Lines c and d are parallel lines cut by transversal p. Horizontal and parallel lines c and d are cut by transversal p. On line c where it intersects with line p, 4 angles are formed. Clockwise, from uppercase left, the angles are: 1, 2, 3, 4. On line d where it intersects with line p, 4 angles are formed. Clockwise, from uppercase left, the angles are: 5, 6, 7, 8. Which must be true by the corresponding angles theorem? ∠1 ≅ ∠7 ∠2 ≅ ∠6 ∠3 ≅ ∠5 ∠5 ≅ ∠7

The correct answer is a: every individual has an equal chance of being selected. A random sample is a subset of a population that is selected in a way that each member of the population has an equal probability of being chosen.

The randomness of the sample helps to ensure that the results obtained from the sample are representative of the entire population.

Option b, the probabilities cannot change during a series of selections, is a requirement for independent events but not necessarily for a random sample.

Option c, there must be sampling with replacement, is not always required for a random sample. Sampling with replacement means that after an individual is selected, they are returned to the population and can be selected again. This is not always necessary for a random sample.

Therefore, the correct answer is a: every individual has an equal chance of being selected.

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Answer:

the other side is 8 cm and the perimeter is 22 cm

Step-by-step explanation:

one side is 3

another side has to be 8 because

(3)(8)= 24 cm^2

the perimeter is 8(2) + 3(2)= 16+6= 22 cm

Answer:

22cm

Step-by-step explanation:

Hope this helps!

We do not have sufficient evidence to conclude that the mean pressure produced by the valve is greater than 4.3 pounds/square inch at a level of significance of 0.01.

What is the mean and standard deviation?

A concise measurement of how far each observation deviates from the mean is the standard deviation. If the differences themselves were tallied up, the positive would perfectly balance the negative and their aggregate would be zero. In order to combine the squares of the differences.

The null hypothesis is that the true mean pressure produced by the valve is equal to the designed mean pressure of 4.3 pounds/square inch:

H₀: µ = 4.3

The alternative hypothesis is that the true mean pressure produced by the valve is greater than 4.3 pounds/square inch:

Ha: µ > 4.3

We will use a one-sample t-test to test this hypothesis. The test statistic is calculated as:

[tex]t = (x - \mu) / (s / \sqrt(n))[/tex]

where x is the sample mean, µ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.

Plugging in the given values, we get:

t = (4.5 - 4.3) / (0.8 / √(19)) = 1.84

Using a t-distribution table with 18 degrees of freedom (n-1), and a significance level of 0.01 (one-tailed test), the critical t-value is 2.552.

Since our calculated t-value of 1.84 is less than the critical t-value of 2.552, we fail to reject the null hypothesis.

Therefore, there is not enough evidence to conclude that the valve performs above the designed specifications at a significance level of 0.01.

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Answer:

1 kilogram = 2.204623 pounds

$17/kilogram = $17/2.204623 pounds

= $7.71/pound

In summary, if the input graph G(V, E) has a negative weight cycle, there are no solutions to the single-source shortest path problem.

In the single-source shortest path problem, the goal is to find the shortest path from a given source vertex to all other vertices in a graph G(V, E), where V is the set of vertices and E is the set of edges.
If the input graph G(V, E) has a negative weight cycle, then there are no correct solutions to the single-source shortest path problem. This is because the presence of a negative weight cycle allows for a path with decreasing total weight, as you can continually traverse the cycle to reduce the path's weight. As a result, the shortest path is undefined in this case.

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For the conversion of polar coordinates into cartesian coordinates means (r, θ) changed into (x,y), the Cartesian equation for the polar equation r = 9 sin(θ), is equals to x² + y² - 9y = 0.

In the polar coordinate system has different characteristics. To convert polar coordinates to Cartesian or rectangular coordinates, solve for the x and y coordinates separately.

We have a polar equation, r = 9 sin(θ). We have to convert this polar equation into cartesian equation. To convert from polar coordinates (r, θ) to cartesian coordinates ( x, y ). So, we use the following equations, x = rcos(θ)

y = r sin(θ)

Now, x² + y² = r² sin²(θ) + r²cos(θ)

= r²( sin²(θ) + cos²(θ))

= r²

Here, we have r = 9 sin(θ)

=>[tex] r = 9 \frac{ 9y}{r}[/tex]

=> r² = 9y

So, we can write as, x² + y² = 9y

=> x² + y² - 9y = 0

Hence, required equation is x² + y² - 9y = 0.

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To find the z-statistic, we can use the formula: z = (p-hat - p) / sqrt(p * (1-p) / n). Therefore, the z-statistic for this p-hat is 2.52.

where p-hat is the sample proportion, p is the hypothesized population proportion, and n is the sample size.

Plugging in the values, we get:

z = (0.48 - 0.4) / sqrt(0.4 * 0.6 / 160)
z = 2.52

Therefore, the z-statistic for this p-hat is 2.52.

To calculate the z-statistic for the given p-hat, we will use the following formula:

z = (p-hat - p) / sqrt((p * (1 - p)) / n)

where p-hat is the sample proportion (0.48), p is the hypothesized proportion (0.4), and n is the sample size (160).

z = (0.48 - 0.4) / sqrt((0.4 * (1 - 0.4)) / 160)
z = (0.08) / sqrt(0.24 / 160)
z = 0.08 / 0.030

The z-statistic for this p-hat is approximately 2.67.

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The probability of selecting a white bead, replacing it, and then selecting a red bead is 0.15 and this is an independent event.

To fill in the blanks:

1)The probability of selecting a white bead, replacing it, and then selecting a red bead is:

P(white, then red) = P(white) * P(red | white) = (3/10) * (5/10) = 0.15

Here, P(white) is the probability of selecting a white bead on the first draw, which is 3/10

And P(red | white) is the probability of selecting a red bead on the second draw, given that a white bead was selected on the first draw.

Since the first bead is replaced before the second draw, the probability of selecting a red bead on the second draw is still 5/10.

2)This is an independent event.

This is because the first draw does not affect the probability of the second draw, since the bead is replaced.

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The populations of the two towns will be equal in the year 2048.

Let's assume that the population of Appleville in the year "t" is "A(t)", and the population of Treehill in the year "t" is "T(t)". We know that the population of Treehill is increasing linearly by 100 per year. Therefore, we can express the population of Treehill in terms of "t" as follows:

T(t) = 41820 + 100t

The population of Appleville is also growing linearly, but we do not know the exact rate of growth. However, we can calculate this rate using the population data we have. The change in population from 2010 to 2020 is 35000 - 28000 = 7000. This change occurred over a period of 10 years. Therefore, the average annual growth rate is:

(35000 - 28000) / 10 = 700

We can use this rate to express the population of Appleville in terms of "t" as follows:

A(t) = 28000 + 700t

Now, we can set these two equations equal to each other to find the year when the populations of the two towns will be equal:

41820 + 100t = 28000 + 700t

Simplifying this equation, we get:

420t = 13820

t = 33

Therefore, the populations of the two towns will be equal in the year 2015 + 33 = 2048.

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Answer:

27 ,200 (1-0.5)7

Step-by-step explanation:

The value of the function 5⋅f(1)+5⋅g(9) is -55

Here the graph of function y = f(x) and function y = g(x) is shoen in attached figure.

We need to find the value of expression 5⋅f(1)+5⋅g(9)      .......(1)

From the graph, the value of function f(x) for x = 1 would be,

y = f(1)

y = -5

From the graph, the value of function g(x) for x = 9 would be,

y = g(9)

y = -6

Substitute these values in expression (1),

5⋅f(1)+5⋅g(9)

= (5 × (-5)) + (5 × (-6))

= -25 - 30

= -55

This is the required value of expression.

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The probability function is a mathematical concept that maps the probability of an event to a certain value.

It is often used to model the likelihood of an event occurring, based on certain conditions or variables. The probability function can take on many different forms, depending on the nature of the problem being studied.

One important tool in probability theory is the counting rule for combinations. This rule allows us to calculate the number of possible combinations of items from a larger set, given certain constraints.

For example, we might want to know how many ways there are to choose three different objects from a set of five, without regard to order. The counting rule for combinations is based on the idea that the order in which objects are chosen doesn't matter.

Therefore, we can calculate the number of combinations by dividing the total number of possible permutations by the number of ways that the objects can be ordered.

This leads to the formula nCr = n/ r * (n-r), where n is the total number of objects, r is the number of objects being chosen, and ! represents the factorial function.

The probability function can make use of the counting rule for combinations in various ways, depending on the nature of the problem being studied.

For example, in the case of a discrete probability distribution, the function might assign probabilities to different outcomes based on the number of combinations that lead to each outcome.

This can be especially useful in situations where the outcomes are not equally likely, or where there are different levels of uncertainty or randomness involved.

Overall, the counting rule for combinations provides an important tool for calculating probabilities in many different contexts. By understanding the relationship between combinations and probability,

we can better understand the underlying structure of many different types of problems in statistics, mathematics, and other fields.

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Standardize every test score from Mr. Bowman's class, the new mean would be 0, the new standard deviation would be 1, and the shape of the distribution would remain the same as before.

Let's first understand what standardizing means.
Standardizing test scores involves transforming the original scores to a new scale, called z-scores.

This process is done by subtracting the mean (average) score from each test score and dividing the result by the standard deviation of the scores. The formula for finding the z-score is:
z = (X - μ) / σ
X is the original test score, μ is the mean, and σ is the standard deviation.
Now, let's discuss the properties of the standardized scores:
New mean:

Standardize the test scores, the new mean (average) of the z-scores will always be 0.

This occurs because we subtract the mean from each test score, making the sum of the differences equal to 0.
New standard deviation:

After standardization, the standard deviation of the z-scores will always be 1.

This is because we divide each score by the original standard deviation, results in the new standard deviation being equal to 1.
Shape of the distribution:

The shape of the distribution (i.e., the pattern of the scores) will not be affected by standardization.

If the original scores followed a bell-shaped curve, the standardized scores would maintain the same bell shape.

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The final cost of filling up the tank with 750 liters of oil at a price of 81.5p per liter with a 7.5% discount is 56,540.62p.

The price of oil is 81.5p per litre. If we want to calculate the cost of filling up the tank, we need to know how many liters of oil we need to add.

To do this, we need to subtract the current amount of oil in the tank (450 liters) from the maximum capacity of the tank (1200 liters):

1200 - 450 = 750 liters

So we need to add 750 liters of oil to fill up the tank.

Next, we need to calculate the cost of adding 750 liters of oil at the price of 81.5p per liter:

750 x 81.5 = 61,125p

To calculate the discount, we first need to find 7.5% of the total cost. We can do this by multiplying the total cost by 7.5% expressed as a decimal:

61,125 x 0.075 = 4,584.38p

So the discount is 4,584.38p.

To find the final cost after the discount, we need to subtract the discount from the total cost:

61,125 - 4,584.38 = 56,540.62p

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For a bridge which is in the shape of a parabolic arch ,Yes a sailboat that is 39 feet tall fit under the bridge 10 feet from the​ center.

What is Parabola?

A parabola in conic section refers to an equation of a curve, such that in which a point on the curve is equidistant from a fixed point, and a fixed line. The fixed point of parabola is called the focus , and the fixed line is called the directrix of the parabola.

The equation of parabola is

(x-h)² = 4a (y-k) -------- (1)

where (h, k) is the vertex of the parabola.

Here (h, k)= (0,40)

Putting the value in equation (1) we get,

(x-0)² = 4a (y-40) ------- (2)

As the paraboloid bridge has a span of 180 feet

So, the ends of the bridge at (±90, 0)

Substituting the point (90, 0) at equation (2) we get,

(90-0)² = 4a (0-40)

⇒ (90)² = - 4a×40

⇒ 8100= - 160a

⇒ a= -(8100/160)

⇒ a= - (405/8)

So the equation (2) can be modified as,

x² = -((4× 405)/8) (y-40)

⇒ x²= (-405/2) (y-40)

When x= 10,

(10)² = (-405/2)(y-40)

⇒ 100 = (-405y+16200)/2

⇒ 200 = -405y+16200

⇒-405y= -16000

⇒y= 16000/405≈ 39.50

Hence, yes a sailboat that is 39 feet tall fit under the bridge 10 feet from the​ center.

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It is not true that if A is an n × n matrix with detA = 2, then detA³ = 6.

det(A³) = det(A × A × A) = det(A) × det(A) × det(A) = (det(A))³ = 2³ = 8

detA³ = 8, not 6.

The determinant of a matrix is a scalar value that encodes various properties of the matrix.

One of the properties is the volume scaling factor that is induced by the matrix transformation.

The determinant has the property that det(kA) = kⁿ × det(A) for any scalar k and n × n matrix A, where n denotes the dimension of the matrix.

Therefore, we have:

det(A³) = det(A × A × A) = det(A) × det(A) × det(A) = (det(A))³ = 2³ = 8

detA³ = 8, not 6.

The determinant of a matrix raised to a power is not simply obtained by raising the determinant to the same power.

The determinant of a matrix raised to a power can be obtained by raising the determinant of the matrix to the power and then multiplying by the scaling factor induced by the matrix transformation.

Specifically, for a matrix A with detA = 2 and an integer k, we have:

[tex]det(A^k) = (det(A))^k \times scaling factor induced by A^k[/tex]

The scaling factor induced by [tex]A^k[/tex] can be computed by considering the effect of [tex]A^k[/tex] on the unit hypercube.

This calculation requires the use of linear algebra and is beyond the scope of this answer.

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We can expect to observe 14 complete interference fringes within the central diffraction peak.

The act of bending light around corners such that it spreads out and illuminates regions where a shadow is anticipated is known as diffraction of light. In general, since both occur simultaneously, it is challenging to distinguish between diffraction and interference.

The number of complete interference fringes within the central diffraction peak depends on the specific experimental setup, such as the distance between the slits and the screen, the distance between the slits, and the wavelength of the light being used.

However, we can use the following equation to calculate the number of interference fringes within the central diffraction peak:

N = (2d/a)(L/λ)

where N is the number of interference fringes, d is the distance between the slits, a is the width of the slits, L is the distance between the slits and the screen, and λ is the wavelength of the light being used.

Assuming that we have a double-slit setup and the distance between the slits and the screen is much larger than the slit width, we can simplify the equation to:

N = 2d/a

Given that d/a = 7, we can substitute this value into the equation to get:

N = 2(7) = 14

Therefore, we can expect to observe 14 complete interference fringes within the central diffraction peak.

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For a sample of 50 customers of last year's customers, the expected number of customers requiring a recall is equals to the 7.5 customers.

The expected value or say mean of a binomial distribution is determined by multiplying the number of trials (n) by the probability of successes (p), that is E(x) = n × p. We have a LaRusso Auto Group of vehicles. Total sales of cars in last year

= 5,000

Number of defective cars in sold cars

= 750

For a random sample of last year's customers, sample size, n= 50

We have to determine the expected number of customers requiring a recall.

Now, the probability of defective cars and customers make a factory recall for dealer correction, p = [tex]\frac{750}{5000} [/tex] = 0.15

So, the sales of cars in Store follows the binomial distribution with parameters n, p. Using expected value or number formula, E(x) = n×p

= 50 × 0.15 = 7.5

Hence, required value is 7.5 customers.

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The variance is calculated by subtract the budgeted production volume from  actual production volume, and multiply that difference by  standard cost. If the result is positive, favorable, and if it is negative, unfavorable.

The difference above the actual and budgeted production volumes, multiplied by the average cost per unit, is known as the production volume variance. The formula for calculating the production volume variance is as follows:

Production Volume Variance = (Actual Production Volume ₋Budgeted Production Volume) × Standard Cost per unit

If the actual production volume is higher than the budgeted production volume, the production volume variance will be favorable because it means that the company produced more than anticipated, which could lead to increased revenue. On the other hand, if the actual production volume is lower than the budgeted production volume, the production volume variance will be unfavorable because it means that the company produced less than anticipated, which could lead to decreased revenue.

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The function has the value of its range decrease as the values in its domain increase is equals to [tex]f(x) = 0.3^ x [/tex]. So, option(B) is right one.

The domain of a function is defined as a set of all possible inputs for the function and range of the function is the set of all values that f takes. For example, the domain of f(x)=x² is all reals and range their corresponding values of f(x).

We have a number of functions and we have to check its range decrease as the values in its domain increase.

A) The function is defined as [tex]f(x) = 3^ x [/tex], As we put values x from reals, x = 0,1,2,3,

=> f(x) = 3⁰, 3¹,3² ,...

so that with increase of input value of x ( domain) the value of range also increase.

B) function is defined, [tex]g(x) = 3.5^ x [/tex]

As we put values x from reals, x = 0,1,2,3,

=> f(x) = 3.5⁰, 3.5,3.5² ,...

so that with increase of input value of x (domain) the value of range also increase.

C) The function is [tex]h(x) =0.3^{x}[/tex]

As we put values x from reals, x = 0,1,2,3,

=> f(x) = 0.3⁰, 0.3,0.3² ,...

=> f(x) = 1, 0.3, 0.09,...

so that with increase of input value of x ( domain) the value of range decrease.

D) The function is [tex]k(x) = \frac{3 ^{x}}{2 } [/tex]

As we put values x from reals, x = 0,1,2,3,

=>[tex] f(x) = \frac{ {3}^{0}}{2} , \frac{3^{1}}{2}....[/tex]

= 0.5, 1.5,...

so that with increase of input value of x ( domain) the value of range is also increases. Hence, required value is

[tex]f(x) = 0.3^ x [/tex]

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The function that has the values of its range decrease as the values in its domain increase is the function (C) h(x) = 0.3^x.

To see why, let's take a look at the

other functions:

(A) f(x) = 3^x: As x increases, 3^x also increases, so the range of f(x) increases as the values in its domain increase.

(B) g(x) = 3.5^x: Similarly to (A), as x increases, 3.5^x also increases, so the range of g(x) increases as the values in its domain increase.(D) k(x) = 1/2(3^x): As x increases, 3^x increases, so 1/2(3^x) also increases, although at a slower rate. Therefore, the range of k(x) increases as the values in its domain increase.

However, for (C) h(x) = 0.3^x: As x increases, 0.3^x decreases, approaching zero. Therefore, the range of h(x) decreases as the values in its domain increase.

Thus, the answer is (C) h(x) = 0.3^x.

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The appropriate statistical technique to compare money spent at the store between two groups in a marketing research study is the independent-samples t-test.

The independent-samples t-test is used to compare the means of two independent groups. In this case, the groups are the ones who received the coupon and the ones who did not, and they are independent because the participants were randomly assigned to each group.

The null hypothesis for the independent-samples t-test is that there is no difference between the means of the two groups. The alternative hypothesis is that there is a significant difference between the means of the two groups.

By conducting an independent-samples t-test on the data, the researcher can determine whether the difference in money spent at the store between the two groups is statistically significant or simply due to chance. If the p-value is less than the significance level (usually 0.05), the researcher can reject the null hypothesis and conclude that there is a significant difference between the two groups in terms of the money spent at the store.

Therefore, the independent-samples t-test is an appropriate statistical technique to compare the means of two independent groups and test for significant differences between them.

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The percent of net change for each corporation are:

The percent of net change is calculated using the formula given below as follows:

a. Berkshire Hathaway:

Percent of net change = (35 / 199740) x 100%

Percent of net change = 0.0175 x 100%

Percent of net change = 1.75%

b. Verizon:

Percent of net change = (0.04 / 51.19) x 100%

Percent of net change = 0.00078 x 100%

Percent of net change = 0.08%

c. McDonalds:

Percent of net change = (-1.13 / 116.34) x 100%

Percent of net change = -0.0097 x 100%

Percent of net change = -0.97%

d. Nike:

Percent of net change = (0.37 / 62.74) x 100%

Percent of net change= 0.0059 x 100%

Percent of net change = 0.59%

e. Delta Airlines:

Percent of net change = (-1.18 / 48.02) x 100%

Percent of net change = -0.0246 x 100%

Percent of net change = -2.46%

f. Toyota:

Percent of net change = (-0.15 / 105.42) x 100%

Percent of net change = -0.0014 x 100%

Percent of net change = -0.14%

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Complete question:

Berkshire Hathaway

Net change = +35

Last = 199740

VZ Verizon

Net change = +0.04

Last = 51.19

MCD McDonalds

Net change = -1.13

Last 116.34

NKE Nike

Net change = +0.37

Last = 62.74

DAL Delta Airlines

Net change = -1.18

Last = 48.02

TM Toyota

Net change = -0.15

Last = 105.42

use the information from exercise 5 to determine the percent of net change from april 21 to april 22 for each of the corporations listed in that question. round answers to the nearest tenth of a percent.

Let h be the height of the shipping container in feet. The volume of a rectangular box can be found by multiplying the length, width, and height of the box. Using this formula, we can set up an equation to solve for h:

8 x 40 x h = 3,040

Simplifying this equation, we get:

320h = 3,040

Dividing both sides by 320, we get:

h = 3,040 / 320

h = 9.5

Therefore, the height of the shipping container is 9.5 feet.

Since X is a compact metric space, it is complete and totally bounded. Therefore, by the Arzelà-Ascoli theorem, it suffices to show that F is uniformly bounded and equicontinuous.

To show that F is uniformly bounded, let M be a positive number such that |f(x)| ≤ M for all x ∈ X and f ∈ F. Since F is compact, there exist finitely many functions f1, f2, ..., fn ∈ F such that for every f ∈ F, there exists i ∈ {1, 2, ..., n} such that ||f - fi|| < ϵ/3, where ||·|| denotes the supremum norm on C(X). Then, for any x ∈ X, we have

|f(x)| ≤ |f(x) - fi(x)| + |fi(x)| + |fi(x)| - M ≤ ||f - fi|| + |fi(x)| + M ≤ ϵ/3 + M + ϵ/3 = 2ϵ/3 + M.

Therefore, F is uniformly bounded by 2ϵ/3 + M, which does not depend on the choice of f and ϵ.

To show that F is equicontinuous, let ϵ > 0 be arbitrary. For each x ∈ X, there exists δx > 0 such that |f(x) - f(y)| < ϵ/3 for all f ∈ F and y ∈ X with |x - y| < δx, since F is compact and therefore uniformly continuous. Since X is compact, there exists a finite cover {B(x1, δx1/2), B(x2, δx2/2), ..., B(xn, δxn/2)} of X, where B(x, r) denotes the open ball centered at x with radius r. Let δ = min{δx/2 : 1 ≤ i ≤ n}. Then, for any x, y ∈ X with |x - y| < δ, there exists i ∈ {1, 2, ..., n} such that x, y ∈ B(xi, δxi/2), so |f(x) - f(y)| < ϵ/3 for all f ∈ F. Moreover, since |x - y| < δxi/2, we have |f(x) - f(y)| < ϵ/3 for all f ∈ F and x, y ∈ B(xi, δxi/2). Therefore, for any x, y ∈ X with |x - y| < δ, we have

|f(x) - f(y)| ≤ |f(x) - f(xi)| + |f(xi) - f(y)| < ϵ/3 + ϵ/3 = 2ϵ/3.

Thus, F is equicontinuous with respect to δ, which does not depend on the choice of f and ϵ.

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the randomly generated integer between 0 and 100 (inclusive). If there is a tie, the winner will be randomly selected from the tied submissions. What is the probability that a person who buys a ticket with a single submission will win the lottery

The probability that a person will win depends on the number of people who buy tickets and submit their integers. Assuming there are n participants, each person's submission has a 1/n chance of being selected as the winner.

Since there are 101 possible integers that can be generated, the probability that any given person's submission is exactly correct is 1/101. Therefore, the probability that a person who buys a ticket with a single submission will win the lottery is 1/n * 1/101.

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The probability that a student lives within 3 miles of the school is 75%, and the probability that a student who lives within 3 miles walks to school is 20%. We can find the probability that a student both lives within 3 miles and walks to school by multiplying these probabilities:

0.75 x 0.20 = 0.15

Therefore, the probability that a student lives within 3 miles and walks to school is 0.15 or 15%.

Answer:

0.15  or   15%

Step-by-step explanation:

computing a confidence interval for the mean of the differences in paired data.

Compute the differences between the two measurements for each pair of data points.

Calculate the mean and standard deviation of the differences.

Compute the standard error of the mean of the differences by dividing the standard deviation of the differences by the square root of the sample size.

Determine the appropriate confidence level and degrees of freedom based on the sample size and type of test being conducted.

Use a t-distribution to find the t-value associated with the desired confidence level and degrees of freedom.

Compute the confidence interval by adding and subtracting the product of the t-value and the standard error of the mean of the differences from the sample mean of the differences.

Regarding the conclusion, if the confidence interval does not include zero, it means that there is a statistically significant difference between the two devices, and the biologist's claim is supported at the chosen level of significance. If the confidence interval includes zero, it means that there is no statistically significant difference between the two devices, and the biologist's claim is rejected at the chosen level of significance.

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Beth had traveled 3 miles from the school. They are traveling at right angles to each other, forming a right triangle.

Let's follow these steps:

1. Sarah is heading due north, and Beth is heading due east. They are traveling at right angles to each other, forming a right triangle.
2. One hour later, they are 5 miles apart. This distance represents the hypotenuse of the right triangle.
3. We are given that Sarah has traveled 4 miles from the school. This distance represents one side of the right triangle (north side).
4. We need to find the distance Beth traveled, which represents the other side of the right triangle (east side).

We can use the Pythagorean theorem to solve this problem:

a² + b² = c²

where a and b are the lengths of the two shorter sides (Sarah and Beth's distances), and c is the length of the hypotenuse (the distance between them).

In this problem, we have:

a = 4 miles (Sarah's distance)
c = 5 miles (distance between them)

We need to find b (Beth's distance). So, we can rewrite the Pythagorean theorem as:

b² = c² - a²

Now, plug in the given values:

b² = 5² - 4²
b² = 25 - 16
b² = 9

To find b, take the square root of both sides:

b = √9
b = 3

So, Beth had traveled 3 miles from the school.

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The score in the 4th test is 80 and the median of the hours worked is 6

Here, we have

Scores = 92, 88, 76

Mean = 84

So, we have

mean = sum/count

This gives

(92 + 88 + 76 + score)/4 = 84

Evaluate

score = 4 * 84 - (92 + 88 + 76)

So, we have

score = 80

Here, we have

Hours = 8, 6, 8, 6, 4

Sort in ascending order

So, we have

Hours = 4, 6, 6, 8, 8

The median of the hours worked is the middle value

So, we have

median = 6

hence, the median is 6

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The expected number of people who will drop out due to either relapse or moving away is 10 + 6 = 16. However, taking into account the standard deviations, it is 16 +/- 5.

To find the expected number of people who will drop out due to either relapse or moving away, we need to add the expected number of people who will drop out due to relapse (10) and the expected number of people who will drop out due to moving away (6).

Expected number of people who will drop out due to either relapse or moving away = 10 + 6 = 16.

However, we also need to take into account the standard deviations for each of these groups. To do this, we can use the square root of the sum of the variances (SD squared) for each group, squared.

Variances:
- Relapse: 4 squared = 16
- Moving away: 3 squared = 9

Square root of the sum of the variances:
- sqrt(16 + 9) = 5

Therefore, the expected number of people who will drop out due to either relapse or moving away, taking into account the standard deviations, is 16 +/- 5.

This means that we can expect anywhere between 11 and 21 people to drop out due to either relapse or moving away during the two-year cohort study.

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