a researcher surveyed 300 people and found that 147 prefer x to y. calculate the 99% confidence interval for the true proportion of people who prefer x to y.

The bearing from the port to the ship is approximately North-28.17 degrees-East. This is calculated by finding the angle formed by the two legs of a right triangle, where the opposite side is 3 miles and the adjacent side is 9 miles.

Let's assume that the ship starts at point S, and the port is at the origin (0,0). Then, the ship's initial direction of North-25 degrees-West means that it's heading towards the point that is 25 degrees clockwise from the West direction, which is equivalent to the point (-cos(25),sin(25)) on the unit circle.

After traveling 3 miles in that direction, the ship ends up at point P1 (-3cos(25),3sin(25)).

Next, the ship turns right and travels 9 miles, which takes it to point

P2 (-3cos(25)+9sin(25),3sin(25)+9cos(25)).

To find the bearing from the port to the ship at this point, we need to find the angle that the line segment from the origin to P2 makes with the positive x-axis. Let θ be this angle. Then, we have

tan(θ) = (3sin(25)+9cos(25))/(-3cos(25)+9sin(25))

Using a calculator, we can find that tan(θ) is approximately -1.32. Therefore, we have

θ = arctan(-1.32) = -52.83 degrees

Since the bearing is measured clockwise from the North direction, the bearing from the port to the ship is

360 - (25 + 52.83) = 282.17 degrees

Rounding to the nearest hundredth, we get the final answer of 282.17 degrees. Therefore, the bearing from the port to the ship is approximately North-28.17 degrees-East.

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The probability that a user who logs on every day is from inside the country is 36.36%.

(a) To find the percent of all users who log on every day, we need to calculate the weighted average of the percentage of users who log on every day from outside the country and inside the country. Let's call this percentage "x".

x = 0.7 * 0.6 + 0.3 * 0.8
x = 0.42 + 0.24
x = 0.66

Therefore, 66% of all users log on every day.

(b) Bayes' Theorem states that the probability of an event A happening given that event B has occurred is equal to the probability of B given A multiplied by the probability of A, divided by the probability of B.

Let's define event A as a user being from inside the country and event B as a user logging on every day. We want to find the probability of A given B.

P(A|B) = P(B|A) * P(A) / P(B)

We already know P(B|A) = 0.8 (the probability of a user logging on every day given that they are from inside the country). We also know P(A) = 0.3 (the probability of a user being from inside the country). We just calculated P(B) in part (a) as 0.66.

P(A|B) = 0.8 * 0.3 / 0.66
P(A|B) = 0.3636

Therefore, the probability that a user who logs on every day is from inside the country is 36.36%.

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To express the answer in terms of r1, r2, r3, r4, r5, r6, r7, r8, c1, c2, c3, l1, and s, you need to first determine the transfer function of the circuit. This can be done using circuit analysis techniques such as Kirchhoff's laws, nodal analysis, or mesh analysis.



For an ideal op-amp circuit analysis, follow these steps:

1. Identify the type of circuit (inverting, non-inverting, integrator, differentiator, etc.) based on the arrangement of resistors, capacitors, and the op-amp.
2. Apply Kirchhoff's current and voltage laws to determine equations relating the input and output voltages.
3. Apply Laplace transform to the derived equations, replacing the time domain with the frequency domain (s-domain).
4. Solve the transformed equations for the output voltage in terms of the input voltage and the given terms (r1-r8, c1-c3, l1, and s).

Please provide the specific problem or circuit so that we can accurately provide the solution in terms of the given terms.

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The value of a polynomial at a given point is the result obtained by substituting the given value for the variable in the polynomial expression and simplifying the resulting expression.

In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, and multiplication. The degree of a polynomial is the highest power of its variable. For example, the polynomial 3x² - 5x + 2 has a degree of 2.

The value of a polynomial at a given point is the result obtained by substituting the given value for the variable in the polynomial expression and simplifying the resulting expression.

For instance, if we want to evaluate the polynomial 3x² - 5x + 2 at x = 4, we would replace each occurrence of x in the expression with 4, yielding 3(4)² - 5(4) + 2 = 42. Therefore, the value of the polynomial 3x² - 5x + 2 at the point x = 4 is 42. The value of a polynomial at a given point is important in many areas of mathematics, including calculus, algebra, and geometry.

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All the true statements about the area of the park include the following:

(a)The park can be decomposed into two parallelograms.

(b)The formula A = bh can be used to find the area of each piece of the park.

(c)The park can be decomposed into a parallelogram, a triangle, and a trapezoid.

(d)The area of the park is 126 m².

In Mathematics and Geometry, a parallelogram refers to a four-sided geometrical figure (shape) and it can be defined as a type of quadrilateral and two-dimensional geometrical figure that is composed of two (2) equal and parallel opposite sides.

By critically observing the image of the park, if it is split along the bottom of the first part, two (2) parallelograms would be created. Similarly, splitting the park along the side of the bottom parallelogram would create a trapezoid, triangle, and a parallelogram.

In Mathematics and Geometry, the area of a parallelogram can be calculated by using the following formula:

Area of a parallelogram, A = base area × height

Area of a park = (5 × 12) + (6 × 11)

Area of a park = 60 + 66

Area of a park = 126 m²

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Complete Question:

There is a park near Raphael’s home. To find its area, Raphael took the measurements shown. Select all the true statements about the area of the park.

(a)The park can be decomposed into two parallelograms.

(b)The formula A = bh can be used to find the area of each piece of the park.

(c)The park can be decomposed into a parallelogram, a triangle, and a trapezoid.

(d)The area of the park is 126 m2.

(e)The area of the park is 96 m2.

The values of the trig ratios is Sin X = a/c, Cos X = b/c and Tan X = a/b.

The value of each of the trig ratio is determined by applying a short formula known as SOH CAH TOA  as shown below;

SOH CAH TOA

SOH = sin θ = opposite /hypothenuse side

TOA = tan θ = opposite side / adjacent side

CAH = cos θ = adjacent side / hypothenuse side

Let the opposite side of angle X = a

Let the adjacent side of angle X = b

Let the hypothenuse side of angle X =  c

The values of the trig ratios is calculated as follows;

Sin X = a/c

Cos X = b/c

Tan X = a/b

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Answer:

45.3 miles

Step-by-step explanation:

a^2 + b^2 = c^2

23^2 + 39^2 = c^2

529 + 1521 = c^2

2050 = c^2

45.27692569068708 = c

approx. 45.3 = c

Answer:

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

The area of the photo is:

Area of the ad is:

Area of the photo is half the area of the entire ad:

Only positive root makes sense:

This gives us a dendrogram with three levels, where the first level has two clusters {{7,10},{20,28}} and {35}, the second level has two clusters {{7,10,20,28},35}, and the third level has only one cluster {{7,10,20,28,35}}.

What is a sequence?

A sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms).

To perform hierarchical clustering using single linkage, we start by treating each point as its own cluster, and then iteratively merge the two closest clusters until only one cluster remains. We use the single linkage method, which defines the distance between two clusters as the minimum distance between any two points in the clusters.

First, we calculate the pairwise distances between each point:

  7    10   20   28   35

7   -    3    13   21   28

10  3    -    10   18   25

20  13   10   -    8    15

28  21   18   8    -    7

35  28   25   15   7    -

Next, we find the two closest points/clusters and merge them:

  7,10  20   28   35

7,10  -    10   18   25

20    10   -    8    15

28    18   8    -    7

35    25   15   7    -

The closest points/clusters are 7 and 10, so we merge them to form a new cluster {7,10}.

  7,10  20,28  35

7,10  -    18     25

20,28 18   -      7

35    25   7      -

The closest points/clusters are now {20,28} and 35, so we merge them to form a new cluster {{20,28},35}.

 7,10  {20,28,35}

7,10  -    7

{20,28,35} 7    -

The closest points/clusters are now {7,10} and {{20,28},35}, so we merge them to form a new cluster {{{7,10},{20,28}},35}.

Hence, This gives us a dendrogram with three levels, where the first level has two clusters {{7,10},{20,28}} and {35}, the second level has two clusters {{7,10,20,28},35}, and the third level has only one cluster {{7,10,20,28,35}}.

The dendrogram can be visualized as in the attached image.

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The vectors {1, x, x²} are linearly independent and span P₂.

Thus, we can conclude that the vectors {1, x, x²} form a basis for P₂.

What is the system of equations?

A system of equations is a collection of one or more equations that are considered together. The system can consist of linear or nonlinear equations and may have one or more variables. The solution to a system of equations is the set of values that satisfy all of the equations in the system simultaneously.

To determine whether the vectors {1, x, x²} form a basis for P₂, we need to check whether they are linearly independent and whether they span P₂.

First, let's check whether the vectors are linearly independent.

To do this, we need to find constants c₀, c₁, and c₂ such that c₀ + c₁x + c₂x² = 0 for all x in P₂.

If we can find such constants, then the vectors are linearly dependent and not a basis for P₂.

Otherwise, the vectors are linearly independent.

Setting x = 0, we get c₀ = 0. Setting x = 1, we get c₀ + c₁ + c₂ = 0. Setting x = -1, we get c₀ - c₁ + c₂ = 0.

Solving this system of equations, we get c₀ = 0, c₁ = -c₂/2, and c₂ = c₂.

Since c₂ can take on any value, we can find values of c₁ and c₂ such that c₀ + c₁x + c₂x² = 0 for all x in P₂ only if c₁ = c₂ = 0.

Therefore, the vectors {1, x, x²} are linearly independent and span P₂.

Thus, we can conclude that the vectors {1, x, x²} form a basis for P₂.

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The null hypothesis is that the mean tuna consumption by a person is equal to 3.6 pounds per year (μ = 3.6). The alternative hypothesis is that the mean tuna consumption by a person is less than 3.6 pounds per year (μ < 3.6).

Null Hypothesis (H₀): The mean tuna consumption is 3.6 pounds per year (µ = 3.6). Alternative Hypothesis (H₁): The mean tuna consumption is not 3.6 pounds per year (µ ≠ 3.6).

Given that the P-value (0.003) is less than the significance level α = 0.07, we reject the null hypothesis. This suggests that there is enough evidence to reject the nutritionist's claim that the mean tuna consumption is 3.6 pounds per year.

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The graph represents the system of inequalities 2x + 3y ≥ 4 and x + 2y ≤ 3. The test point (0, 1) satisfies both of the inequalities in that system. So, the correct answer is C). 2x + 3y is greater than or equal to 4 and x + 2y is less than or equal to 3.

The graph represents the system of inequalities: 2x + 3y ≥ 4 and x + 2y ≤ 3. To determine the test point that satisfies both of the inequalities in the system, we can pick any point that lies within the shaded region on the graph. One such point is (1, 1).

Plugging this point into both inequalities, we get

2(1) + 3(1) ≥ 4 → 5 ≥ 4 (true)

1 + 2(1) ≤ 3 → 3 ≤ 3 (true)

Since both inequalities are true for the point (1, 1), it satisfies both of the inequalities in the system. So, the correct option is C).

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--The given question is incomplete, the complete question is given

" Which system of inequalities does the graph represent? Which test point satisfies both of the inequalities in that system?

The graph represents the system of inequalities__________

A) 2x + 3y is greater than or equal to 4 and x+2y is greater than or equal to 3

B) 2x + 3y is less than or equal to 4 and x + 2y is less than or equal to 3

C) 2x + 3y is greater than or equal to 4 and x + 2y is less than or equal to 3

D) 2x +3y is less than or equal to 4 and x + 2y is greater than or equal to 3

E) 2x + 3y is less than or equal to 4 and 2x + 2y is less than or equal to to 3"--

The mean is higher than the median because the data is skewed to the right. The median is more resistant to the skew in the data.

To calculate the standard deviation of the number of lines in use that this support center expects to have at noon, we would need to have a dataset of the number of lines in use at different times.

If we have this dataset, we can use the following formula to calculate the standard deviation:

Standard deviation = √(sum((x - mean)²) / n)

Where:

x is the number of lines in use at a given time

mean is the mean of the number of lines in use across all times

n is the total number of times in the dataset

We can calculate the mean of the number of lines in use by adding up all the values and dividing by the total number of times. Once we have the mean, we can calculate the standard deviation using the formula above. However, without access to the dataset, it is not possible to provide a specific answer.

Therefore, The mean is higher than the median because the data is skewed to the right. The median is more resistant to the skew in the data.

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Complete question:

Let the random variable X represent the number of telephone lines in use by the technical support center of a software manufacturer at noon each day. The probability distribution of X is shown in the table below.

In a sentence of two, comment on the relationship between the mean and the median relative to the shape of this distribution.

An observation that causes the values of the slope and the intercept in the line of best fit to be considerably different from what they would be if the observation were removed from the data set is said to be an outlier.

An outlier is an observation that is significantly different from other observations in a dataset. It is a data point that is located far away from the other data points, and it may have a disproportionate influence on the analysis and conclusions drawn from the data.

what is slope?

Slope is a measure of the steepness of a line. It is calculated as the change in the y-coordinate (vertical change) divided by the change in the x-coordinate (horizontal change) between two points on the line. The slope represents the rate at which the line is rising or falling.

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Option A is correct i.e. Angle C has a measurement of 38°.

A circle's arc is described as a section or segment of the circumference of a circle.

It is given that Arc AB = 76°. We must determine the value of angle C.

As we know that The angle at the circumference is double of the angle at the center.

So, we have:

2 × Angle C = 76°

Angle C = 1 / 2 × 76°

Angle C = 76° / 2

Angle C = 38°

Hence, Option A is correct i.e. Angle C has a measurement of 38°.

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Given Question is incomplete, the complete question is below:

Angle ACB intercepts arc AB. Arc AB has a measure of 76 degrees. What is the measure of Angle C?

38°

76°

90°

152°

The new perimeter of the triangle after it is dilated with a scale factor of 3 is 36 units.

If James has a triangle with a perimeter of 12, it means that the sum of the lengths of all three sides of the triangle is 12. Let's call the lengths of the three sides a, b, and c, where a + b + c = 12.

When the triangle is dilated with a scale factor of 3, it means that all the sides of the triangle are multiplied by 3. Let's call the new lengths of the sides A, B, and C, where A = 3a, B = 3b, and C = 3c.

The new perimeter of the triangle is the sum of the lengths of the new sides, which is:

A + B + C = 3a + 3b + 3c

We know that a + b + c = 12, so we can substitute this into the above equation:

A + B + C = 3(12)

A + B + C = 36

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(a) The probability that the current component is Brand X is 1/2, since both brands are equally likely to fail at any given time and the replacement component is always from the opposite brand.

(b) The age of the current component has a uniform distribution on [0,1] if it is a Brand Y component (since it was just replaced) and on [0,2] if it is a Brand X component (since it has been in use for some time).

(c) The total lifetime of the current component has a mixture distribution, where the probability density function is given by:

    f(t) = (1/4) for 1 ≤ t ≤ 2

    f(t) = (1/6) for 2 ≤ t ≤ 3

(d) If the replacement component is chosen randomly with a probability 1/2 for each brand, then the probability that the current component is Brand X is still 1/2.

This is because if the current component is a Brand X component, it has been in use for a time between 0 and 2 (uniformly distributed) and then it will fail at a time between 1 and 2 (uniformly distributed), for a total lifetime between 1 and 2 (with probability 1/2) or between 2 and 3 (with probability 1/2).

If the current component is a Brand Y component, it has been in use for a time between 0 and 1 (uniformly distributed) and then it will fail at a time between 1 and 3 (uniformly distributed), for a total lifetime between 1 and 2 (with probability 1/3), between 2 and 3 (with probability 1/3), or between 3 and 4 (with probability 1/3).

However, the distribution of the age and total lifetime of the current component will be different. The age of the current component will have a mixture distribution, where the probability density function is given by:

f(t) = (1/4) for 1 ≤ t ≤ 2

f(t) = (1/6) for 2 ≤ t ≤ 3

f(t) = (1/12) for 3 ≤ t ≤ 4

This is because if the current component is a Brand X component, it has been in use for a time between 0 and 2 (uniformly distributed) and then it will fail at a time between 1 and 2 (uniformly distributed), for a total lifetime between 1 and 2 (with probability 1/2). If the current component is a Brand Y component, it has been in use for a time between 0 and 3 (uniformly distributed) and then it will fail at a time between 1 and 3 (uniformly distributed), for a total lifetime between 1 and 2 (with probability 1/6), between 2 and 3 (with probability 1/3), or between 3 and 4 (with probability 1/6). The total lifetime of the current component will also have a mixture distribution, where the probability density function is the same as in part (c).

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Answer:

(40 yds/min)(30 min) = 1,200 yds

1,200/150 = y/900, so y = 7,200 yards

7,200 yds/(40 yds/min) = 180 minutes

For estimating the average number of miles per gallon of gasoline a car,

a)About 68% of possible sample means will be in the range between 24.711 and 25.288.

b) About 95% of possible sample means will be in the range between 24.422 and 25.578.

c) About 99.7% of possible sample means will be in the range between 24.133 and 25.867.

Let's we are interested in estimating average number of miles per gallon of gasoline a car. Firstly, sample size, n = 12

mean = 25 miles per gallon

standard deviation = 1

Shape of distribution is bell-shaped. Using Empirical rule,

Standard error =[tex] \frac{ \sigma}{\sqrt{n}}[/tex]

[tex] = \frac{ 1}{\sqrt{12}}[/tex]

= 0.289

a) The 68% of observed data points will lie inside one standard deviation of the mean, i.e [tex]\mu ± \sigma [/tex]

= 25 ± 0.289

= ( 24.711, 25.288)

b) 95% will fall within two standard deviations,i.e., [tex]\mu ± 2 \sigma [/tex]

= 25 ± 2×0.289

= (24.422, 25.578)

c) 99.7% will occur within three standard deviations.i.e., [tex]\mu ± 3 \sigma [/tex]

= 25 ± 3×0.289

= (24.133, 25.867)

Hence, required value is (24.133, 25.867)

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Complete question:

Suppose that you are interested in estimating the average number of miles per gallon of gasoline your car can get. You calculate the miles per gallon for each of the next twelve times you fill the tank. Suppose that in truth, the values for your car are bell-shaped, with a mean of 25 miles per gallon and a standard deviation of 1. Find the possible sample means you are likely to get based on your sample of twelve observations. Consider the intervals into which 68%, 95%, and almost all of the potential sample means will fall, using the Empirical Rule. (Round all answers to the nearest thousandth.)

a) About 68% of possible sample means will be in the range between ____ and ____ .

b) About 95% of possible sample means will be in the range between ____ and ____ .

c) About 99.7% of possible sample means will be in the range between ____ and ____ .

The expected mean of the sampling distribution, with groups of size 30, will also be 47. This is because the Central Limit Theorem states that as sample size increases.

The sampling distribution of the mean approaches a normal distribution with a mean equal to the population mean. Therefore, with a large enough sample size of 30, the expected mean of the sampling distribution will be the same as the population mean of 47.
Given that the population has a mean of 47, when creating a sampling distribution for the mean using groups of size 30, the expected mean of the sampling distribution will be the same as the population mean. The expected mean of the sampling distribution with groups of size 30 will be 47.

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(i) T-interval

(ii) F. Because I don't have a large number of professors and I don't know the standard deviation of all FSU professors' heights.

What is standard deviation?

Standard deviation is a measure of the amount of variation or dispersion of a set of data values from the mean value.

When constructing a confidence interval for the mean of a population using a sample, we use either a Z-interval or a T-interval based on the sample size and whether we know the population standard deviation.

If the sample size is large (usually taken to be greater than or equal to 30) and/or we know the population standard deviation, then we can use a Z-interval.

However, if the sample size is small (usually less than 30) and/or we don't know the population standard deviation, we should use a T-interval.

In this case, we don't have a large sample size (just the professors the student has this term), and we don't know the standard deviation of all FSU professors' heights, so we would use a T-interval.

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(a) The theoretical probability of rolling an odd number is 3/6 or 1/2.

(b) The experimental probability of rolling an odd number is 0.6.

(c) The Statement is True.

(a) The theoretical probability of rolling an odd number is 3/6 or 1/2.

(b) The experimental probability of rolling an odd number is

= (3+7+2)/20

= 12/20

= 0.6.

(c) As the number of rolls increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal.

Therefore, the statement "As the number of rolls increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal" is true.

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The ordered pairs of the linear expression 3x + 3y = 9 is (0, 3)

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

The linear expression 3x+3y=9

To determine the ordered pairs of the linear expression, we set x to any value say x = 0 0 and then calculate the value of y

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

3(0) + 3y = 9

Evauate

3y = 9

Divide both sides by 3

y = 3

This means that the value of y is equal to 3

So, we have (0, 3)

Hence, the ordered pairs of the linear expression is (0, 3)

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The expected count of members living less than 25 miles from the waterfall who rated the cultural importance as high is 60.

To determine the expected count of members living less than 25 miles from the waterfall who rated the cultural importance as high, we need to use the information provided in the table.

Here we need to find the total number of respondents in each group For those living less than 25 miles from the waterfall,

The total is 150.

For those living more than 25 miles from the waterfall,

the total is 100.

Again,we need to find the proportion of respondents in each group who rated the cultural importance as high.

For those living less than 25 miles from the waterfall,

the proportion is 60/150 = 0.4.

For those living more than 25 miles from the waterfall,

the proportion is 40/100 = 0.4.

Now, we can find the expected count of members living less than 25 miles from the waterfall who rated the cultural importance as high by multiplying the total number of respondents in that group (150) by the proportion who rated the cultural importance as high (0.4). Expected count = 150 x 0.4 = 60

Therefore, the expected count of members living less than 25 miles from the waterfall who rated the cultural importance as high is 60.

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We can conclude that the equation has exactly one real root (namely, x = 0) and two complex conjugate roots.

What is the quadratic equation?

The solutions to the quadratic equation are the values of the unknown variable x, which satisfy the equation. These solutions are called roots or zeros of quadratic equations. The roots of any polynomial are the solutions for the given equation.

I can give an example to demonstrate how to use inspection and Descartes' rule to find a real root of an equation and show that there are no other real roots.

Consider the equation x³ - 3x² + 2x = 0. By inspection, we can see that x = 0 is a real root of the equation since the left-hand side of the equation evaluates to 0 when x = 0.

To use Descartes' rule, we need to count the number of sign changes in the coefficients of the polynomial f(x) = x³ - 3x²  + 2x.

There are two sign changes: from positive to negative in the coefficient of x² and from negative to positive in the constant term.

Therefore, according to Descartes' rule, the equation has either two or zero positive real roots.

Next, we need to count the number of sign changes in the coefficients of f(-x), which is obtained by replacing x with -x in f(x).

We have f(-x) = -x³ - 3x² - 2x, which has one sign change: from negative to positive in the coefficient of x².

Therefore, according to Descartes' rule, the equation has either one or three negative real roots.

Since the total number of positive and negative real roots must add up to the degree of the polynomial (which is 3 in this case),

Hence, we can conclude that the equation has exactly one real root (namely, x = 0) and two complex conjugate roots.

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The sample that could constitute a discrete random variable is the number of near collisions of aircraft in a year.

This is because it is a countable, finite number and not a continuous measurement like the height of the ocean's tide. The total number of points scored in a football game could also be considered a discrete random variable because it is a countable, finite number. However, the height of the ocean's tide is a continuous measurement and cannot be counted as a discrete random variable.
The sample that could constitute a discrete random variable is: i. total number of points scored in a football game.
A discrete random variable represents a countable number of distinct values or outcomes. In this case, the total number of points scored in a football game can be counted and listed, making it a discrete random variable.
On the other hand, the height of the ocean's tide at a given location (ii) is a continuous random variable, as it can take any value within a given range, and the number of near collisions of aircraft in a year (iii) could also be considered as a discrete random variable, but it's not one of the options in your question.

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The 95% confidence interval for the proportion of the population who prefer brand named items is:

CI = (0.102, 0.232)

What is confidence interval?

A confidence interval is a statistical tool used to estimate the range of possible values in which a population parameter, such as the mean or proportion, is expected to lie with a certain level of confidence based on the observed sample data.

To find the 95% confidence interval for the population proportion, we use the formula:

CI = p′ ± z*[tex]\sqrt{(p'q'/n)[/tex]

where:

CI: confidence interval

p′: sample proportion

q′: 1 - p′

z: z-score from the standard normal distribution for the desired confidence level (95% in this case)

n: sample size

Substituting the given values, we get:

CI = 0.167 ± 1.96[tex]\sqrt{((0.1670.833)/180)[/tex]

Simplifying, we get:

CI = 0.167 ± 0.065

Therefore, the 95% confidence interval for the proportion of the population who prefer brand named items is:

CI = (0.102, 0.232)

This means that we can be 95% confident that the true population proportion of people who prefer brand named items falls within this range.

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True. The central limit theorem states that if the sample size n is large enough (usually considered to be at least 30), then the sampling distribution of the mean will be approximately normal, regardless of the shape of the population distribution.

The sampling distribution of the mean can be approximated by the normal distribution if the sample size (n) is at least 30. This statement is based on the Central Limit Theorem, which states that the sampling distribution of the mean of a random sample drawn from any population will approach a normal distribution as the sample size increases, regardless of the shape of the population distribution. A sample size of 30 is often considered the threshold for approximating a normal distribution in such cases.

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The new price of the playground slide after a 45 1/8% markup is $476.25.

To find the new price after a 45 1/8% markup, we need to first calculate the markup amount and then add it to the original price.

Markup amount = original price x markup rate

Markup rate = 45 1/8% = 45.125%

We need to convert the percentage to a decimal by dividing by 100

Markup rate = 45.125% ÷ 100 = 0.45125

Now we can calculate the markup amount

Markup amount = $328 x 0.45125 = $148.25

To find the new price, we add the markup amount to the original price

New price = original price + markup amount

New price = $328 + $148.25 = $476.25

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The area of the rhombus is 108 meters²

A rhombus is a type of quadrilateral with four sides of equal length. It is also known as a diamond or a lozenge.

The diagonals of a rhombus bisect each other at right angles, and they also bisect the angles of the rhombus. The area of a rhombus can be found by multiplying the lengths of its diagonals and dividing by 2.  

Hence, the Area of the rhombus (A) = d₁d₂/2  

Where d₁ and d₂ are lengths of diagonals

Here we have

A drawing of a rhombus with both diagonals bisecting each other at right angles. The vertical diagonal is divided into two lengths of 6 meters each and the horizontal diagonal is divided into two lengths of 9 meters each

From the given data,

Length of vertical diagonal (d₁) = 2 × 6 = 12 meters

Length of horizontal diagonal (d₂) = 2 × 9 = 18 meters  

Using the formula, Area of the rhombus (A) = d₁d₂/2

= (12)(18)/2 = 6 (18) = 108 meters²  

Therefore,

The area of the rhombus is 108 meters²  

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