For the IVP: (t-4) cos ty" – In(t-1)y'+√7+5y=e-', y(2) = 1, y'(2) = 1 determine the largest interval in which the solution is certain to exist
a. (-5,4)
b. (π/2,4)
c. (1,[infinity])
d. (1,π/2)

The range of the temperature data in degrees Fahrenheit is 15.

Option A is the correct answer.

We have,

From the scatterplot,

The highest average temperature = 45

The lowest temperature = 30

Now,

Range.

= Highest temperature - Lowest temperature

= 45 - 30

= 15

Thus,

The range of the temperature data in degrees Fahrenheit is 15.

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Answer:

To solve the differential equation 2y^2 cos(x)dx + (4 + 4y sin(x))dy = 0, we can use the method of integrating factors.

First, we can rearrange the equation as:

2y^2 cos(x)dx = - (4 + 4y sin(x))dy

Dividing both sides by y^2(4 + 4sin(x)), we get:

-2cos(x)/y^2 dx + (1 + sin(x))/y dy = 0

Now we can identify the coefficients of dx and dy as -2cos(x)/y^2 and (1 + sin(x))/y, respectively.

To find the integrating factor, we can use the formula:

μ(x) = exp[∫P(x)dx]

where P(x) is the coefficient of dx. In this case, we have:

P(x) = -2cos(x)/y^2

So we need to integrate P(x) with respect to x:

∫P(x)dx = -2∫cos(x)/y^2 dx = 2sin(x)/y^2 + C

where C is an arbitrary constant.

Therefore, the integrating factor is:

μ(x) = exp[2sin(x)/y^2 + C]

Multiplying both sides of the differential equation by the integrating factor, we get:

-2cos(x) exp[2sin(x)/y^2 + C] dx/y^2 + (1 + sin(x)) exp[2sin(x)/y^2 + C] dy/y = 0

Now we can rewrite this equation as a total derivative:

d/dx [exp[2sin(x)/y^2 + C]/y] = 0

Integrating both sides with respect to x, we get:

exp[2sin(x)/y^2 + C]/y = D

where D is a constant of integration.

Solving for y, we get:

y = sqrt[2sin(x)/(D - exp[2sin(x)/y^2 + C])]

This is the general solution to the differential equation. The constant D and C can be determined from initial or boundary conditions, if given.

The general solution to the differential equation is:

-y^2 ln|4 + 4y sin(x)| = y + C

where C = C1 + C2.

To solve the differential equation 2y^2cos(x)dx + (4 + 4y sin(x))dy = 0, we first need to check whether it is a homogeneous equation or not. A homogeneous equation is one where all the terms have the same degree. In this case, we have a term with x and a term with y, so it is not homogeneous.

Next, we can check whether it is a separable equation or not. A separable equation is one where we can write it in the form f(x)dx = g(y)dy. We can rearrange the equation as:

2y^2cos(x)dx = - (4 + 4y sin(x))dy

Dividing both sides by (4 + 4y sin(x)) and rearranging, we get:

-2y^2cos(x) / (4 + 4y sin(x)) dx = dy

Now, we can integrate both sides with respect to their respective variables:

∫ -2y^2cos(x) / (4 + 4y sin(x)) dx = ∫ dy

To solve the integral on the left-hand side, we can use the substitution u = 4 + 4y sin(x), which gives du/dx = 4y cos(x) and du = 4y cos(x)dx. Substituting this into the integral, we get:

∫ -y^2 / u du = -y^2 ln|u| + C1

Substituting back u = 4 + 4y sin(x), we get:

∫ -y^2 / (4 + 4y sin(x)) du = -y^2 ln|4 + 4y sin(x)| + C1

Integrating the right-hand side with respect to y, we get:

∫ dy = y + C2

Therefore, the general solution to the differential equation is:

-y^2 ln|4 + 4y sin(x)| = y + C

where C = C1 + C2.

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a) The mean value of the variable x is defined as the weighted average of x over the interval [x, x + dx], where the weights are given by the probability density function P(x). Mathematically, it is expressed as x = ∫x(x+dx) P(x) dx

b) The variance of the variable x, denoted by σ², is defined as the weighted average of the squared deviations of x from its mean value, where the weights are given by the probability density function P(x). Mathematically, it is expressed as σ² = ∫(x-x)2 P(x) dx

c) The standard deviation of the variable x, denoted by o, is the square root of the variance. Mathematically, it is expressed as σ = √σ² These definitions hold true for any normalized probability density function of the variable x over the interval [x, x + dx].

Given a normalized probability density function P(x) of finding the variable x in the interval [x, x + dx], the definitions for the mean value, variance, and standard deviation are as follows:

a) The mean value (µ) of the variable x is defined as the expected value, which can be calculated using the integral:
µ = ∫xP(x)dx, where the integral is taken over the entire range of x.

b) The variance (σ²) is defined as the average squared deviation from the mean value (µ). It can be calculated using the integral:
σ² = ∫(x - µ)²P(x)dx, where the integral is taken over the entire range of x.

c) The standard deviation (σ) of the variable x is defined as the square root of the variance:
σ = sqrt(σ²)

These definitions will help you analyze the given probability density function and understand its central tendency and dispersion.

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Answer:

Math is a broad field that encompasses several branches, each with its own units of measurement. Some examples of units in math include:

In geometry:- Units of length, such as meters, centimeters, and inches

Units of area, such as square meters, square centimeters, and square feet

Units of volume, such as cubic meters, cubic centimeters, and cubic feet- Units of weight or mass, such as kilograms, grams, and pounds - Units of time, such as seconds, minutes, and hours

Units of temperature, such as Celsius and

Fahrenheit

Units of angle measurement, such as degrees and radians

Units of speed or velocity, such as meters per second or miles per hour

Units of frequency, such as Hertz or cycles per second

Units of energy or work, such as joules, calories, and foot-pounds

Units of power, such as watts and horsepower

These are just a few examples of the many units used in math. The type of unit used depends on the specific problem or application.

HOPE IT HELPS

PLEASE MARK ❣️‼️ ME AS BRAINLIEST .

Cristobal needs to spend more than $122.70 for the membership at the 2nd bookstore to be better value.

For first bookstore, as Cristobal pays $24.27 for an annual membership, save 15% on all his purchases, the amount he saves on purchases will be represented as 0.15x.

So total cost of being a member of the first bookstore is:

$24.27 + $0.15x.

For second bookstore, as Cristobal pays $36.54 for an annual membership, save 25% on all his purchases, the amount he saves on purchases will be represented as 0.25x.

So the total cost of being a member of the second bookstore is:

= $36.54 + $0.25x.

To determine when membership at second bookstore is better value, we must set total cost of second bookstore less than first bookstore and then, we will solve for x:

$36.54 + $0.25x < $24.27 + $0.15x

$12.27 < $0.10x

x > $122.70.

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The approximate amount of area in the circle is 113.04 square feet.

The area of a circle is given through the expression [tex]A = \pi r^2[/tex], in which π is about equal to 3.14, and r is the radius of the circle.

In this instance, we are given the diameter of the circle, that is 12 feet. The radius of the circle is half of the periphery, so the radius is

r = 12 / 2 = 6 feet

Now, we're suitable to use the methodology for the area of a circle to discover the approximate amount of space within the circle

[tex]A = \pi r^2 = 3.14 * 6^2 = 3.14 * 36 \approx 113.04[/tex] square feet[tex]A = \pi r^2[/tex]

Accordingly, the approximate amount of area in the circle is 113.04 square feet.

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The radius of the figure is 2√2.

We have,

From the figure,

The right angle triangle.

One angle is 90 and the other two angles will be the same. ie. 45

Now,

The sides opposite to the equal angles are the same.

From the figure,

Side = 2

Now,

Applying the Pythagorean theorem,

radius² = side² + side²

radius² = 2² + 2²

radius² = 4 + 4

radius = √8 = 2√2

Thus,

The radius of the figure is 2√2.

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The real values of $c$ for which $x^2 + 16x + c$ is the square of a binomial are $64$ and $0$.

To find these values, we can use the concept of completing the square. For a quadratic expression to be the square of a binomial, the coefficient of the linear term ($16x$) must be twice the product of the square root of the constant term ($c$) and the square root of the coefficient of the quadratic term ($1$). In this case, the coefficient of the linear term is $16$ and the coefficient of the quadratic term is $1$. So, we have $16 = 2\sqrt{c}\sqrt{1}$.

Simplifying this equation gives $16 = 2\sqrt{c}$. Dividing both sides by $2$ yields $\sqrt{c} = 8$. Squaring both sides gives $c = 64$. Thus, $c = 64$ is one possible value.

Additionally, if we consider the case when $c = 0$, the quadratic expression becomes $x^2 + 16x + 0 = (x + 8)^2$. Therefore, $c = 0$ is another possible value.

In summary, the real values of $c$ for which $x^2 + 16x + c$ is the square of a binomial are $64$ and $0$.

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Answer:

[tex]12w \geqslant 150[/tex]

[tex]w \geqslant 12.5[/tex]

So Jordan needs to save $12 a week for at least 13 consecutive weeks.

The solutions to the equation 2n +6=2(3+n) is infinite many

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

2n +6=2(3+n).

Open the brackets

So, we have

2n + 6 = 2n + 6

Evaluate the like terms

0 = 0

This means that the equation has infinite many solutions

Can you find more solutions?

Yes, this is because any real value can be used for n

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The value of the equation 5- c  for c = 3 will be  2.

Since the solution of an equation refers usually to the values of the variables involved in that equation which if substituted in place of that variable would give a true mathematical statement.

We need to find the solutions does the equation 5 - c for c = 3;

Now solving for c;

5-c

for c = 3

5 - 3 = 2

Therefore, the value is 2.

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Answer:

The angle of elevation from you to the top of the skyscraper is approximately 56.2 degrees.

Step-by-step explanation:

The function f(x) is increasing for all x < 3. Then the correct option is A.

Given that:

Function, f(x) = (x + 6) / (x² - 9x + 18)

A function is an assertion, concept, or principle that establishes an association between two variables. Functions may be found throughout mathematics and are essential for the development of significant links.

Simplify the function, then we have

f(x) = (x + 6) / (x² - 9x + 18)

f(x) = (x + 6) / (x² - 6x - 3x + 18)

f(x) = (x + 6) / [x(x - 6) - 3(x - 6)]

f(x) = (x + 6) / (x - 6)(x - 3)

The graph is given below.

The function f(x) is increasing for all x < 3. Then the correct option is A.

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Answer:

x = [tex]\sqrt{51}[/tex] , y = 7

Step-by-step explanation:

since PA is a tangent, then angle between tangent and radius at point of contact A is 90°

the triangle with radius x is right.

using Pythagoras' identity in the right triangle

x² + 7² = 10²

x² + 49 = 100 ( subtract 49 from both sides )

x² = 51 ( take square root of both sides )

x = [tex]\sqrt{51}[/tex]

since PB is a tangent then ∠ B = 90° and triangle with y is right

note that the segment from B to the centre is the radius and is equal to x

using Pythagoras' identity in this right triangle

y² + x² = 10²

y² + ([tex]\sqrt{51}[/tex] )² = 100

y² + 51 = 100 ( subtract 51 from both sides )

y² = 49 ( take square root of both sides )

y = [tex]\sqrt{49}[/tex] = 7

then x = [tex]\sqrt{51}[/tex] and x = 7

We have shown that A(-) = F(a + dF(a(-)), as required.

To prove that A(-) = F(a + dF(a(-)), we need to show that the affine function A coincides with the function F at every point x in RP.

Let x be an arbitrary point in RP. We can write x = a + t, where t is a vector in the tangent space of RP at a. Since U is open in RP, we can choose a small enough neighborhood of a in U such that a + t is also in U.

Since F is differentiable at a, we can apply the multivariable chain rule to get:

dF(a + t) = dF(a) + J(a)t + o(||t||)

where J(a) is the Jacobian matrix of F at a, and o(||t||) is a term that goes to zero faster than ||t|| as t approaches zero.

Since A is affine, we can write:

A(x) = A(a + t) = A(a) + Bt

where B is a constant matrix. Since A(a) = F(a) and dA(a) = dF(a), we have:

A(x) = F(a) + dF(a)t + o(||t||)

Comparing the two expressions for A(x), we see that we can choose B = dF(a) and the remainder term o(||t||) is the same in both expressions. Therefore:

A(x) = F(a) + dF(a)t + o(||t||) = F(a + t) + o(||t||) = F(x) + o(||t||)

Since o(||t||) goes to zero faster than ||t|| as t approaches zero, we have:

A(x) = F(x)

for all x in RP. Therefore, we have shown that A(-) = F(a + dF(a(-)), as required.

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After purchasing a carpet that is 5 feet long and 6 feet wide, Felicia cut off a section of 8 square feet so the area of the remaining piece of carpet is 22 square feet.

To find the area of the remaining piece of carpet, we need to subtract the area that Felicia cut off from the total area of the carpet.

The total area of the carpet is the product of its length and width, which is:

5 feet x 6 feet = 30 square feet

Felicia cut off 8 square feet from the carpet, so the area of the remaining piece of carpet is:

30 square feet - 8 square feet = 22 square feet

Therefore, the area of the remaining piece of carpet is 22 square feet.

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Using SOH CAH TOA, the value of hypotenuse, h, is 29.1 cm

From the question, we are to calculate the value of the hypotenuse.

In the diagram, h represents the hypotenuse

Using SOH CAH TOA

sin (angle) = Opposite / Hypotenuse

cos (angle) = Adjacent / Hypotenuse

tan (angle) = Opposite / Adjacent

From the given information,

Angle = 31°

Opposite = 15 cm

Hypotenuse = h

Thus,

sin (31°) = 15 cm / h

0.515038 = 15 cm / h

Then,

h = 15 / 0.515038 cm

h = 29.12406 cm

h ≈ 29.1 cm

Hence,

The value of h is 29.1 cm

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Regarding the cylindrical designs, it is recommended that Kevin choose the first design, which takes around 108.35 square inches of plastic. Kevin does not have enough plastic to build the second design since it needed around 431.97 square.

Here we used the surface area formula for cylinders.

Surface Area = 2πr² + 2πrh

R is the base and h is the height.

For First Design we have

Diameter (d) = 2r = 3

so r = 1.5

So Surface Area = 2π(1.5)² + 2π(1.5) (10)

SA First Cylinder = 108.35

Repeating the same step for the second cylinder we have:

SA 2ndCylinder = 431.97

Thus, the conclusion we have above is the correct one because:

108.35in² <  205in² > 431.97in²

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The Total surface area of the given prism is: 312 ft²

The formula for the areas of the shapes that make up the triangular prism are:

Area of triangle = ¹/₂bh

where:

b is base

h is height

Area of rectangle = L * W

where:

L is length

W is width

Thus:

Total surface area = 2(¹/₂ * 6 * 4) + 2(5 * 18) + (18 * 6)

Total surface area = 24 + 180 + 108

Total surface area = 312 ft²

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All the items on Conner's social media profile that may give a criminal too much information are:

Exposing one's birthday might lead to identity theft since it is a private data that can be utilized by malicious individuals to acquire access to other crucial information.

While an image of Conner together with his buddies taken outside The Hub may look exciting and a great memory, it might reveal his position, making it simple for perpetrators to trail their movements and prey on them or their acquaintances.

Conner's username in his account is similarly critical; if it is exceptional and not extensively known, criminals can simulate him or deceptively target him through methods like social engineering tricks to attain his confidential details.

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The probability that among 4 randomly selected motorists, the officer will find at least one motorist driving more than 5 miles per hour over the speed limit is 0.8704, rounded to the nearest ten-thousandth.

To solve this problem, we can use the complement rule, which states that the probability of an event happening is equal to 1 minus the probability of the event not happening.

First, let's find the probability that none of the 4 randomly selected motorists will be driving more than 5 miles per hour over the speed limit.

Since the officer estimates that 40% of motorists will be driving more than 5 miles per hour over the speed limit, then the probability of a motorist not driving more than 5 miles per hour over the speed limit is 1 - 0.4 = 0.6.

The probability that none of the 4 motorists will be driving more than 5 miles per hour over the speed limit is therefore:

0.6 x 0.6 x 0.6 x 0.6 = 0.1296

Now we can use the complement rule to find the probability that at least one of the 4 motorists will be driving more than 5 miles per hour over the speed limit:

1 - 0.1296 = 0.8704

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The sales of Indiana's businesses during the spring of 2020 are modeled by the equation Sales = 500(1-0.10)^t, where t is the number of days and sales are in millions of dollars. If sales reach $23.5 million, it will be considered a statewide economic crisis.

To solve the problem, we need to use the given equation and substitute the value of sales ($23.5 million) into it. Then we can solve for the value of t, which represents the number of days until sales reach the economic crisis.

500(1-0.10)^t = 23.5

(1-0.10)^t = 0.047

Taking the natural logarithm of both sides,

ln[(1-0.10)^t] = ln(0.047)

t ln(0.90) = -3.057

t = -3.057 / ln(0.90)

Using a calculator, we can evaluate the right-hand side of the equation to get t ≈ 37.28 days.

Therefore, it will take approximately 37.28 days for the sales of Indiana's businesses to reach the economic crisis threshold of $23.5 million.

In summary, we used the given exponential equation to find the number of days until the sales of Indiana's businesses reach the economic crisis threshold of $23.5 million. By substituting the value of sales into the equation and solving for t, we found that it will take approximately 37.28 days for this critical point to be reached. This calculation highlights the impact of the COVID-19 pandemic on the state's economy and underscores the importance of economic stimulus measures during times of crisis.

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It would also increase the computational complexity of the algorithm.

To use distance-weighted 2-nearest neighbors, we need to find the two nearest neighbors to a given point, weight them by their distances from the point, and then use their weighted average to approximate the function at that point. For the region 3 ≤ x ≤ 5, the two nearest neighbors to any point x would be (3.0, 5.0) and (5.0, 5.0).

The distance-weighted average approximation of f(x) in this region is:

f(x) ≈ (w1f(3) + w2f(5)) / (w1 + w2)

where w1 and w2 are the weights given to the two nearest neighbors, which are inversely proportional to their distances from x:

w1 = 1 / |x - 3.0|^2

w2 = 1 / |x - 5.0|^2

Substituting in the given values, we get:

f(x) ≈ [(1/|x-3.0|^2)*5.0 + (1/|x-5.0|^2)*5.0] / [(1/|x-3.0|^2) + (1/|x-5.0|^2)]

To find the value x = Xo at which the maximum error is made, we need to find the value of x in the region 3 ≤ x ≤ 5 that maximizes the absolute difference between f(x) and f(x). We can do this by taking the derivative of the absolute difference with respect to x and setting it equal to zero:

d/dx |f(x) - f(x)| = d/dx |8x - 10 - x^2 - f(x)| = 0

Solving for x, we get:

x = 3.8 or x = 4.2

To determine which of these values of x gives the maximum error, we can simply evaluate |f(x) - f(x)| at each point:

|x=3.8| = |(1/0.04)*3.0 + (1/0.04)5.0 - (1/0.16)(-1.24)| = 10.74

|x=4.2| = |(1/0.04)*5.0 + (1/0.04)5.0 - (1/0.04)(-3.56)| = 13.96

Therefore, the maximum error occurs at x = 4.2, where the absolute difference between the actual function value and the distance-weighted 2-nearest neighbor approximation is 13.96.

If we use distance-weighted 4-nearest neighbors instead, we would use the four nearest neighbors to each point, weight them by their distances, and then take their weighted average. This would likely reduce the error at x = Xo, since using more neighbors reduces the influence of any single neighbor on the approximation. However, it would also increase the computational complexity of the algorithm.

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The 90% confidence interval for the mean of all tree-ring dates from this archaeological site is (1233, 1303) A.D. (rounded to nearest whole number).

To find the sample mean year x and sample standard deviation s, we can use the calculator's mean and standard deviation functions:

x = 1268 (rounded to nearest whole number)

s = 43 (rounded to nearest whole number)

To find the critical value for a 90% confidence interval, we can use a t-distribution with n-1 degrees of freedom (where n is the sample size). Since the sample size is not given, we'll assume it's 9 (the number of years listed in the data set). Using a t-table or calculator, the critical value for a 90% confidence interval with 8 degrees of freedom is approximately 1.860 (rounded to three decimal places).

The maximal margin of error for a 90% confidence interval can be found using the formula:

E = tc * s / sqrt(n)

where tc is the critical value, s is the sample standard deviation, and n is the sample size. Plugging in the values we have, we get:

E = 1.860 * 43 / sqrt(9) = 35.13 (rounded to nearest whole number)

To find the 90% confidence interval for the mean of all tree-ring dates from this archaeological site, we can use the formula:

(lower limit, upper limit) = (x - E, x + E)

Plugging in the values we have, we get:

(lower limit, upper limit) = (1268 - 35, 1268 + 35) = (1233, 1303)

So the 90% confidence interval for the mean of all tree-ring dates from this archaeological site is (1233, 1303) A.D. (rounded to nearest whole number).

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Answer: :)

The correct answer is C. A scalene triangle is a triangle with no sides that are the same length. This means that all three sides of a scalene triangle have different lengths. In addition, a scalene triangle does not have any angles that are congruent. This is in contrast to an isosceles triangle, which has two sides of equal length, and an equilateral triangle, which has all three sides of equal length.

Step-by-step explanation:

Every composite number has at least one natural number factor other than 1 and itself. The correct answer is option D.

A composite number is a natural number larger than one that is not a prime number, implying that it has at least one element other than 1 and itself.

As a result, every composite number has at least one natural number factor that is neither 1 nor itself.

The smallest natural number that can divide all of the numbers in the integer list is 1.

When we divide a number by itself, we obtain 1 as the component.

Hence, the correct answer is option D.

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The complete question is as follows:

What is 1 of a natural number factor?

A. every

B. odd

C. even

D. composite

Answer:

To find the surface area of a regular hexagonal pyramid, we need to find the area of the six triangular faces and the area of the hexagonal base, and then add them together.

The area of each triangular face is given by the formula:

(1/2) x base x height

In this case, the base of each triangle is the side length of the hexagon (8), and the height is the slant height of the pyramid (16). Therefore, the area of each triangular face is:

(1/2) x 8 x 16 = 64

The hexagonal base can be divided into six equilateral triangles, each with side length 8. The area of each equilateral triangle is:

(1/4) x sqrt(3) x side length^2

Plugging in the values, we get:

(1/4) x sqrt(3) x 8^2 = 16sqrt(3)

To find the total surface area, we add the area of the six triangular faces and the area of the hexagonal base:

6 x 64 + 16sqrt(3) = 384 + 16sqrt(3)

Rounding to the nearest tenth, the surface area of the regular hexagonal pyramid is:

398.6 square units (rounded to one decimal place)

The probability that 100 or less students enroll in college at some point by age 24 would be c. 97.8%

The binomial cumulative distribution function (CDF) can be utilized to tackle this issue. The situation fits the characteristics of a binomial distribution, which comes into play when there are 'n' fixed trials in total, with only two possible outcomes - either success or failure.

Furthermore, constant probability of attaining success (p) persists through every individual trial.

The formula is:

P ( X ≤ 100 ) = ∑ [ C ( n , k ) x p^ k x q ^ ( n - k ) ] for k = 0 to 100

Using a binomial calculator, we find out that:

P ( X ≤ 100 ) = 0. 978 or 97. 8 %

In conclusion, option C is correct.

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Part A: The given evidence is convincing to provide the true proportion regarding the attendees.

Part B: The error is a type 1 error in the hypothesis testing.

Type 1 Error and Type 2 Error

A type 1 error in hypothesis testing occurs when a null hypothesis is rejected when it is true.

A type II error in hypothesis testing occurs when the investigator fails to reject the null hypothesis that is actually false.  

Given that,

the total number of attendees who ordered fish is 1000.

And, The random selection for the sample size of the attendees who ate fish is 80 of which 64 people got sick.

Hence, The number of attendees who ate the fish and got sick is calculated as given below.

No. of attendees = 64/80

% of No. of attendees = 64/80 x 100

% of No. of attendees = 80%

The random selection for the sample size of the attendees who did not eat fish is 60 of which 39 people got sick.

The number of attendees who did not eat the fish and got sick is calculated as given below.

No. of attendees  = 39/60

% of No. of attendees = 39/60 x 100

% of No. of attendees = 65%

Hence, For Part A;

The given evidence is convincing to provide the true proportion of all attendees who ate the fish that got sick is more than the true proportion of all attendees who did not eat the fish that got sick.

For Part B;

The mistake here is that the doctor's theory (hypothesis) got rejected regarding the number of attendees who ate the fish got sick than those who did not eat the fish.

This error is a type 1 error in the hypothesis testing.

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