Answer:

Similar triangles are triangles that have the same shape but different sizes. In other words, if two triangles are similar, then their corresponding angles are congruent and their corresponding sides are in equal proportion.

For Question:

In Δ MSK and ΔQRT

∡S=∡R right angle

∡K=∡T=180°-90°-30°=60° Given

∡M=∡Q=180°-90°-60°=30° Given

Therefore,

Δ MSK  [tex]\bold{\sim}[/tex]  ΔQRT

By AA similarity.

Hence Proved:

The algebric expression -1/4 + 3/5 is equal to 7/20 when simplified.

To solve the expression -1/4 + 3/5, we need to find a common denominator for the fractions and then perform the addition.

The common denominator for 4 and 5 is 20. We can rewrite the fractions with this denominator:

-1/4 = -5/20

3/5 = 12/20

Now that the fractions have the same denominator, we can add them:

-5/20 + 12/20 = (-5 + 12)/20 = 7/20

Therefore, -1/4 + 3/5 is equal to 7/20.

To further simplify the fraction, we can check if there is a common factor between the numerator and denominator. In this case, 7 and 20 have no common factors other than 1, so the fraction is already in its simplest form.

Thus, the final answer is 7/20.

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Matching of the equations of the Ellipses to their equivalent equations in standard form are:

The general form of  x²/3² + (y - 2)²/2² = 1 is 4x² - 36y + 9y² = 0

The general form of (x + 7)²/7² + y²/6² = 1 is 36x² + 504x + 49y² = 0

The general form of x²/6² + (y + 4)²/4² = 1  is 16x² + 288y + 36y² = 0

The general form of (x - 3)²/3² + y²/5² = 1  is 25x² - 150x + 9y² = 0

The general form and the standard form of the ellipse are:

- The general form is: Ax² + Bxy + Cy² + Dx + Ey + F = 0

- The standard form is: (x - h)²/a² + (y - k)²/b² = 1

1) x²/3² + (y - 2)²/2² = 1

Expanding gives:

x²/9 + (y - 2)²/4 = 1

Multiply through by 36 to get:

4x² + 9(y - 2)² = 36

Expand bracket to get:

4x² + 9y² - 36y + 36 = 36

Subtract 36 from both sides to get:

4x² - 36y + 9y² = 0

The general form of  x²/3² + (y - 2)²/2² = 1 is 4x² + 9y² - 36y = 0

2) (x + 7)²/7² + y²/6² = 1

(x + 7)²/49 + y²/36 = 1

Multiply through by 1764 to get:

36(x + 7)² + 49y² = 1764

Expand bracket to get:

36x² + 504x + 1764 + 49y² = 1764

Subtract 1764 from both sides to get:

36x² + 504x + 49y² = 0

The general form of (x + 7)²/7² + y²/6² = 1 is 36x² + 504x + 49y² = 0

3) x²/6² + (y + 4)²/4² = 1

x²/36 + (y + 4)²/16 = 1

Multiply through by 576 to get:

16x² + 36(y + 4)² = 576

Expand bracket to get:

16x² + 36y² + 288y + 576 = 576

Subtract 576 from both sides to get:

16x² + 288y + 36y² = 0

The general form of x²/6² + (y + 4)²/4² = 1  is 16x² + 288y + 36y² = 0

4)  (x - 3)²/3² + y²/5² = 1

(x - 3)²/9 + y²/25 = 1

Multiply through by 225 to get:

25(x - 3)² + 9y² = 225

Expand the bracket power to get:

25x² - 150x + 225 + 9y² = 225

Subtract 225 from both sides

25x² - 150x + 9y² = 0

The general form of (x - 3)²/3² + y²/5² = 1  is 25x² - 150x + 9y² = 0

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Match the equations of ellipses to their equivalent equations in standard form.

25x² - 150x + 9y² = 0

16x² + 288y + 36y² = 0

49x² + 686 + 36y² = 0

4x² - 36y + 9y² = 0

36x² + 504x + 49y² = 0

9x² - 54x + 25y² = 0

x²/3² + (y - 2)²/2² = 1  ⇒

(x + 7)²/7² + y²/6² = 1 ⇒

x²/6² + (y + 4)²/4² = 1 ⇒

(x - 3)²/3² + y²/5² = 1 ⇒  

The number of $10 tickets sold is 1210 tickets.

The number of $20 tickets sold is 1344 tickets.

The number of $30 tickets sold is 827 tickets.

In order to write a system of linear equations to describe this situation, we would assign variables to the number of each tickets sold, and then translate the word problem into an algebraic equation as follows:

Since the basketball team sold 3381 tickets overall, 134 more $20 tickets than $10 tickets and the total sales are $63,790, a system of three linear equations to model this situation is given by;

x + y + z = 3381

y - x = 134

10x + 20y + 30z = 63790  ⇒ x + 2y + 3z = 6379

Next, we would solve the system of three linear equations simultaneously as follows;

x + x + 134 + z = 3381

2x + 134 + z = 3381

z = 3247 - 2x

x + 2(x + 134) + 3(3247 - 2x) = 6379

x = 1210 tickets.

For the value of y, we have:

y = x + 134

y = 1210 + 134

y = 1344 tickets.

For the z-value, we have:

z = 3247 - 2x

z = 3247 - 2(1210)

z = 827 tickets.

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Complete Question:

A basketball team sells tickets that cost $10, $20, or, for VIP seats, $30. The team has sold 3381 tickets overall. It has sold 134 more $20 tickets than $10 tickets. The total sales are $63,790. How many tickets of each kind have been sold?

How many $10 tickets were sold?

How many $20 tickets were sold?

how many $30 tickets sold?

Answer:

Kirsten makes $12.50 per hour

Step-by-step explanation:

The total is $432.50

$5 is taken out of her paycheck, so that will be negative.

35 hours is the time she spent working.

The unknown variable we are looking for is the amount she makes in one hour.

Setting up the equation, we get:

$432.50 = $35x - $5

where x is the amount she makes every hour

Solve for x:

$437.50=$35x

$437.50 / $35 = x

Therefore, x = $12.50

Answer:

Step-by-step explanation:

According to the triangle proportionality theorem (or thales theorem), the lines are parallel if AD/BD = AE/CE

15/12= 1.25

10/8 = 1.25

Because the ratios are equal, the lines are parallel to each other.  

Answer:

3.36643191324

-1.36643191324

Step-by-step explanation:

subtract 23 from both sides so that everything is on one side

5x²-10x-23=0

you can't factor this, so you need to use the quadratic formula:

(-b±√(b²-4*a*c))/2a

in our case, a= 5 b= -10 and c = -23

(-(-10)±√(10²-4*5*(-23)))/(2*5)

(10±√(100+460))/10

(10±√560)/10

we solve this twice. once where ± is + and again where ± is -

(10+√560)/10 = 3.36643191324

(10-√560)/10= -1.36643191324

Para calcular la población de bacterias al cabo de 4 horas, primero necesitamos determinar cuántos intervalos de 20 minutos hay en 4 horas.

4 horas son equivalentes a 240 minutos. Dividiendo 240 minutos entre 20 minutos por intervalo, obtenemos 12 intervalos.

Dado que la población de bacterias aumenta un 25% cada 20 minutos, podemos aplicar este crecimiento a cada intervalo de 20 minutos.

Para cada intervalo de 20 minutos, la población de bacterias aumentará en un 25%.

Por lo tanto, podemos calcular la población de bacterias al final de los 12 intervalos de la siguiente manera:

Población final = Población inicial * (1 + tasa de crecimiento)^número de intervalos

Población inicial = 32,200 bacterias

Tasa de crecimiento = 25% = 0.25

Número de intervalos = 12

Población final = 32,200 * (1 + 0.25)^12

Realizando los cálculos, obtenemos la población final de bacterias al cabo de 4 horas.

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Answer:

7: 63

8: 73

arithmetic sequence

y = 10x - 7

or f(n) = 10x -7

or

[tex]a_{n}[/tex] = 3 + (n-1)10

Step-by-step explanation:

the output increases by 10 every time that the input increases by 1.  That gives us our common difference or slope.  The y intercept is -7. That is the value is you worked backwards until you get to n = 0.  The initial value  is 3.  That is when n is 1.

When n is 3, f(n) is 23

When n is 2, f(n) is 13

When n is 1, f(n) is 3

When n is 0, f(n) is -7

I am not sure if this is clear.  I am assuming that you have a lot of knowledge of linear equations and how to write arithmetic sequence.  If my explanation is confusing it is me and not you.

Answer:

a. 63,73

b. Arithmetic sequence

c.t(n)=10n-7

Explanation:

a. Here is the completed table:

n | t(n)

4 | 33

5 | 43

6 | 53

7 | 63

8 | 73

b.

The type of sequence is arithmetic.

An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant.

In this case, the difference between any two consecutive terms is 10.

c.

The equation for the arithmetic sequence is:

t(n)=a+(n-1)d

where:

For Question:

Now

equation becomes:

t(4) = a+(4-1)10

33=a+30

a=33-30

a=3

Now, the Equation becomes

t(n) = 3+(n-1)10

t(n) = 3+10n-10

t(n)=10n-7

If Q is an orthogonal matrix with an eigenvalue λ1 = 1, then x, the eigenvector corresponding to λ1, is also an eigenvector of QT with an eigenvalue λ2 = λ1 * (QT * x).

To show that x is also an eigenvector of QT, we need to demonstrate that QT * x is a scalar multiple of x.

Given that Q is an orthogonal matrix, we know that QT * Q = I, where I is the identity matrix. This implies that Q * QT = I as well.

Let's denote x as the eigenvector corresponding to the eigenvalue λ1 This means that Q * x = λ1 * x.

Now, let's consider QT * x. We can multiply both sides of the equation Q * x = λ1 * x by QT:

QT * (Q * x) = QT * (λ1 * x)

Applying the associative property of matrix multiplication, we have:

(QT * Q) * x = λ1 * (QT * x)

Using the fact that Q * QT = I, we can simplify further:

I * x = λ1 * (QT * x)

Since I * x equals x, we have:

x = λ1 * (QT * x)

Now, notice that λ1 * (QT * x) is a scalar multiple of x, where the scalar is λ1. Therefore, we can rewrite the equation as:

x = λ2 * x

where λ2 = λ1 * (QT * x).

This shows that x is indeed an eigenvector of QT, with the eigenvalue λ2 = λ1 * (QT * x).

In conclusion, if Q is an orthogonal matrix with an eigenvalue λ1 = 1, then x, the eigenvector corresponding to λ1, is also an eigenvector of QT with an eigenvalue λ2 = λ1 * (QT * x).

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The equation of the line that passes through the points (1, 0) and (3, 6) is y = 3x - 3.

To find the equation of a line passing through two points, we can use the slope-intercept form of a linear equation: y = mx + b, where m is the slope of the line and b is the y-intercept.

Given points:

Point 1: (1, 0)

Point 2: (3, 6)

Step 1: Calculate the slope (m) using the formula:

m = (y2 - y1) / (x2 - x1)

Substituting the coordinates:

m = (6 - 0) / (3 - 1)

m = 6 / 2

m = 3

Step 2: Substitute one of the given points and the slope into the equation y = mx + b to find the y-intercept (b).

Using Point 1 (1, 0):

0 = 3(1) + b

0 = 3 + b

b = -3

Step 3: Write the equation of the line using the slope (m) and the y-intercept (b):

y = 3x - 3

Therefore, the equation of the line that passes through the points (1, 0) and (3, 6) is y = 3x - 3.

This equation represents a line with a slope of 3, indicating that for every increase of 1 unit in the x-coordinate, the y-coordinate increases by 3 units. The y-intercept of -3 means that the line crosses the y-axis at the point (0, -3). By substituting any x-value into the equation, we can determine the corresponding y-value on the line.

Hence, the equation of the line passing through (1, 0) and (3, 6) is y = 3x - 3.

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a) The triangles in this problem are congruent.

b) The reason is the SAS congruence theorem.

The Side-Angle-Side (SAS) congruence theorem states that if two sides of two similar triangles form a proportional relationship, and the angle measure between these two triangles is the same, then the two triangles are congruent.

The equal sides for this problem are given as follows:

Hypotenuses and PS.

As angle P is bisected, we have <1 = <2, which are the angles between the equal sides, hence the SAS theorem holds true for this problem.

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The equations that represent a line parallel to 3x - 4y = 7 and passes through the point (-4, -2) are:

0 = 3x - 4y - 4

3x - 4y = 4

To determine which equations represent a line that is parallel to the given line and passes through the point (-4, -2), we need to find the equations that have the same slope as the given line.

The equation 3x - 4y = 7 can be rewritten in slope-intercept form as y = (3/4)x - 7/4. From this form, we can see that the slope of the given line is 3/4.

Now let's analyze the options:

Oy = -3x + 1: This equation has a slope of -3, which is not equal to the slope of the given line (3/4). Therefore, this option does not represent a line parallel to the given line.

0 = 3x - 4y - 4: This equation can be rewritten as 3x - 4y = 4. Comparing this to the given line, we can see that it has the same coefficients of x and y, which means it has the same slope of 3/4. Therefore, this option represents a line parallel to the given line.

4x - 3y = -3: This equation has a slope of 4/3, which is not equal to the slope of the given line (3/4). Therefore, this option does not represent a line parallel to the given line.

Oy - 2 = -(x - 4): This equation can be rewritten as y = -x + 2. The slope of this line is -1, which is not equal to the slope of the given line (3/4). Therefore, this option does not represent a line parallel to the given line.

Oy + 2 = 2(x + 4): This equation can be rewritten as y = 2x + 6. The slope of this line is 2, which is not equal to the slope of the given line (3/4). Therefore, this option does not represent a line parallel to the given line.

In conclusion, the equations that represent a line parallel to 3x - 4y = 7 and passes through the point (-4, -2) are:

0 = 3x - 4y - 4

3x - 4y = 4

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To shift the graph left, we need to subtract 1 from x. This gives us F(x) = (x - 1)², which matches option D. Therefore, the correct answer is D.

The graph of F(x) is a parabolic graph, just like the graph of G(x)=x². F(x) is shifted to the left by 1 unit.

The equation for the graph of F(x) is F(x) = (x - 1)². This is because when a parabolic graph is shifted left or right, it affects the x-value of the vertex of the parabola.

The standard form of the equation for a parabolic graph is y = a(x - h)² + k, where (h, k) is the vertex of the graph and a determines whether the graph is facing up or down. In this case, the vertex of the graph of G(x)=x² is (0, 0).

When we shift this graph left by 1 unit, the vertex becomes (-1, 0). Therefore, the equation for F(x) is F(x) = a(x + 1)².

To determine the value of a, we can look at the y-intercept of the graph, which is (0, 1). Plugging these values into the equation, we get 1 = a(0 + 1)², which simplifies to 1 = a.

Therefore, the equation for F(x) is F(x) = (x + 1)². However, if we expand this equation, we get F(x) = x² + 2x + 1, which is not in the form given in the options.

To shift the graph left, we need to subtract 1 from x. This gives us F(x) = (x - 1)², which matches option D. Therefore, the correct answer is D.

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In 2025, both colleges will have an enrollment of 21,200 students.

To determine when the colleges will have the same enrollment, we need to set up an equation based on the projected enrollment increases and declines.

Let's assume the number of years from 2020 is represented by 'x'.

For College A, the projected enrollment can be represented by: 18,700 + 500x.

For College B, the projected enrollment can be represented by: 26,200 - 1000x.

To find when the colleges will have the same enrollment, we need to solve the equation:

18,700 + 500x = 26,200 - 1000x

Combining like terms, we get:

1500x = 7,500

Dividing both sides by 1500, we find:

x = 5

Therefore, the colleges will have the same enrollment in 5 years, or in the year 2025.

To determine the enrollment at that time, we substitute x = 5 into either equation. Let's use College A's equation:

Enrollment in College A in 2025:

18,700 + 500(5) = 18,700 + 2500 = 21,200 students

Enrollment in College B in 2025:

26,200 - 1000(5) = 26,200 - 5000 = 21,200 students

Both colleges will have an enrollment of 21,200 students in the year 2025 when they have the same enrollment.

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Answer:

The length should be 11 feet and the width should be 6 feet. 11 feet is 5 feet more than 6 feet. 11 feet times 6 feet is 66 square feet.

Step-by-step explanation:

The values of w, x, y, z are 120°, 60°, 120° , 60° respectively.

Angles in parallel lines are angles that are created when two parallel lines are intersected by another line called a transversal.

lines PQ and RS are parallel to each other and AB and CD are also parallel to each other. A pair of parallel line will serve as transversal.

Angles on parallel lines can be ;

Vertically opposite, alternate, corresponding and in each case the angles are equal.

w = 120°( vertically opposite angles)

x = 180-120 = 60( corresponding angles)

x = z = 60° ( corresponding angles)

y = 180-60 = 120°( angles on a straight line)

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Answer: Read the solution

Step-by-step explanation:

A. For diagram A, triangles ANC and BDE are similar. Thus, we can use similarity ratios to find the length of AC. (x+1)/12 = (x+5)/15. 15x+15=12x+60. Thus 3x=45, and x=15. Since we need to find AC, AC = 15+1 = 16.

B. For diagram B, triangles SRT and VUT are similar. Again, using similarity ratios, (4x-1)/14=(x+2)/6 or 14x+28=24x-6. 10x=34, x=3.4.

The value of dy/dt  at x= 1 if y = [tex]3x^2[/tex] - 3 and dx/dt = 3 is 18.

To find dy/dt at x = 1, we need to differentiate the given equation y = [tex]3x^2[/tex] - 3 with respect to t and then substitute x = 1 and dx/dt = 3.

Let's begin by differentiating y = [tex]3x^2[/tex]- 3 with respect to t using the chain rule. Since y is a function of x and x is a function of t, we have:

dy/dt = (dy/dx) * (dx/dt)

Given dx/dt = 3, we can rewrite the equation as:

dy/dt = [tex](dy/dx) * 3[/tex]

Now, let's find the derivative of y = [tex]3x^2 - 3[/tex] with respect to x:

dy/dx = d/dx[tex](3x^2 - 3)[/tex]

= 6x

Substituting this back into the equation for dy/dt, we get:

dy/dt = [tex](6x) * 3[/tex]

= 18x

Finally, we can evaluate dy/dt at x = 1:

dy/dt at x = 1 = 18(1)

= 18

Therefore, the value of dy/dt at x = 1 is 18.

It's important to note that the value of dx/dt = 3 was given, and we used it to find dy/dt using the chain rule and the derivative of the equation with respect to x.


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Navern Corporation's manufacturing cycle efficiency (MCE) for its elevators is approximately 11.36%.

Value-added time is the time spent on activities that directly add value to the product.

Value-added time: Process time = 5 days (the process of manufacturing the elevators)

Total cycle time: Wait time + Inspection time + Process time + Move time + Queue time

= 12 days + 12 days + 5 days + 6 days + 9 days

= 44 days

MCE = (Value-added time / Total cycle time) * 100

= (5 days / 44 days) * 100

≈ 11.36%

Therefore, Navern Corporation's manufacturing cycle efficiency (MCE) for its elevators is approximately 11.36%.

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Nave Corporation's manufacturing cycle efficiency (MCE) for its elevators is approximately 27.27%.

To calculate the manufacturing cycle efficiency (MCE) for Nave Corporation's elevators, we need to determine the ratio of value-added time to the total lead time.

Value-added time refers to the time spent on activities that directly contribute to the production or customization of the elevators, while lead time refers to the total time from order placement to installation.

Given the breakdown of time:

12 days for manufacturing and customization

5 days for waiting or non-value-added time

The value-added time is 12 days, and the total lead time is 44 days.

To calculate MCE, we divide the value-added time by the total lead time and multiply by 100 to express it as a percentage:

MCE = (Value-added time / Total lead time) x 100

MCE = (12 / 44) x 100

MCE = 27.27%

Therefore, Nave Corporation's manufacturing cycle efficiency (MCE) for its elevators is approximately 27.27%.

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Answer:

If Delaware was proportional to Alabama in enrollment to the district, it would have approximately 37 districts.

The ratio of students to districts in Alabama is 630,683/137 = 4,603 students per district. If Delaware had the same ratio of students to districts, it would have 120,475/4,603 = 26.16 districts. Rounding to the nearest whole number, Delaware would have approximately 26 districts if it was proportional to Alabama in enrollment to the district.

Answer: true

Step-by-step explanation:

Answer:

Step-by-step explanation:

                                                   error

Answer:

x ≈ 75.2

Step-by-step explanation:

using the tangent ratio in the right triangle

tan62° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{x}{40}[/tex] ( multiply both sides by 40 )

40 × tan62° = x , then

x ≈ 75.2 ( to the nearest tenth )

The complete factorization of the polynomial x³+2x²+x+2 is(x-1)(x+1)(x+2).

To factorize the polynomial x³+2x²+x+2 completely, one may use the rational root theorem, and synthetic division method.

Therefore, first we list the factors of 2 and divide it by the factors of 1 as the leading coefficient:Factors of 2 = ±1, ±2Factors of 1 = ±1Therefore, the possible rational roots for the given polynomial equation are± 1, ±2From the given polynomial, x³+2x²+x+2,

we can easily see that there is no remainder when divided by (x-1). Hence, (x-1) is a factor of the polynomial and we can use synthetic division to factorize the polynomial.x³+2x²+x+2 = (x-1)(x²+3x+2)Now we can use the quadratic formula or factoring to find the remaining factors. We get,x²+3x+2 = (x+1)(x+2).

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State the null and alternative hypotheses.

c. H0: µ = 0, Ha: µ > 3

The null hypothesis (H0) states that the population mean rating is equal to 3, while the alternative hypothesis (Ha) suggests that the population mean rating is greater than 3.

Specify the rejection region for α = 0.01. Reject H0 if

a. z > 2.33

The rejection region for a one-tailed test with a significance level of 0.01 (α) is in the upper tail of the distribution. In this case, we reject the null hypothesis if the test statistic (z-score) is greater than 2.33.

Calculate the test statistic

c. 0.44

To calculate the test statistic, we use the formula:

z = (sample mean - population mean) / (sample standard deviation / √n)

Plugging in the given values, we get:

z = (3.31 - 3) / (0.70 / √172) ≈ 0.44

What is your conclusion?

b. Fail to Reject H0

Since the calculated test statistic (0.44) does not exceed the critical value (2.33) in the rejection region, we fail to reject the null hypothesis. Therefore, we do not have sufficient evidence to conclude that the average rating is significantly greater than 3 in the population.

Obtain the lower bound of a 99% confidence interval for the mean rating.

d. 3.17

To obtain the lower bound of a 99% confidence interval, we subtract the margin of error from the sample mean. The margin of error can be calculated by multiplying the critical value (obtained from the z-table for a 99% confidence level) with the standard error (sample standard deviation divided by the square root of the sample size).

The lower bound is given by:

3.31 - (2.33 * (0.70 / √172)) ≈ 3.17

Obtain the upper bound of a 99% confidence interval for the mean rating.

c. 3.44

To obtain the upper bound of a 99% confidence interval, we add the margin of error to the sample mean. Using the same calculation as above, the upper bound is given by:

3.31 + (2.33 * (0.70 / √172)) ≈ 3.44

What assumption(s) do you need to make in order to answer the above questions?

b. The population distribution is assumed to be normal.

To perform hypothesis testing and construct confidence intervals, it is typically assumed that the population distribution is approximately normal. Additionally, assumptions such as random sampling and independence of observations are generally made.

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