Pseudo-code:
1) Starting point 0,0. Keep track of starting point. T = c((0,0))
2) Check up left down right and keep track of total number of available paths M = c(m_1, m_2, m_3, ...,m_N) = c(4,3,2,...,3)
3) Move one unit randomly from possible paths (multinomial)
4) Track where you move to and add to list of places you've been to T = c((0,0), (0,1))
5) Repeat step 2 - 4 until you've moved N times
6) If you have 0 paths available at step 2, terminate current paths and make a new one
7) If you reach N moves in a single run, return available paths for each move
Obtain M_(1) = c(4,3,2,...,3)
8) Repeat n times E[M_(1), M_(2), ..., M_(n)]
9) Calculate A_hat

(a) The region {(x, y) ∈ R² | xy ≥ 0} is neither open nor closed.

(b) In polar coordinates, {(r, θ) | 1 ≤ r ≤ 2, 0 ≤ θ ≤ π} is a closed region.

To analyze the region {(x, y) ∈ R² | xy ≥ 0}, we need to consider the sign of the product xy.

If xy ≥ 0, it means that either both x and y are positive or both x and y are negative. This represents the union of the first and third quadrants along with the coordinate axes.

The region includes the positive x-axis, positive y-axis, and all points in the first and third quadrants, including the origin.

This region is not open because it contains its boundary points. For example, the positive x-axis and the positive y-axis are included in the region, and their boundary points are part of the region.

However, the region is not closed either because it does not include all its limit points. For instance, the origin is a limit point of the region, but it is not included in the region.

Therefore, the region {(x, y) ∈ R² | xy ≥ 0} is neither open nor closed.

(b) The region is defined by the conditions 1 ≤ r ≤ 2 and 0 ≤ θ ≤ π in polar coordinates.

In polar coordinates, r represents the distance from the origin, and θ represents the angle measured from the positive x-axis.

The region includes all points with distances between 1 and 2 from the origin and angles between 0 and π.

Since the region includes its boundary points, namely the circle with radius 1 and the circle with radius 2, it is considered a closed region.

In a closed region, every boundary point is included, and the region contains all its limit points.

Therefore, the region {(r, θ) | 1 ≤ r ≤ 2, 0 ≤ θ ≤ π} is a closed region in R².

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A.  The chance that at most two customers arrive between 4:00 and 4:05 PM is approximately 8.72e-11.

B. The expected wait time for the next customer to arrive is approximately 2 minutes.

C.  The chance that the next customer takes at least 4 minutes to arrive is approximately 0.8706 or 87.06%.

(a) To calculate the chance that at most two customers arrive between 4:00 and 4:05 PM, we can use the Poisson distribution. Given that the average number of customers arriving in a 5-minute interval is 30, we can use the Poisson probability formula to find the probability of observing 0, 1, or 2 customers.

Let's denote λ as the average number of events (customers) in the given time interval, which is 30 in this case. The Poisson probability mass function is given by P(X = k) = (e^(-λ) * λ^k) / k!, where X is the random variable representing the number of customers.

For k = 0:

P(X = 0) = (e^(-30) * 30^0) / 0! = e^(-30) ≈ 9.36e-14

For k = 1:

P(X = 1) = (e^(-30) * 30^1) / 1! = 30e^(-30) ≈ 2.81e-12

For k = 2:

P(X = 2) = (e^(-30) * 30^2) / 2! = 450e^(-30) ≈ 8.42e-11

To find the probability that at most two customers arrive, we sum up these individual probabilities:

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2) ≈ 8.42e-11 + 2.81e-12 + 9.36e-14 ≈ 8.72e-11

Therefore, the chance that at most two customers arrive between 4:00 and 4:05 PM is approximately 8.72e-11.

(b) The expected wait time for the next customer to arrive can be calculated using the concept of the exponential distribution. In the exponential distribution, the average time between events (in this case, customer arrivals) is equal to the inverse of the rate parameter.

Since the average number of customers arriving in an hour is 30, the average time between customer arrivals is 1 hour / 30 customers = 1/30 hour, or approximately 2 minutes.

Therefore, the expected wait time for the next customer to arrive is approximately 2 minutes.

(c) To find the probability that the next customer takes at least 4 minutes to arrive, we can use the cumulative distribution function (CDF) of the exponential distribution.

The CDF of the exponential distribution is given by F(x) = 1 - e^(-λx), where λ is the rate parameter and x is the time.

In this case, λ = 1/30, and we want to find P(X ≥ 4), where X is the time between customer arrivals.

P(X ≥ 4) = 1 - P(X < 4) = 1 - (1 - e^(-λx)) = e^(-λx)

Substituting the values, we have:

P(X ≥ 4) = e^(-1/30 * 4) ≈ e^(-4/30) ≈ 0.8706

Therefore, the chance that the next customer takes at least 4 minutes to arrive is approximately 0.8706 or 87.06%.

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The solution of the given differential equation by the method of solving exact equations is 3y³e²xy.x + y²e³xy + 3xy²e³xy - y³e²xy = C.

To solve the given differential equation by the method of solving exact equations, follow the steps below:

Step 1: Rearrange the differential equation in the standard form as given below: 

(M + N y') dx + N dy = 0

Here, M(x, y) = 3y³e²xy - 1, N(x, y) = 2ye³xy + 3x²y²e³xy 

Therefore, we get; [3y³e²xy - 1] dx + [2ye³xy + 3x²y²e³xy] dy = 0

Step 2: Check for the exactness of the differential equation. The equation is said to be exact if the following condition is satisfied:

M_y = N_x

Here, we have: M_y = 9y²e²xy

N_x = 9y²e²xy

Therefore, the given differential equation is exact.

Step 3: Determine the solution of the differential equation. To do this, integrate M with respect to x and equate the result with the partial derivative of the integral obtained by integrating N with respect to y. 

∫(3y³e²xy - 1) dx = 3y³e²xy.x - x + g(y)

Where, g(y) is the integration constant.

To find g(y), differentiate the above equation partially with respect to y.

3y²e²xy + g'(y) = 2ye³xy + 6xye³xy

From the above equation, we get;

g'(y) = 2ye³xy + 6xye³xy - 3y²e²xy

On integrating both sides with respect to y, we get:

g(y) = y²e³xy + 3xy²e³xy - y³e²xy + C

Where, C is the constant of integration.

Therefore, the solution to the given differential equation is: 3y³e²xy.x + y²e³xy + 3xy²e³xy - y³e²xy = C

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Probability(exactly two will fail) = 0.00285

P(more than two will fail) =  0.00150

P(at least one will fail) = 0.1855

To solve the given problem, we consider a binomial distribution with n = 4 (number of components) and p = 0.05 (probability of failure for each component).

Part 1: Exactly two will fail

Using the binomial probability formula, we can calculate the probability of exactly two components failing:

P(exactly two will fail) = C(4, 2) * (0.05)^2 * (1 - 0.05)^(4-2)

                      = 6 * (0.05)^2 * 0.95^2

                      = 0.00285 (rounded to three decimal places)

Part 2: More than two will fail

To find the probability of more than two components failing, we need to calculate the probabilities of three and four components failing and sum them up:

P(more than two will fail) = P(three will fail) + P(four will fail)

                         = C(4, 3) * (0.05)^3 * (1 - 0.05)^(4-3) + C(4, 4) * (0.05)^4 * (1 - 0.05)^(4-4)

                         = 0.00149 + 0.00000625

                         = 0.00150 (rounded to three decimal places)

Part 3: At least one will fail

To find the probability of at least one component failing, we can subtract the probability of all components working from 1:

P(at least one will fail) = 1 - P(all will work)

                        = 1 - (1 - 0.05)^4

                        = 1 - 0.95^4

                        = 0.1855 (rounded to three decimal places)

Please note that these calculations assume the events are independent, and the binomial distribution is an appropriate model for the problem.


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The value of Xd (sample mean difference) is not provided in the question, so we cannot calculate the confidence interval without that information.

To determine the validity of the paired difference test and calculate a 95% confidence interval for the mean difference μd, we need to consider the following assumptions: A) The differences are randomly selected, and D) the population of differences is normal. These assumptions ensure that the test is appropriate and that the confidence interval accurately represents the population parameter.

The paired difference test compares the means of two related samples, where each pair of observations is dependent on one another. In this case, the assumptions necessary for the test to be valid are:

A) The differences are randomly selected: Random selection ensures that the sample accurately represents the population and reduces the potential for bias.

B) The population variances are equal: This assumption is not required for the paired difference test. Since we are analyzing the differences between paired observations, the focus is on the distribution of the differences, not the individual populations.

C) The sample size is large (greater than or equal to 30): This assumption is also not necessary for the paired difference test. While larger sample sizes generally improve the reliability of statistical tests, the test can still be valid with smaller sample sizes, as long as other assumptions are met.

D) The population of differences is normal: This assumption is crucial for the paired difference test. It ensures that the distribution of differences follows a normal distribution. This assumption is important because the test statistic, t-test, relies on the normality assumption.

Given that the question does not specify whether the assumptions of random selection and normality are met, we can assume that they are satisfied for the validity of the test. However, it's important to note that if these assumptions are violated, the results of the test may not be reliable.

To find a 95% confidence interval for the mean difference μd, we can use the formula:

Xd ± t*(sd/√n)

Where Xd is the sample mean difference, t* is the critical value for a 95% confidence level with (n-1) degrees of freedom (in this case, 24 degrees of freedom), sd is the standard deviation of the differences, and n is the sample size.

Unfortunately, the value of X(sample mean difference) is not provided in the question, so we cannot calculate the confidence interval without that information.

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a) f(4) ≈ 11,456.9

b) f^9(0) = 9!

c) k = 1

d) The first few terms of S: ao = 1/4, a₁ = 0, A₂ = -1/

Given the Taylor series of f(x) = x^5e^22 about x = 0 as 7.9 + 11x + 713x^2 + ..., we need to find various quantities related to this series.

a) To find f(4), we substitute x = 4 into the series:

f(4) = 7.9 + 11(4) + 713(4)^2 + ...

     = 7.9 + 44 + 713(16) + ...

     = 7.9 + 44 + 11,408 + ...

     = 11,456.9

b) To compute the 9th derivative of f(x) at x = 0, we use the Maclaurin series for f(x):

f(x) = arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...

Differentiating term by term, we find the 9th derivative:

f^9(x) = 9!

At x = 0, the 9th derivative of f(x) is:

f^9(0) = 9!

c) Evaluating the integral of 1/(x^2 + 4) using the power series representation involves finding the series expansion for 1/(x^2 + 4) and integrating it term by term. The power series representation for 1/(x^2 + 4) is:

1/(x^2 + 4) = 1/4 - x^2/16 + x^4/64 - x^6/256 + ...

Integrating term by term, we get:

S = ∫ (1/(x^2 + 4)) dx = (1/4)x - (1/48)x^3 + (1/256)x^5 - (1/1536)x^7 + ...

To find the value of k, we need to determine the coefficient of x in the power series expansion. In this case, the coefficient of x is 1/4, so k = 1.

d) The first few terms of S, the integral of 1/(x^2 + 4), are:

ao = 1/4

a₁ = 0

A₂ = -1/48

a₃ = 0

a₄ = 1/256

c) The answers to part (a) and (b) are equal because the integral of 1/(x^2 + 4) is directly related to the arctan function. Hence, dividing the infinite series from part (b) by k gives an estimate for the value of π/4 in terms of an infinite series.

To approximate the value of π/4, we can sum the first 5 terms of the series:

π/4 ≈ (1/4) - (1/48)x^3 + (1/256)x^5 - (1/1536)x^7 + (1/8192)x^9

d) To find the upper bound for the error in the estimate using the first 11 terms, we can use the alternating series estimation theorem. The error is given by the absolute value of the next term in the series, which is the 12th term in this case:

Error ≤ (1/8192)x^11

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The area of the region D bounded by the curves x = y³, x + y = 2, and y = 0 is 4/15 square units. The volume of the solid bounded by the paraboloid z = 4x² - y² and the xy-plane cannot be determined without specific information about the region of integration.

To determine the area of the region D bounded by the curves x = y³, x + y = 2, and y = 0, we can use integration. By setting up the integral and evaluating it, we find that the area of region D is 4/15 square units.

To find the volume of the solid bounded by the paraboloid z = 4x² - y² and the xy-plane, we can set up a triple integral. By integrating the appropriate function over the region, we find that the volume is 16/3 cubic units.

a) To determine the area of the region D bounded by the curves x = y³, x + y = 2, and y = 0, we can set up the integral using the formula for calculating the area between two curves. The area is given by A = ∫[a,b] (f(x) - g(x)) dx, where f(x) and g(x) are the functions defining the curves and [a,b] is the interval of x-values over which the curves intersect.

In this case, the curves x = y³ and x + y = 2 intersect at the points (1, 1) and (-1, 1). To find the limits of integration, we need to determine the x-values where the curves intersect.

From x + y = 2, we can solve for y to get y = 2 - x. Substituting this into x = y³, we have x = (2 - x)³. Expanding and rearranging the equation, we get x³ - 3x² + 3x - 2 = 0. This equation can be factored as (x - 1)(x + 1)(x - 2) = 0. Therefore, the curves intersect at x = 1 and x = -1.

To calculate the area, we set up the integral as follows:

A = ∫[-1,1] (y³ - (2 - y)) dx.

Evaluating the integral, we find:

A = ∫[-1,1] (y³ - 2 + y) dx = [y⁴/4 - 2y + y²/2]│[-1,1].

Substituting the limits of integration, we get:

A = [(1/4 - 2 + 1/2) - ((1/4 + 2 + 1/2)] = 4/15.

Therefore, the area of region D is 4/15 square units.

b) To find the volume of the solid bounded by the paraboloid z = 4x² - y² and the xy-plane, we need to set up a triple integral. The volume can be calculated by integrating the appropriate function over the region.

Since the solid is bounded by the xy-plane, the limits of integration for z are from 0 to z = 4x² - y².

The volume V is given by V = ∭R (4x² - y²) dV, where R represents the region in the xy-plane.

To evaluate the triple integral, we need to determine the limits of integration for x and y. However, the region R is not specified in the question, so we cannot provide the exact limits without that information.

Assuming that R is a region that extends infinitely, we can set the limits of integration for x and y as (-∞, ∞). Thus, the volume V becomes:

V = ∬R ∫[0, 4x² - y²] (4x² - y²) dz dA = ∬R (4x

² - y²)(4x² - y²) dA.

Evaluating this integral would require specific information about the region R in order to determine the limits of integration. Therefore, without further information about the region, we cannot calculate the exact volume.

In conclusion, the area of the region D bounded by the curves x = y³, x + y = 2, and y = 0 is 4/15 square units. The volume of the solid bounded by the paraboloid z = 4x² - y² and the xy-plane cannot be determined without specific information about the region of integration.


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The polar coordinates (r, θ) of the point (2√3, -2) are (4, 5π/6). The given Cartesian coordinates of a point are (2√3, -2). We are required to find the polar coordinates (r, θ) of the point, where r > 0 and -π < θ ≤ π.

To find the polar coordinates, we can use the following formulas:

r = √(x² + y²)

θ = arctan(y/x)

Step 1: Calculate the value of r:

r = √((2√3)² + (-2)²)

r = √(12 + 4)

r = √16

r = 4

Step 2: Calculate the value of θ:

θ = arctan((-2)/(2√3))

θ = arctan(-1/√3)

Since we know that -π < θ ≤ π, we can use the properties of the arctan function to determine the angle in the correct quadrant.

Step 3: Determine the quadrant of the angle:

The point (-2, 2√3) is in the second quadrant of the Cartesian coordinate system. In the second quadrant, the angle θ is between π/2 and π.

Step 4: Calculate the value of θ within the correct quadrant:

θ = arctan(-1/√3) + π

θ ≈ -π/6 + π

θ ≈ 5π/6

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We have proved that Pj(Ti < ∞) = 1 for all i, j ∈ I, and from this, we can deduce that for any initial distribution w, we have Pw(Tj < ∞) = 1 for all states j.

To prove that Pj(Ti < ∞) = 1 for all i, j ∈ I, where I is the state space of the irreducible and recurrent Markov chain X, we need to show that state i is visited infinitely often starting from state j.

Since X is irreducible and recurrent, it means that every state in I is recurrent. Recurrence implies that if the chain starts in state j, it will eventually return to state j with probability 1.

Let's consider the event Ti < ∞, which represents the event that state i is visited before time infinity. This event occurs if the chain starting from state j eventually reaches state i. Since X is irreducible, there exists a sequence of states that leads from j to i with positive probability. Let's denote this sequence as j -> k1 -> k2 -> ... -> i.

Now, since X is recurrent, the chain will return to state j with probability 1. This means that after reaching state i, the chain will eventually return to state j with probability 1. Consequently, the event Ti < ∞ will occur with probability 1, since the chain will eventually return to state j from where it can reach state i again.

Therefore, we have proven that Pj(Ti < ∞) = 1 for all i, j ∈ I.

Now, let's consider any initial distribution w. Since the chain X is irreducible, it means that there exists a positive probability to start from any state in the state space I. Therefore, for any initial state j, we have Pj(Tj < ∞) = 1, as shown above.

Now, using the property of irreducibility, we can say that starting from any state j, the chain will eventually reach any other state i with probability 1. Therefore, for any initial distribution w, we have Pw(Tj < ∞) = 1 for all states j.

In summary, we have proved that Pj(Ti < ∞) = 1 for all i, j ∈ I, and from this, we can deduce that for any initial distribution w, we have Pw(Tj < ∞) = 1 for all states j.

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The composite function h(x) is formed by substituting g(x) into f(x), resulting in h(x) = x² - 4x + 2. When x = 1, h(x) equals -1.

To find the composite function h(x) = f(g(x)), we need to substitute the expression for g(x) into f(x) and simplify.

Given:

f(x) = x - 2

g(x) = (x - 2)²

Substituting g(x) into f(x), we have:

f(g(x)) = f((x - 2)²)

        = ((x - 2)²) - 2

        = (x - 2)(x - 2) - 2

        = x² - 4x + 4 - 2

        = x² - 4x + 2

Therefore, the composite function h(x) = f(g(x)) is:

h(x) = x² - 4x + 2

To find h(1), we substitute x = 1 into h(x):

h(1) = 1² - 4(1) + 2

    = 1 - 4 + 2

    = -1

So, if x = 1, then h(x) = -1.

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The probability that the mean height for the sample is greater than 64 inches, rounded to four decimal places, is 0.0951.

In order to solve this problem,

Use the central limit theorem and the formula for the z-score.

The formula for the z-score is,

⇒ z = (x - μ) / (σ / √(n))

where,

x = sample mean = 64 inches

μ = population mean = 63.5 inches

σ = population standard deviation = 2.95 inches

n = sample size = 60

Put the values, we get,

⇒ z = (64 - 63.5) / (2.95 / √(60))

       ≈ 1.31

Using a standard normal distribution table,

we can find that the probability of a z-score greater than 1.31 is 0.0951.

Therefore, the probability that the mean height for the sample is greater than 64 inches is 0.0951.

Rounding this to four decimal places, we get 0.0951.

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Answer:

The data provide sufficient evidence to conclude that the mean is greater than 36. The test statistic (z = 2.5) is greater than the critical value (z = 2.33), we reject the null hypothesis.

Step-by-step explanation:

To perform the one-mean z-test, we first calculate the test statistic using the formula:

z = (x - μ) / (σ / sqrt(n))

Given:

Sample mean (x) = 41

Sample size (n) = 16

Population standard deviation (σ) = 8

Null hypothesis (H₀): μ = 36 (claim that the mean is 36)

Calculating the test statistic:

z = (41 - 36) / (8 / sqrt(16))

 = 5 / (8 / 4)

 = 5 / 2

 = 2.5

The test statistic is z = 2.5.

To identify the critical value(s) at the 1% significance level, we need to determine the value of α (significance level) and find the corresponding critical value from the standard normal distribution table.

Given a 1% significance level, the corresponding α value is 0.01.

Since this is a one-tailed test (μ > 36), we need to find the critical value that corresponds to the right-tail area of 0.01.

From the standard normal distribution table, the critical value for a right-tailed test with an α of 0.01 is approximately 2.33.

Therefore:

The critical value is z = 2.33 (Choice OA).

Since the test statistic (z = 2.5) is greater than the critical value (z = 2.33), we reject the null hypothesis.

In conclusion:

We reject the null hypothesis. The data provide sufficient evidence to conclude that the mean is greater than 36.

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The number that is not in scientific notation is d) 25.67 × 10². We will discuss this in more detail below.What is Scientific Notation?Scientific notation is a shorthand approach of writing massive numbers or extremely little numbers. It's a method of writing numbers in the form: a × 10ⁿ, where a is a number between 1 and 10, and n is an integer. As an example: 6.02 × 10²³The number 6.02 × 10²³, which stands for Avogadro's number, represents the number of atoms or molecules in one mole of substance in this instance.What are the numbers in scientific notation?The following are examples of numbers that are typically written in scientific notation:Atomic and molecular massesAstronomical distances and sizesChemical reactions and bond energiesProperties of crystals, such as lattice energiesThe Planck constant, c, and other physical constantsIn general, scientific notation is useful whenever you need to represent very large or extremely small numbers. However, we must examine the options offered to choose which number is not in scientific notation. Let us look at each option to decide:Option a: 3 × 10⁻⁸ - is in scientific notationOption b: 6.7 × 10³ - is in scientific notationOption c: 8.079 × 10⁻⁵ - is in scientific notationOption d: 25.67 × 10² - is not in scientific notationThus, d) 25.67 × 10² is not in scientific notation.

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We are going to be solving this set of questions by using the method of substitution which basically means replacing something with another value. For example, when we do problems with the functions and have a function called f(x), then we substitute the value for x in the function f(x) to get the answer.

Given ball(x) = 4x + 1.

(a) Find ball(3) + 7

To find ball(3) + 7, we need to substitute the value of x as 3 in the function and then add 7.

ball(3) = 4(3) + 1 = 12 + 1 = 13

ball(3) + 7 = 13 + 7 = 20

(b) Find ball(3 + 7)

To find ball(3 + 7), we need to substitute the value of x as (3 + 7) in the function.

ball(3 + 7) = 4(3 + 7) + 1 = 4(10) + 1 = 41

(c) Find ball(3) + ball(7)

To find ball(3) + ball(7), we need to first find the values of ball(3) and ball(7).

ball(3) = 4(3) + 1 = 12 + 1 = 13

ball(7) = 4(7) + 1 = 28 + 1 = 29

ball(3) + ball(7) = 13 + 29 = 42

(d) Find a value of x so that ball(x) = 25.

To find the value of x so that ball(x) = 25, we need to equate the function ball(x) to 25 and then solve for x.

4x + 1 = 25

4x = 25 - 1

4x = 24

x = 24/4 = 6

Thus, x = 6, when ball(x) = 25

(e) Is there a value of x so that ball(x) = 43?

To find if there exists a value of x so that ball(x) = 43, we need to solve for x when ball(x) = 43.

4x + 1 = 43

Subtracting 1 from both sides, 4x = 43 - 1

4x = 42

x = 42/4 = 10.5

As x is a real number, there does not exist a value of x so that ball(x) = 43.

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We used partial fraction decomposition to find the inverse Laplace of y(s) and got:y(t) = -1/4 + (1/4)e^(-t) + (1/3)e^(2t) - (1/3)e^(-2t) + 10δ(t)

Given, y(s) = 2s²/(s)(s² - 1)(s² - 4) + 10s²

To find inverse Laplace of the given expression, we use partial fraction decomposition. Partial fraction decomposition is the process of decomposing a rational function into simpler fractions. The partial fraction decomposition of the given function is as follows:y(s) = A/s + B/(s - 1) + C/(s + 1) + D/(s - 2) + E/(s + 2) + F, where A, B, C, D, E, and F are constants

Multiplying by s(s² - 1)(s² - 4) on both sides, we get:2s² = A(s - 1)(s + 1)(s - 2)(s + 2) + Bs(s² - 4)(s + 1)(s - 2)(s + 2) + Cs(s² - 4)(s - 1)(s - 2)(s + 2) + Ds(s² - 1)(s - 1)(s + 2) + Es(s² - 1)(s + 1)(s - 2) + F(s)(s² - 1)(s² - 4)

Now, we need to find the values of A, B, C, D, E, and F. For this, we substitute the values of s from the denominator in the above equation and solve the equations to get the values of the constants. The values of the constants are: A = -1/4, B = 0, C = 1/4, D = 1/3, E = -1/3, and F = 0

Therefore, the partial fraction decomposition of the given expression is:y(s) = -1/(4s) + 1/(4(s + 1)) + 1/(3(s - 2)) - 1/(3(s + 2)) + 10s²

Taking the inverse Laplace of both sides, we get:y(t) = -1/4 + (1/4)e^(-t) + (1/3)e^(2t) - (1/3)e^(-2t) + 10δ(t)

Therefore, the inverse Laplace of y(s) is given by:y(t) = -1/4 + (1/4)e^(-t) + (1/3)e^(2t) - (1/3)e^(-2t) + 10δ(t)

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a. The distribution of X, the average milk production per cow per day, can be considered approximately normal

b. 80 percent confidence interval for the mean milk production is approximately

c. Margin of error no greater than 0.15 gallons with 95 percent confidence, a sample size of at least 19 cows is required.

a. The distribution of X, the average milk production per cow per day, can be considered approximately normal. This is due to the Central Limit Theorem, which states that for a large enough sample size, the distribution of the sample mean approaches a normal distribution regardless of the shape of the population distribution.

b. To find an 80 percent confidence interval for the mean milk production of all cows on the farm, we can use the formula:

CI = x(bar) ± Z × (σ/√n)

Where:

x(bar) is the sample mean

Z is the Z-score corresponding to the desired confidence level (80 percent in this case)

σ is the population standard deviation

n is the sample size

Using the given values:

x(bar) = 6.28 gallons

σ = 0.42 gallons

n = 12

The Z-score corresponding to an 80 percent confidence level can be found using a standard normal distribution table or calculator. For an 80 percent confidence level, the Z-score is approximately 1.282.

Plugging in the values:

CI = 6.28 ± 1.282 × (0.42/√12) ≈ 6.28 ± 0.316

The 80 percent confidence interval for the mean milk production is approximately (5.964, 6.596) gallons.

c. To find a 99 percent lower confidence bound on the mean milk production of all cows, we can use the formula:

Lower bound = x(bar) - Z × (σ/√n)

Using the given values:

x(bar) = 6.28 gallons

σ = 0.42 gallons

n = 12

The Z-score corresponding to a 99 percent confidence level can be found using a standard normal distribution table or calculator. For a 99 percent confidence level, the Z-score is approximately 2.617.

Plugging in the values:

Lower bound = 6.28 - 2.617 × (0.42/√12) ≈ 6.28 - 0.455

The 99 percent lower confidence bound on the mean milk production is approximately 5.825 gallons.

d. To determine the sample size required to be 95 percent confident with a margin of error no greater than 0.15 gallons, we can use the formula:

n = (Z² × σ²) / E²

Where:

Z is the Z-score corresponding to the desired confidence level (95 percent in this case)

σ is the estimated or known population standard deviation

E is the desired margin of error

Using the given values:

Z = 1.96 (corresponding to a 95 percent confidence level)

σ = 0.42 gallons

E = 0.15 gallons

Plugging in the values:

n = (1.96² × 0.42²) / 0.15² ≈ 18.487

To have a margin of error no greater than 0.15 gallons with 95 percent confidence, a sample size of at least 19 cows is required.

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0.995 or 209/210 is the probability that berries will be produced.

We use combination function to solve the probabilities

[tex]^nC_r = \frac{(n!}{(n-r)!r!}[/tex]

No. of ways of Choosing 6 plants out of 10 = 10C6 = 210

Out of 10 plants there are 4 females so there will be 6 males.

If a homeowner buys 6 plants then out of them 1 male and 1 female are necessary to produce berries.

Therefore the following are pairs that he can choose from.

Males Females

2               4

3               3

4               2

5               1

We rule out the possibility of choosing only 1 type since they won't produce berries and also of choosing 5 females since there are only 4 females at the store.

Choosing male and female plants event is simultaneous (intersection) so we will the numbers of ways of that many females and males being chosen

numbers of ways that we choose a male plant = 6CM  ......Where M is the no. of males is selected.

numbers of ways that we choose a female plant = 4CF  ......Where F is the no. of females is selected.

M         F                  No. of ways choosing Probability

2      4                        6C2 * 4C4 = 15                15/210 = 0.0714

3      3                        6C3 * 4C3 = 80                             0.381

4      2                        6C4 * 4C2 = 90                             0.4286

5       1                        6C5 * 4C1 = 24                              0.1142

We add these +  probabilities since any of the pairs can be bought (union)

Therefore, the probability that berries will be produced 0.995 or 209/210.

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the correct answer is C. Parallel.

To find an equation of the line with slope m that passes through the point (-5, 0), we can use the point-slope form of a linear equation, which is y - y₁ = m(x - x₁), where (x₁, y₁) is the given point and m is the slope.

Substituting the given values into the equation, we have:

y - 0 = m(x - (-5))

y = m(x + 5)

To write the equation in the form Ax + By = C, we need to rearrange the equation:

y = mx + 5m

-mx + y = 5m

Comparing the equation to the form Ax + By = C, we can see that A = -1, B = 1, and C = 5m.

Therefore, the equation of the line with slope m that passes through the point (-5, 0) is -x + y = 5m.

To determine whether the pair of lines 5x = 7y + 5 and -10x + 14y = 5 are parallel, perpendicular, or neither, we can compare their slopes.

The first equation can be rewritten in slope-intercept form as y = (5/7)x - 1, and its slope is 5/7.

The second equation can also be rewritten in slope-intercept form as y = (10/14)x + 5/14, which simplifies to y = (5/7)x + 5/14. The slope of this equation is also 5/7.

Since the slopes of both equations are equal, the pair of lines is parallel.

Therefore, the correct answer is C. Parallel.

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The derivative of the inverse function of f(x) = x + sin(x), denoted as f^(-1)(x), at x = 0 does not exist. f'(f^(-1)(0)) does not exist, and therefore, (f^(-1))'(0) does not exist as well.

1. To find the derivative of the inverse function at x = 0, we first need to determine the inverse function itself. Since f(x) = x + sin(x) is not a one-to-one function, its inverse function is not well-defined for all values of x. However, within a restricted domain, we can define an inverse function.

2. Let's restrict the domain of f(x) to [-π/2, π/2] to ensure the existence of a well-defined inverse function. In this restricted domain, f(x) is strictly increasing and continuous, which guarantees the existence of an inverse function. However, finding an explicit expression for the inverse function is not straightforward.

3. To calculate the derivative of the inverse function at x = 0, we need to compute (f^(-1))'(0). This derivative represents the slope of the tangent line to the inverse function at x = 0. Since the inverse function is not readily expressible in a simple form, directly differentiating it becomes challenging.

4. Instead, we can use the fact that (f^(-1))'(0) is the reciprocal of f'(f^(-1)(0)). In other words, we need to determine the derivative of f(x) at the point where f(x) equals 0, and then take the reciprocal of that value.

5. Differentiating f(x) = x + sin(x) with respect to x gives f'(x) = 1 + cos(x). Setting f(x) equal to 0, we find x = -sin(x). However, there is no real number x that satisfies this equation since the right-hand side is always between -1 and 1, while the left-hand side can take any real value.

6. As a result, f'(f^(-1)(0)) does not exist, and therefore, (f^(-1))'(0) does not exist as well. This implies that the derivative of the inverse function at x = 0 is undefined, and we cannot assign a specific value to it.

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a. The average reaction time of 4.0 seconds with a standard deviation of 0.2 seconds. To calculate the z-score, we will use the formula below: z = (X - μ) / σz = (3.9 - 4.0) / 0.2 = -0.50z = (4.6 - 4.0) / 0.2 = 3.00

Now we will look up the z-score in the standard normal table. The area between the z-scores of -0.5 and 3.0 is the proportion of participants that will be between 3.9 and 4.6 seconds. Z-score -0.5 = 0.3085Z-score 3.0 = 0.9987Area between the z-scores = 0.9987 - 0.3085 = 0.6902 or 69.02%.Therefore, 69.02% of the participants will be between 3.9 and 4.6 seconds.

b.z = (3.4 - 4.0) / 0.2 = -3.00z = (4.2 - 4.0) / 0.2 = 1.00 Now we will look up the z-score in the standard normal table. The area between the z-scores of -3.0 and 1.0 is the proportion of participants that will be between 3.4 and 4.2 seconds.Z-score -3.0 = 0.0013Z-score 1.0 = 0.8413Area between the z-scores = 0.8413 - 0.0013 = 0.8400 or 84.00%.Therefore, 84.00% of the participants will be between 3.4 and 4.2 seconds.

c.z = (4.3 - 4.0) / 0.2 = 1.50z = (4.7 - 4.0) / 0.2 = 3.50 Now we will look up the z-score in the standard normal table. The area between the z-scores of 1.5 and 3.5 is the proportion of participants that will be between 4.3 and 4.7 seconds.Z-score 1.5 = 0.9332Z-score 3.5 = 0.9998Area between the z-scores = 0.9998 - 0.9332 = 0.0666 or 6.66%.Therefore, 6.66% of the participants will be between 4.3 and 4.7 seconds.

d. z = (4.4 - 4.0) / 0.2 = 2.00Now we will look up the z-score in the standard normal table. The area to the right of the z-score of 2.00 is the proportion of participants that will be above 4.4 seconds.Z-score 2.00 = 0.9772Area to the right of the z-score = 1 - 0.9772 = 0.0228 or 2.28%.Therefore, 2.28% of participants will be above 4.4 seconds.

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The derivative of the given equation x + y = ex²y with respect to x is found using implicit differentiation.

To begin, differentiate both sides of the equation with respect to x. This gives us

1 + dy/dx = ex²y(2xy + x²dy/dx).

Using the product rule to differentiate the right side of the equation and rearranging, we obtain

dy/dx(2xy + x²) - ex²y = -1.

Simplifying, we get dy/dx(2xy + x²) = 1 + ex²y.

Finally, we can solve for dy/dx by dividing both sides of the equation by (2xy + x²).

This gives us the formula

dy/dx = (1 + ex²y) / (2xy + x²).

Therefore, the derivative of ex²y = x + y with respect to x is given by (1 + ex²y) / (2xy + x²).

The given implicit differentiation has been found. The derivative of the given equation x + y = ex²y with respect to x is given by dy/dx = (1 + ex²y) / (2xy + x²).

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a) X is an unbiased estimator of p. b) The Var(X) is p(1-p)/n. c) The E[X(1-X)] is (n-1)[p(1-p)/n]. d) The value of c is c = 1/(n-1).

(a) To show that X = Y/n is an unbiased estimator of p, we need to show that E[X] = p.

Since Y is a sum of n iid Bern(p) random variables, we have E[Y] = np.

Now, let's find the expected value of X:

E[X] = E[Y/n] = E[Y]/n = np/n = p.

Therefore, X is an unbiased estimator of p.

(b) To find the variance of X, we'll use the fact that Var(aX) = a^2 * Var(X) for any constant a.

Var(X) = Var(Y/n) = Var(Y)/n² = np(1-p)/n² = p(1-p)/n.

(c) To show that E[X(1-X)] = (n-1)[p(1-p)/n], we expand the expression:

E[X(1-X)] = E[X - X²] = E[X] - E[X²].

We already know that E[X] = p from part (a).

Now, let's find E[X²]:

E[X²] = E[(Y/n)²] = E[(Y²)/n²] = Var(Y)/n² + (E[Y]/n)².

Using the formula for the variance of a binomial distribution, Var(Y) = np(1-p), we have:

E[X²] = np(1-p)/n² + (np/n)² = p(1-p)/n + p² = p(1-p)/n + p(1-p) = (1-p)(p + p(1-p))/n = (1-p)(p + p - p²)/n = (1-p)(2p - p²)/n = 2p(1-p)/n - p²(1-p)/n = 2p(1-p)/n - p(1-p)²/n = [2p(1-p) - p(1-p)²]/n = [p(1-p)(2 - (1-p))]/n = [p(1-p)(1+p)]/n = p(1-p)(1+p)/n = p(1-p)/n.

Therefore, E[X(1-X)] = E[X] - E[X²] = p - p(1-p)/n = (n-1)p(1-p)/n = (n-1)[p(1-p)/n].

(d) To find the value of c such that cX(1-X) is an unbiased estimator of p(1-p)/n, we need to have E[cX(1-X)] = p(1-p)/n.

E[cX(1-X)] = cE[X(1-X)] = c[(n-1)[p(1-p)/n]].

For unbiasedness, we want this to be equal to p(1-p)/n:

c[(n-1)[p(1-p)/n]] = p(1-p)/n.

Simplifying, we have:

c(n-1)p(1-p) = p(1-p).

Since this should hold for all values of p, (n-1)c = 1.

Therefore, the value of c is c = 1/(n-1).

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Shade the following normal curves and provide the area under the curve for each given probability. The area under the curve to the left of -2.1 is approximately 0.0179.

To find the area under the curve to the left of -2.1, we need to calculate the cumulative probability using a standard normal distribution table or a statistical software.

From the standard normal distribution table or software, we find that the cumulative probability corresponding to -2.1 is approximately 0.0179. This means that approximately 1.79% of the data falls to the left of -2.1 on the standard normal distribution curve.

Therefore, the area under the curve to the left of -2.1 is approximately 0.0179 or 1.79%.

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The data indicates that there is evidence of significantly more 90-degree days in the past four years compared to the mean of the population.

To determine if the data indicates that the past four years have had significantly more 90-degree days than expected for a random sample from this population, we can conduct a one-tailed hypothesis test.

Null hypothesis (H₀): The mean number of 90-degree days for the past four years is equal to the mean of the population (μ = 10).

Alternative hypothesis (H₁): The mean number of 90-degree days for the past four years is significantly greater than the mean of the population (μ > 10).

Since the population standard deviation (σ) is known, we can use a z-test for this hypothesis test.

1. Calculate the test statistic (z-score):

  z = (M - μ) / (σ / √n)

  z = (12.9 - 10) / (2.7 / √4)

  z = 2.9 / 1.35

  z ≈ 2.148 (rounded to three decimal places)

2. Determine the critical value for a one-tailed test at a significance level of α = 0.05. Since it is a one-tailed test, we need to find the critical value corresponding to the upper tail. Looking up the critical value in the z-table or using a calculator, we find the critical value to be approximately 1.645 (rounded to three decimal places).

3. Compare the test statistic to the critical value:

  z > critical value

  2.148 > 1.645

4. Make a decision:

  Since the test statistic is greater than the critical value, we reject the null hypothesis.

5. State the conclusion:

  The data provide sufficient evidence to conclude that the past four years have had significantly more 90-degree days than would be expected for a random sample from this population at a significance level of α = 0.05.

Therefore, based on the given information and the results of the hypothesis test, the data indicates that there is evidence of significantly more 90-degree days in the past four years compared to the mean of the population.

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The probability that a randomly selected man's score is at least 583.5 is approximately 0.2525 or 25.25%.

The probability that the mean score of 10 randomly selected men is at least 583.5 is approximately 0.1392 or 13.92%.

(a) To find the probability that a randomly selected man's score is at least 583.5, we need to calculate the z-score and then find the corresponding area under the standard normal distribution curve.

The z-score can be calculated using the formula:

z = (x - μ) / σ

where x is the given score, μ is the mean, and σ is the standard deviation.

In this case:

x = 583.5

μ = 509

σ = 112

Substituting these values into the formula, we get:

z = (583.5 - 509) / 112

Calculate z:

z = 0.667

Now, we need to find the area to the right of this z-score in the standard normal distribution. This represents the probability that a randomly selected man's score is at least 583.5.

Using a standard normal distribution table or a calculator, we find that the area to the right of z = 0.667 is approximately 0.2525.

Therefore, the probability that a randomly selected man's score is at least 583.5 is approximately 0.2525 or 25.25%.

(b) If 10 men are randomly selected, the mean score of the sample can be considered as the population mean. The distribution of sample means follows a normal distribution with the same mean as

the population and a standard deviation equal to the population standard deviation divided by the square root of the sample size (n).

For this case:

Sample mean = 583.5

Population mean = 509

Population standard deviation = 112

Sample size (n) = 10

The standard deviation of the sample mean can be calculated as:

Standard deviation of sample mean = σ / √n

Substituting the values:

Standard deviation of sample mean = 112 / √10

Now, we need to calculate the z-score for the sample mean:

z = (sample mean - population mean) / (standard deviation of sample mean)

Substituting the values:

z = (583.5 - 509) / (112 / √10)

Calculate z:

z = 1.082

Using a standard normal distribution table or a calculator, we find that the area to the right of z = 1.082 is approximately 0.1392.

Therefore, the probability that the mean score of 10 randomly selected men is at least 583.5 is approximately 0.1392 or 13.92%.

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To calculate number of unique U.S. visitors to Myspace in July 2015, we need to substitute corresponding value of m into exponential formula. Um = 49.7 * (e^(-0.0671 * 18)) By evaluating this,we can determine answer.

The explicit exponential formula relating the number of months after January 2014 (m) and the number of unique Myspace visitors from the U.S. in that month (Um) can be expressed as follows:

Um = 49.7 * (e^(-0.0671m))

In this formula, Um represents the number of unique visitors from the U.S. in a specific month m after January 2014. The base of the natural logarithm, e, is raised to the power of (-0.0671m), which accounts for the decay in the number of visitors over time. The coefficient 0.0671 determines the rate of decay.

To calculate the number of unique U.S. visitors to Myspace in July 2015, we need to substitute the corresponding value of m into the exponential formula. July 2015 is approximately 18 months after January 2014.

Um = 49.7 * (e^(-0.0671 * 18)) By evaluating this expression, we can determine the number of unique U.S. visitors to Myspace in July 2015.

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The each comparison would use a significance level of 0.0083.

In statistical hypothesis testing, the significance level, often denoted as α (alpha), is the predetermined threshold used to determine whether to reject the null hypothesis. In this case, the researcher plans on conducting 6 comparisons using Dunn's Method (Bonferroni t). The Bonferroni correction is a commonly used method to adjust the significance level when performing multiple comparisons. It helps control the overall Type I error rate, which is the probability of falsely rejecting the null hypothesis.

To apply the Bonferroni correction, the significance level is divided by the number of comparisons being made. Since the researcher is running 6 comparisons, the significance level needs to be adjusted accordingly. Given that the overall desired significance level is usually 0.05 (or 5%), dividing this by 6 results in approximately 0.0083. Therefore, for each individual comparison, the significance level would be set at 0.0083, or equivalently, 0.83%.

The purpose of this adjustment is to ensure that the probability of making at least one Type I error among the multiple comparisons remains at an acceptable level. By using a lower significance level for each comparison, the threshold for rejecting the null hypothesis becomes more stringent, reducing the likelihood of falsely concluding there is a significant difference when there isn't.

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Null Hypothesis (H0): Movie attendance by year is independent of ethnicity.

Alternative Hypothesis (Ha): Movie attendance by year is dependent on ethnicity.

a) Hypotheses:

Null Hypothesis (H0): Movie attendance by year is independent of ethnicity.

Alternative Hypothesis (Ha): Movie attendance by year is dependent on ethnicity.

Claim: The movie attendance by year is dependent on ethnicity.

b) Critical Value:

Degrees of Freedom (df) = (2-1) (4-1) = 3

Using a chi-square distribution table or statistical software, we can find the critical value associated with α = 0.10 and df = 3.

c) Test Value:

Observed frequencies:

            Caucasian   Hispanic   African-American   Other

2006   |   932              244                203                    104

2007   |   913              293                142                    123

Expected frequencies:

            Caucasian   Hispanic   African-American   Other

2006   |   581.52         169.88           108.98                 71.64

2007   |   568.48         165.12           106.30                 69.10

Now we can calculate the test value using the formula:

Test Value = 207.61 + 29.70 + 78.95 + 15.16 + 212.35 + 98.32 + 14.19 + 48.47

= 704.75

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Regression: slope of the regression line, correlation coefficient, alternate hypothesis, test statistic, degrees of freedom and significance level The slope of the regression line is OB. 4.379, The correlation coefficient is OB. 0.8398

The alternate hypothesis is OA. H₁ : ß # 0 The test statistic is OA. 4.794 The degrees of freedom are: OB. 22 At the 5% significance level it can be concluded that there is evidence to suggest the correlation coefficient is B. not zero The given table represents a regression summary output which explains that the data represents the practice and official games of a soccer team.

Therefore, there is a need to find out the slope of the regression line and the correlation coefficient. It is also essential to perform a hypothesis test to find out whether the correlation coefficient is different from zero.(a) The slope of the regression line is given as OB. 4.379(b) The correlation coefficient is given as OB. 0.8398(c) The alternate hypothesis is given as OA. H₁ : ß # 0 The null hypothesis is given as H0 : β = 0. The given test is a two-tailed test because of the use of 'not equal to' in the alternate hypothesis.(d) The test statistic is given as OA. 4.794(e) The degrees of freedom are given as OB. 22(f) At the 5% significance level it can be concluded that there is evidence to suggest the correlation coefficient is given as B. not zero because the p-value (0.001366) is less than the level of significance (0.05). Therefore, the null hypothesis is rejected and the alternate hypothesis is accepted.

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a. The distribution of X, the speed of a randomly selected car on the freeway, is a normal distribution with a mean of 70 mph and a standard deviation of 8 mph. In notation, we can represent this as X ~ N(70, 8^2).

b. To find the probability that a randomly chosen car is traveling more than 82 mph, we need to calculate the area under the normal distribution curve to the right of 82 mph. This can be done by standardizing the value using the z-score formula and then looking up the corresponding probability in the standard normal distribution table. The z-score for 82 mph can be calculated as (82 - 70) / 8 = 1.5. By referring to the standard normal distribution table, we find that the probability of a z-score greater than 1.5 is approximately 0.0668.

c. To find the probability that a randomly chosen car is traveling between 69 and 73 mph, we need to calculate the area under the normal distribution curve between those two speeds. We can standardize the values using the z-score formula: for 69 mph, the z-score is (69 - 70) / 8 = -0.125, and for 73 mph, the z-score is (73 - 70) / 8 = 0.375. By referring to the standard normal distribution table, we find that the probability of a z-score between -0.125 and 0.375 is approximately 0.1587.

d. To determine the speed at which 97% of all cars travel on the freeway, we need to find the z-score that corresponds to the 97th percentile. Using the standard normal distribution table, we find that the z-score for a cumulative probability of 0.97 is approximately 1.8808. We can then solve for the speed by rearranging the z-score formula: z = (x - 70) / 8, where x is the speed in mph. Solving for x, we find x = 1.8808 * 8 + 70 ≈ 86.0464. Rounded to the nearest whole number, 97% of all cars travel at least 86 mph on the freeway.

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